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布尔网络

2021-02-06T10:58:50Z

923397935：/* Classification */

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本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network '''由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间 ''t'' 的网络状态上的函数来确定整个网络在时间 ''t+1''的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 Regulatory Networks '''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有 ''2k'' 可能的规则中随机选择的，有 ''K'' 个输入。在 ''K=2'' 时，两类行为往往占主导地位。但对于 ''K>2'' ，人们看到的行为很快就会接近随机映射的典型特征，其中代表 ''N'' 个底层节点的 ''2k'' 种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

'''随机布尔网络 Random Boolean Network（RBN） '''是指从所有可能的特定大小的布尔网络 ''N'' 的集合中随机选取的网络。然后，人们可以从统计学上研究，这种网络的预期特性如何依赖于所有可能网络的集合的各种统计特性。例如，人们可以研究RBN行为如何随着''' Average Connectivity 平均连通性 '''的改变而改变。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

1969年，Stuart A. Kauffman提出了第一个布尔网络，作为遗传调控网络的随机模型，但其数学理解在2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络只有 2N 种可能的状态，一个轨迹迟早会到达以前访问过的状态，因此，由于动力学是确定性的，轨迹将落入一个稳定状态或周期，称为吸引子（不过在更广泛的动力学系统领域，一个周期只有当来自它的扰动导致回到它时才是吸引子）。如果吸引子只有一个状态，则称为点吸引子，如果吸引子由一个以上的状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。只在轨迹开始时出现的状态（没有轨迹导致它们），称为'''伊甸园状态
garden-of-Eden states '''网络的动态从这些状态流向吸引子。到达吸引子所需的时间称为'''瞬时 transient time '''。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机能力的不断提高，对看似简单的模型的理解也越来越深刻，不同的作者对吸引子的平均数量和长度给出了不同的估计，这里简单总结一下主要的出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力系统理论中，网络的结构和吸引子的长度对应于网络的动态阶段。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定、临界或混乱的行为。这种现象受节点平均连接数的临界值（<math>K_{c}</math>）支配，可以用汉明距离作为距离度量。在非稳定体制下，两个初始接近状态之间的平均距离在时间上呈指数级增长，而在稳定体制下则呈指数级减小。在这其中，用 "最初接近的状态 "意味着汉明距离与网络中的节点数（<math>N</math>）相比是很小的。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于N-K模型，如果 <math>K<K_{c}</math> ，网络是稳定的；如果 <math>K=K_{c}</math> ，网络是临界的；如果 <math>K>K_{c}</math> ，网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

一个给定节点的状态 <math>n_{i}</math> 根据其真值表进行更新，真值表的输出是随机填充的。<math>p_{i}</math> 表示将关闭输出分配给给定系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果 <math>p_{i}=p=const.</math> 对于每一个节点，稳定和混沌范围之间的转换取决于 <math>p</math> 。根据伯纳德-德里达和伊夫-波莫的观点
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为 <math>K_{c}=1/[2p(1-p)]</math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果 <math>K</math> 不是常数，且内度和外度之间没有相关性，则稳定性的条件由 <math>\langle K^{in}\rangle</math> 决定，如果 <math>\langle K^{in}\rangle <K_{c}</math> ，网络是稳定的。如果 <math>\langle K^{in}\rangle =K_{c}</math> ，则为临界；如果<math>\langle K^{in}\rangle >K_{c}</math> ，则为不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于'''无标度拓扑 scale-free topology '''的网络来说，稳定性的条件是一样的，其中的出入度分布是幂律分布。<math>P(K) \propto K^{-\gamma}</math>, 和 <math>\langle K^{in}\rangle=\langle K^{out} \rangle</math> ，因为从一个节点发出的每一条外链都是到另一个节点的内链。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度显示了给定节点的布尔函数的输出在其输入变化时发生变化的概率。对于随机布尔网络。
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由最大的特征值<math>\lambda_{Q}</math>来控制。的矩阵 <math>Q</math>，其中<math> Q_{ij}=q_{i}A_{ij}</math>，<math>A</math> 是网络的邻接矩阵。如果 <math>\lambda_{Q}<1</math>，网络是稳定的；如果 <math>\lambda_{Q}=1</math>，网络是临界的；如果 <math>\lambda_{Q}>1</math>，网络是不稳定的。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑结构。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*'''同质情况 Homogeneous Case '''只是指网格，它只是对著名的'''伊辛模型 lsing model '''的还原。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时也称为CRBN，即经典随机布尔网络）是同步更新的。受基因通常不会同时改变其状态这一事实的激励，人们引入了不同的替代方案。常见的分类如下:
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
当 ''t≡Qi(''mod''Pi)'' 其中 ''t'' 是时间步长时，''i'' 节点将被更新。'''确定性异步更新布尔网络'''('''DRBNs''')不是同步更新，但确定性解仍然存在。当 ''t≡Qi(''mod''Pi)'' 时，''i'' 节点将被更新，其中 ''t'' 是时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般异步随机布尔网络）。在这里，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了一种考虑潜在模型不确定性的轨迹最优分类，还提出了一种基于粒子的轨迹分类，对于大型网络具有高度的可扩展性，复杂度比最优解低得多。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2021-02-06T10:56:51Z

923397935：/* Other updating schemes */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network '''由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间 ''t'' 的网络状态上的函数来确定整个网络在时间 ''t+1''的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 Regulatory Networks '''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有 ''2k'' 可能的规则中随机选择的，有 ''K'' 个输入。在 ''K=2'' 时，两类行为往往占主导地位。但对于 ''K>2'' ，人们看到的行为很快就会接近随机映射的典型特征，其中代表 ''N'' 个底层节点的 ''2k'' 种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

'''随机布尔网络 Random Boolean Network（RBN） '''是指从所有可能的特定大小的布尔网络 ''N'' 的集合中随机选取的网络。然后，人们可以从统计学上研究，这种网络的预期特性如何依赖于所有可能网络的集合的各种统计特性。例如，人们可以研究RBN行为如何随着''' Average Connectivity 平均连通性 '''的改变而改变。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

1969年，Stuart A. Kauffman提出了第一个布尔网络，作为遗传调控网络的随机模型，但其数学理解在2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络只有 2N 种可能的状态，一个轨迹迟早会到达以前访问过的状态，因此，由于动力学是确定性的，轨迹将落入一个稳定状态或周期，称为吸引子（不过在更广泛的动力学系统领域，一个周期只有当来自它的扰动导致回到它时才是吸引子）。如果吸引子只有一个状态，则称为点吸引子，如果吸引子由一个以上的状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。只在轨迹开始时出现的状态（没有轨迹导致它们），称为'''伊甸园状态
garden-of-Eden states '''网络的动态从这些状态流向吸引子。到达吸引子所需的时间称为'''瞬时 transient time '''。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机能力的不断提高，对看似简单的模型的理解也越来越深刻，不同的作者对吸引子的平均数量和长度给出了不同的估计，这里简单总结一下主要的出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力系统理论中，网络的结构和吸引子的长度对应于网络的动态阶段。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定、临界或混乱的行为。这种现象受节点平均连接数的临界值（<math>K_{c}</math>）支配，可以用汉明距离作为距离度量。在非稳定体制下，两个初始接近状态之间的平均距离在时间上呈指数级增长，而在稳定体制下则呈指数级减小。在这其中，用 "最初接近的状态 "意味着汉明距离与网络中的节点数（<math>N</math>）相比是很小的。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于N-K模型，如果 <math>K<K_{c}</math> ，网络是稳定的；如果 <math>K=K_{c}</math> ，网络是临界的；如果 <math>K>K_{c}</math> ，网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

一个给定节点的状态 <math>n_{i}</math> 根据其真值表进行更新，真值表的输出是随机填充的。<math>p_{i}</math> 表示将关闭输出分配给给定系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果 <math>p_{i}=p=const.</math> 对于每一个节点，稳定和混沌范围之间的转换取决于 <math>p</math> 。根据伯纳德-德里达和伊夫-波莫的观点
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为 <math>K_{c}=1/[2p(1-p)]</math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果 <math>K</math> 不是常数，且内度和外度之间没有相关性，则稳定性的条件由 <math>\langle K^{in}\rangle</math> 决定，如果 <math>\langle K^{in}\rangle <K_{c}</math> ，网络是稳定的。如果 <math>\langle K^{in}\rangle =K_{c}</math> ，则为临界；如果<math>\langle K^{in}\rangle >K_{c}</math> ，则为不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于'''无标度拓扑 scale-free topology '''的网络来说，稳定性的条件是一样的，其中的出入度分布是幂律分布。<math>P(K) \propto K^{-\gamma}</math>, 和 <math>\langle K^{in}\rangle=\langle K^{out} \rangle</math> ，因为从一个节点发出的每一条外链都是到另一个节点的内链。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度显示了给定节点的布尔函数的输出在其输入变化时发生变化的概率。对于随机布尔网络。
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由最大的特征值<math>\lambda_{Q}</math>来控制。的矩阵 <math>Q</math>，其中<math> Q_{ij}=q_{i}A_{ij}</math>，<math>A</math> 是网络的邻接矩阵。如果 <math>\lambda_{Q}<1</math>，网络是稳定的；如果 <math>\lambda_{Q}=1</math>，网络是临界的；如果 <math>\lambda_{Q}>1</math>，网络是不稳定的。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑结构。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*'''同质情况 Homogeneous Case '''只是指网格，它只是对著名的'''伊辛模型 lsing model '''的还原。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时也称为CRBN，即经典随机布尔网络）是同步更新的。受基因通常不会同时改变其状态这一事实的激励，人们引入了不同的替代方案。常见的分类如下:
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
当 ''t≡Qi(''mod''Pi)'' 其中 ''t'' 是时间步长时，''i'' 节点将被更新。'''确定性异步更新布尔网络'''('''DRBNs''')不是同步更新，但确定性解仍然存在。当 ''t≡Qi(''mod''Pi)'' 时，''i'' 节点将被更新，其中 ''t'' 是时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般异步随机布尔网络）。在这里，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2021-02-06T10:55:47Z

923397935：/* Other updating schemes */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network '''由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间 ''t'' 的网络状态上的函数来确定整个网络在时间 ''t+1''的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 Regulatory Networks '''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有 ''2k'' 可能的规则中随机选择的，有 ''K'' 个输入。在 ''K=2'' 时，两类行为往往占主导地位。但对于 ''K>2'' ，人们看到的行为很快就会接近随机映射的典型特征，其中代表 ''N'' 个底层节点的 ''2k'' 种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

'''随机布尔网络 Random Boolean Network（RBN） '''是指从所有可能的特定大小的布尔网络 ''N'' 的集合中随机选取的网络。然后，人们可以从统计学上研究，这种网络的预期特性如何依赖于所有可能网络的集合的各种统计特性。例如，人们可以研究RBN行为如何随着''' Average Connectivity 平均连通性 '''的改变而改变。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

1969年，Stuart A. Kauffman提出了第一个布尔网络，作为遗传调控网络的随机模型，但其数学理解在2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络只有 2N 种可能的状态，一个轨迹迟早会到达以前访问过的状态，因此，由于动力学是确定性的，轨迹将落入一个稳定状态或周期，称为吸引子（不过在更广泛的动力学系统领域，一个周期只有当来自它的扰动导致回到它时才是吸引子）。如果吸引子只有一个状态，则称为点吸引子，如果吸引子由一个以上的状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。只在轨迹开始时出现的状态（没有轨迹导致它们），称为'''伊甸园状态
garden-of-Eden states '''网络的动态从这些状态流向吸引子。到达吸引子所需的时间称为'''瞬时 transient time '''。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机能力的不断提高，对看似简单的模型的理解也越来越深刻，不同的作者对吸引子的平均数量和长度给出了不同的估计，这里简单总结一下主要的出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力系统理论中，网络的结构和吸引子的长度对应于网络的动态阶段。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定、临界或混乱的行为。这种现象受节点平均连接数的临界值（<math>K_{c}</math>）支配，可以用汉明距离作为距离度量。在非稳定体制下，两个初始接近状态之间的平均距离在时间上呈指数级增长，而在稳定体制下则呈指数级减小。在这其中，用 "最初接近的状态 "意味着汉明距离与网络中的节点数（<math>N</math>）相比是很小的。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于N-K模型，如果 <math>K<K_{c}</math> ，网络是稳定的；如果 <math>K=K_{c}</math> ，网络是临界的；如果 <math>K>K_{c}</math> ，网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

一个给定节点的状态 <math>n_{i}</math> 根据其真值表进行更新，真值表的输出是随机填充的。<math>p_{i}</math> 表示将关闭输出分配给给定系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果 <math>p_{i}=p=const.</math> 对于每一个节点，稳定和混沌范围之间的转换取决于 <math>p</math> 。根据伯纳德-德里达和伊夫-波莫的观点
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为 <math>K_{c}=1/[2p(1-p)]</math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果 <math>K</math> 不是常数，且内度和外度之间没有相关性，则稳定性的条件由 <math>\langle K^{in}\rangle</math> 决定，如果 <math>\langle K^{in}\rangle <K_{c}</math> ，网络是稳定的。如果 <math>\langle K^{in}\rangle =K_{c}</math> ，则为临界；如果<math>\langle K^{in}\rangle >K_{c}</math> ，则为不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于'''无标度拓扑 scale-free topology '''的网络来说，稳定性的条件是一样的，其中的出入度分布是幂律分布。<math>P(K) \propto K^{-\gamma}</math>, 和 <math>\langle K^{in}\rangle=\langle K^{out} \rangle</math> ，因为从一个节点发出的每一条外链都是到另一个节点的内链。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度显示了给定节点的布尔函数的输出在其输入变化时发生变化的概率。对于随机布尔网络。
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由最大的特征值<math>\lambda_{Q}</math>来控制。的矩阵 <math>Q</math>，其中<math> Q_{ij}=q_{i}A_{ij}</math>，<math>A</math> 是网络的邻接矩阵。如果 <math>\lambda_{Q}<1</math>，网络是稳定的；如果 <math>\lambda_{Q}=1</math>，网络是临界的；如果 <math>\lambda_{Q}>1</math>，网络是不稳定的。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑结构。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*'''同质情况 Homogeneous Case '''只是指网格，它只是对著名的'''伊辛模型 lsing model '''的还原。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时也称为CRBN，即经典随机布尔网络）是同步更新的。受基因通常不会同时改变其状态这一事实的激励，人们引入了不同的替代方案。常见的分类如下:
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
当 ''t≡Qi(''mod''Pi)'' 其中 ''t'' 是时间步长时，''i'' 节点将被更新。'''确定性异步更新布尔网络'''('''DRBN''''s)不是同步更新，但确定性解仍然存在。当 ''t≡Qi(''mod''Pi)'' 时，''i'' 节点将被更新，其中 ''t'' 是时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般异步随机布尔网络）。在这里，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2021-02-06T10:51:37Z

923397935：/* Other topologies */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network '''由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间 ''t'' 的网络状态上的函数来确定整个网络在时间 ''t+1''的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 Regulatory Networks '''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有 ''2k'' 可能的规则中随机选择的，有 ''K'' 个输入。在 ''K=2'' 时，两类行为往往占主导地位。但对于 ''K>2'' ，人们看到的行为很快就会接近随机映射的典型特征，其中代表 ''N'' 个底层节点的 ''2k'' 种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

'''随机布尔网络 Random Boolean Network（RBN） '''是指从所有可能的特定大小的布尔网络 ''N'' 的集合中随机选取的网络。然后，人们可以从统计学上研究，这种网络的预期特性如何依赖于所有可能网络的集合的各种统计特性。例如，人们可以研究RBN行为如何随着''' Average Connectivity 平均连通性 '''的改变而改变。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

1969年，Stuart A. Kauffman提出了第一个布尔网络，作为遗传调控网络的随机模型，但其数学理解在2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络只有 2N 种可能的状态，一个轨迹迟早会到达以前访问过的状态，因此，由于动力学是确定性的，轨迹将落入一个稳定状态或周期，称为吸引子（不过在更广泛的动力学系统领域，一个周期只有当来自它的扰动导致回到它时才是吸引子）。如果吸引子只有一个状态，则称为点吸引子，如果吸引子由一个以上的状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。只在轨迹开始时出现的状态（没有轨迹导致它们），称为'''伊甸园状态
garden-of-Eden states '''网络的动态从这些状态流向吸引子。到达吸引子所需的时间称为'''瞬时 transient time '''。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机能力的不断提高，对看似简单的模型的理解也越来越深刻，不同的作者对吸引子的平均数量和长度给出了不同的估计，这里简单总结一下主要的出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力系统理论中，网络的结构和吸引子的长度对应于网络的动态阶段。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定、临界或混乱的行为。这种现象受节点平均连接数的临界值（<math>K_{c}</math>）支配，可以用汉明距离作为距离度量。在非稳定体制下，两个初始接近状态之间的平均距离在时间上呈指数级增长，而在稳定体制下则呈指数级减小。在这其中，用 "最初接近的状态 "意味着汉明距离与网络中的节点数（<math>N</math>）相比是很小的。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于N-K模型，如果 <math>K<K_{c}</math> ，网络是稳定的；如果 <math>K=K_{c}</math> ，网络是临界的；如果 <math>K>K_{c}</math> ，网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

一个给定节点的状态 <math>n_{i}</math> 根据其真值表进行更新，真值表的输出是随机填充的。<math>p_{i}</math> 表示将关闭输出分配给给定系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果 <math>p_{i}=p=const.</math> 对于每一个节点，稳定和混沌范围之间的转换取决于 <math>p</math> 。根据伯纳德-德里达和伊夫-波莫的观点
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为 <math>K_{c}=1/[2p(1-p)]</math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果 <math>K</math> 不是常数，且内度和外度之间没有相关性，则稳定性的条件由 <math>\langle K^{in}\rangle</math> 决定，如果 <math>\langle K^{in}\rangle <K_{c}</math> ，网络是稳定的。如果 <math>\langle K^{in}\rangle =K_{c}</math> ，则为临界；如果<math>\langle K^{in}\rangle >K_{c}</math> ，则为不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于'''无标度拓扑 scale-free topology '''的网络来说，稳定性的条件是一样的，其中的出入度分布是幂律分布。<math>P(K) \propto K^{-\gamma}</math>, 和 <math>\langle K^{in}\rangle=\langle K^{out} \rangle</math> ，因为从一个节点发出的每一条外链都是到另一个节点的内链。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度显示了给定节点的布尔函数的输出在其输入变化时发生变化的概率。对于随机布尔网络。
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由最大的特征值<math>\lambda_{Q}</math>来控制。的矩阵 <math>Q</math>，其中<math> Q_{ij}=q_{i}A_{ij}</math>，<math>A</math> 是网络的邻接矩阵。如果 <math>\lambda_{Q}<1</math>，网络是稳定的；如果 <math>\lambda_{Q}=1</math>，网络是临界的；如果 <math>\lambda_{Q}>1</math>，网络是不稳定的。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑结构。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*'''同质情况 Homogeneous Case '''只是指网格，它只是对著名的'''伊辛模型 lsing model '''的还原。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2021-02-06T10:47:58Z

923397935：/* Variations of the model */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network '''由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间 ''t'' 的网络状态上的函数来确定整个网络在时间 ''t+1''的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 Regulatory Networks '''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有 ''2k'' 可能的规则中随机选择的，有 ''K'' 个输入。在 ''K=2'' 时，两类行为往往占主导地位。但对于 ''K>2'' ，人们看到的行为很快就会接近随机映射的典型特征，其中代表 ''N'' 个底层节点的 ''2k'' 种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

'''随机布尔网络 Random Boolean Network（RBN） '''是指从所有可能的特定大小的布尔网络 ''N'' 的集合中随机选取的网络。然后，人们可以从统计学上研究，这种网络的预期特性如何依赖于所有可能网络的集合的各种统计特性。例如，人们可以研究RBN行为如何随着''' Average Connectivity 平均连通性 '''的改变而改变。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

1969年，Stuart A. Kauffman提出了第一个布尔网络，作为遗传调控网络的随机模型，但其数学理解在2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络只有 2N 种可能的状态，一个轨迹迟早会到达以前访问过的状态，因此，由于动力学是确定性的，轨迹将落入一个稳定状态或周期，称为吸引子（不过在更广泛的动力学系统领域，一个周期只有当来自它的扰动导致回到它时才是吸引子）。如果吸引子只有一个状态，则称为点吸引子，如果吸引子由一个以上的状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。只在轨迹开始时出现的状态（没有轨迹导致它们），称为'''伊甸园状态
garden-of-Eden states '''网络的动态从这些状态流向吸引子。到达吸引子所需的时间称为'''瞬时 transient time '''。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机能力的不断提高，对看似简单的模型的理解也越来越深刻，不同的作者对吸引子的平均数量和长度给出了不同的估计，这里简单总结一下主要的出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力系统理论中，网络的结构和吸引子的长度对应于网络的动态阶段。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定、临界或混乱的行为。这种现象受节点平均连接数的临界值（<math>K_{c}</math>）支配，可以用汉明距离作为距离度量。在非稳定体制下，两个初始接近状态之间的平均距离在时间上呈指数级增长，而在稳定体制下则呈指数级减小。在这其中，用 "最初接近的状态 "意味着汉明距离与网络中的节点数（<math>N</math>）相比是很小的。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于N-K模型，如果 <math>K<K_{c}</math> ，网络是稳定的；如果 <math>K=K_{c}</math> ，网络是临界的；如果 <math>K>K_{c}</math> ，网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

一个给定节点的状态 <math>n_{i}</math> 根据其真值表进行更新，真值表的输出是随机填充的。<math>p_{i}</math> 表示将关闭输出分配给给定系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果 <math>p_{i}=p=const.</math> 对于每一个节点，稳定和混沌范围之间的转换取决于 <math>p</math> 。根据伯纳德-德里达和伊夫-波莫的观点
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为 <math>K_{c}=1/[2p(1-p)]</math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果 <math>K</math> 不是常数，且内度和外度之间没有相关性，则稳定性的条件由 <math>\langle K^{in}\rangle</math> 决定，如果 <math>\langle K^{in}\rangle <K_{c}</math> ，网络是稳定的。如果 <math>\langle K^{in}\rangle =K_{c}</math> ，则为临界；如果<math>\langle K^{in}\rangle >K_{c}</math> ，则为不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于'''无标度拓扑 scale-free topology '''的网络来说，稳定性的条件是一样的，其中的出入度分布是幂律分布。<math>P(K) \propto K^{-\gamma}</math>, 和 <math>\langle K^{in}\rangle=\langle K^{out} \rangle</math> ，因为从一个节点发出的每一条外链都是到另一个节点的内链。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度显示了给定节点的布尔函数的输出在其输入变化时发生变化的概率。对于随机布尔网络。
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由最大的特征值<math>\lambda_{Q}</math>来控制。的矩阵 <math>Q</math>，其中<math> Q_{ij}=q_{i}A_{ij}</math>，<math>A</math> 是网络的邻接矩阵。如果 <math>\lambda_{Q}<1</math>，网络是稳定的；如果 <math>\lambda_{Q}=1</math>，网络是临界的；如果 <math>\lambda_{Q}>1</math>，网络是不稳定的。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*同质情况只是指网格，而网格只是对著名的[[Ising模型]]的简化。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2021-02-06T10:47:08Z

923397935：/* Stability */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network '''由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间 ''t'' 的网络状态上的函数来确定整个网络在时间 ''t+1''的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 Regulatory Networks '''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有 ''2k'' 可能的规则中随机选择的，有 ''K'' 个输入。在 ''K=2'' 时，两类行为往往占主导地位。但对于 ''K>2'' ，人们看到的行为很快就会接近随机映射的典型特征，其中代表 ''N'' 个底层节点的 ''2k'' 种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

'''随机布尔网络 Random Boolean Network（RBN） '''是指从所有可能的特定大小的布尔网络 ''N'' 的集合中随机选取的网络。然后，人们可以从统计学上研究，这种网络的预期特性如何依赖于所有可能网络的集合的各种统计特性。例如，人们可以研究RBN行为如何随着''' Average Connectivity 平均连通性 '''的改变而改变。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

1969年，Stuart A. Kauffman提出了第一个布尔网络，作为遗传调控网络的随机模型，但其数学理解在2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络只有 2N 种可能的状态，一个轨迹迟早会到达以前访问过的状态，因此，由于动力学是确定性的，轨迹将落入一个稳定状态或周期，称为吸引子（不过在更广泛的动力学系统领域，一个周期只有当来自它的扰动导致回到它时才是吸引子）。如果吸引子只有一个状态，则称为点吸引子，如果吸引子由一个以上的状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。只在轨迹开始时出现的状态（没有轨迹导致它们），称为'''伊甸园状态
garden-of-Eden states '''网络的动态从这些状态流向吸引子。到达吸引子所需的时间称为'''瞬时 transient time '''。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机能力的不断提高，对看似简单的模型的理解也越来越深刻，不同的作者对吸引子的平均数量和长度给出了不同的估计，这里简单总结一下主要的出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力系统理论中，网络的结构和吸引子的长度对应于网络的动态阶段。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定、临界或混乱的行为。这种现象受节点平均连接数的临界值（<math>K_{c}</math>）支配，可以用汉明距离作为距离度量。在非稳定体制下，两个初始接近状态之间的平均距离在时间上呈指数级增长，而在稳定体制下则呈指数级减小。在这其中，用 "最初接近的状态 "意味着汉明距离与网络中的节点数（<math>N</math>）相比是很小的。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于N-K模型，如果 <math>K<K_{c}</math> ，网络是稳定的；如果 <math>K=K_{c}</math> ，网络是临界的；如果 <math>K>K_{c}</math> ，网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

一个给定节点的状态 <math>n_{i}</math> 根据其真值表进行更新，真值表的输出是随机填充的。<math>p_{i}</math> 表示将关闭输出分配给给定系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果 <math>p_{i}=p=const.</math> 对于每一个节点，稳定和混沌范围之间的转换取决于 <math>p</math> 。根据伯纳德-德里达和伊夫-波莫的观点
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为 <math>K_{c}=1/[2p(1-p)]</math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果 <math>K</math> 不是常数，且内度和外度之间没有相关性，则稳定性的条件由 <math>\langle K^{in}\rangle</math> 决定，如果 <math>\langle K^{in}\rangle <K_{c}</math> ，网络是稳定的。如果 <math>\langle K^{in}\rangle =K_{c}</math> ，则为临界；如果<math>\langle K^{in}\rangle >K_{c}</math> ，则为不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于'''无标度拓扑 scale-free topology '''的网络来说，稳定性的条件是一样的，其中的出入度分布是幂律分布。<math>P(K) \propto K^{-\gamma}</math>, 和 <math>\langle K^{in}\rangle=\langle K^{out} \rangle</math> ，因为从一个节点发出的每一条外链都是到另一个节点的内链。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度显示了给定节点的布尔函数的输出在其输入变化时发生变化的概率。对于随机布尔网络。
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由最大的特征值<math>\lambda_{Q}</math>来控制。的矩阵 <math>Q</math>，其中<math> Q_{ij}=q_{i}A_{ij}</math>，<math>A</math> 是网络的邻接矩阵。如果 <math>\lambda_{Q}<1</math>，网络是稳定的；如果 <math>\lambda_{Q}=1</math>，网络是临界的；如果 <math>\lambda_{Q}>1</math>，网络是不稳定的。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*同质情况只是指网格，而网格只是对著名的[[Ising模型]]的简化。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2021-02-06T10:40:42Z

923397935：/* Attractors */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network '''由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间 ''t'' 的网络状态上的函数来确定整个网络在时间 ''t+1''的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 Regulatory Networks '''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有 ''2k'' 可能的规则中随机选择的，有 ''K'' 个输入。在 ''K=2'' 时，两类行为往往占主导地位。但对于 ''K>2'' ，人们看到的行为很快就会接近随机映射的典型特征，其中代表 ''N'' 个底层节点的 ''2k'' 种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

'''随机布尔网络 Random Boolean Network（RBN） '''是指从所有可能的特定大小的布尔网络 ''N'' 的集合中随机选取的网络。然后，人们可以从统计学上研究，这种网络的预期特性如何依赖于所有可能网络的集合的各种统计特性。例如，人们可以研究RBN行为如何随着''' Average Connectivity 平均连通性 '''的改变而改变。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

1969年，Stuart A. Kauffman提出了第一个布尔网络，作为遗传调控网络的随机模型，但其数学理解在2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络只有 2N 种可能的状态，一个轨迹迟早会到达以前访问过的状态，因此，由于动力学是确定性的，轨迹将落入一个稳定状态或周期，称为吸引子（不过在更广泛的动力学系统领域，一个周期只有当来自它的扰动导致回到它时才是吸引子）。如果吸引子只有一个状态，则称为点吸引子，如果吸引子由一个以上的状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。只在轨迹开始时出现的状态（没有轨迹导致它们），称为'''伊甸园状态
garden-of-Eden states '''网络的动态从这些状态流向吸引子。到达吸引子所需的时间称为'''瞬时 transient time '''。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机能力的不断提高，对看似简单的模型的理解也越来越深刻，不同的作者对吸引子的平均数量和长度给出了不同的估计，这里简单总结一下主要的出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力系统理论中，网络的结构和吸引子的长度对应于网络的动态阶段。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定、临界或混乱的行为。这种现象受节点平均连接数的临界值（<math>K_{c}</math>）支配，可以用汉明距离作为距离度量。在非稳定体制下，两个初始接近状态之间的平均距离在时间上呈指数级增长，而在稳定体制下则呈指数级减小。在这其中，用 "最初接近的状态 "意味着汉明距离与网络中的节点数（<math>N</math>）相比是很小的。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于N-K模型，如果 <math>K<K_{c}</math> ，网络是稳定的；如果 <math>K=K_{c}</math> ，网络是临界的；如果 <math>K>K_{c}</math> ，网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

一个给定节点的状态 <math>n_{i}</math> 根据其真值表进行更新，真值表的输出是随机填充的。<math>p_{i}</math> 表示将关闭输出分配给给定系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果 <math>p_{i}=p=const.</math> 对于每一个节点，稳定和混沌范围之间的转换取决于 <math>p</math> 。根据伯纳德-德里达和伊夫-波莫的观点
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为 <math>K_{c}=1/[2p(1-p)]</math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果 <math>K</math> 不是常数，且内度和外度之间没有相关性，则稳定性的条件由 <math>\langle K^{in}\rangle</math> 决定，如果 <math>\langle K^{in}\rangle <K_{c}</math> ，网络是稳定的。如果 <math>\langle K^{in}\rangle =K_{c}</math> ，则为临界；如果<math>\langle K^{in}\rangle >K_{c}</math> ，则为不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于具有无标度拓扑的网络，其稳定性条件是相同的，其中进出度分布是幂律分布：<math> P(K) \propto K^{-\gamma} </math>和<math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>，因为节点的每个出度链接都是到另一个节点的入度链接。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度表示给定节点的布尔函数的输出更改（如果其输入更改）的概率。对于随机布尔网络，
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由矩阵<math> Q </math>的最大特征值<math> \lambda_{Q} </math>决定，其中<math> Q_{ij}=q_{i}A_{ij} </math>和<math> A </math>是网络的邻接矩阵。如果<math>\lambda_{Q}<1</math>是稳定的网络，如果<math>\lambda_{Q}=1</math>是关键的网络，如果<math>\lambda_{Q}>1</math>，则不稳定。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*同质情况只是指网格，而网格只是对著名的[[Ising模型]]的简化。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2021-02-06T10:40:18Z

923397935：/* Classical model */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network '''由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间 ''t'' 的网络状态上的函数来确定整个网络在时间 ''t+1''的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 Regulatory Networks '''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有 ''2k'' 可能的规则中随机选择的，有 ''K'' 个输入。在 ''K=2'' 时，两类行为往往占主导地位。但对于 ''K>2'' ，人们看到的行为很快就会接近随机映射的典型特征，其中代表 ''N'' 个底层节点的 ''2k'' 种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

'''随机布尔网络 Random Boolean Network（RBN） '''是指从所有可能的特定大小的布尔网络 ''N'' 的集合中随机选取的网络。然后，人们可以从统计学上研究，这种网络的预期特性如何依赖于所有可能网络的集合的各种统计特性。例如，人们可以研究RBN行为如何随着''' Average Connectivity 平均连通性 '''的改变而改变。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

1969年，Stuart A. Kauffman提出了第一个布尔网络，作为遗传调控网络的随机模型，但其数学理解在2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络只有 2N 种可能的状态，一个轨迹迟早会到达以前访问过的状态，因此，由于动力学是确定性的，轨迹将落入一个稳定状态或周期，称为吸引子（不过在更广泛的动力学系统领域，一个周期只有当来自它的扰动导致回到它时才是吸引子）。如果吸引子只有一个状态，则称为点吸引子，如果吸引子由一个以上的状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。只在轨迹开始时出现的状态（没有轨迹导致它们），称为'''伊甸园状态
garden-of-Eden states '''网络的动态从这些状态流向吸引子。到达吸引子所需的时间称为'''瞬时 transient time '''。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机能力的不断提高，对看似简单的模型的理解也越来越深刻，不同的作者对吸引子的平均数量和长度给出了不同的估计，这里简单总结一下主要的出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力系统理论中，网络的结构和吸引子的长度对应于网络的动态阶段。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定、临界或混乱的行为。这种现象受节点平均连接数的临界值（<math>K_{c}</math>）支配，可以用汉明距离作为距离度量。在非稳定体制下，两个初始接近状态之间的平均距离在时间上呈指数级增长，而在稳定体制下则呈指数级减小。在这其中，用 "最初接近的状态 "意味着汉明距离与网络中的节点数（<math>N</math>）相比是很小的。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于N-K模型，如果 <math>K<K_{c}</math> ，网络是稳定的；如果 <math>K=K_{c}</math> ，网络是临界的；如果 <math>K>K_{c}</math> ，网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

一个给定节点的状态 <math>n_{i}</math> 根据其真值表进行更新，真值表的输出是随机填充的。<math>p_{i}</math> 表示将关闭输出分配给给定系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果 <math>p_{i}=p=const.</math> 对于每一个节点，稳定和混沌范围之间的转换取决于 <math>p</math> 。根据伯纳德-德里达和伊夫-波莫的观点
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为 <math>K_{c}=1/[2p(1-p)]</math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果 <math>K</math> 不是常数，且内度和外度之间没有相关性，则稳定性的条件由 <math>\langle K^{in}\rangle</math> 决定，如果 <math>\langle K^{in}\rangle <K_{c}</math> ，网络是稳定的。如果 <math>\langle K^{in}\rangle =K_{c}</math> ，则为临界；如果<math>\langle K^{in}\rangle >K_{c}</math> ，则为不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于具有无标度拓扑的网络，其稳定性条件是相同的，其中进出度分布是幂律分布：<math> P(K) \propto K^{-\gamma} </math>和<math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>，因为节点的每个出度链接都是到另一个节点的入度链接。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度表示给定节点的布尔函数的输出更改（如果其输入更改）的概率。对于随机布尔网络，
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由矩阵<math> Q </math>的最大特征值<math> \lambda_{Q} </math>决定，其中<math> Q_{ij}=q_{i}A_{ij} </math>和<math> A </math>是网络的邻接矩阵。如果<math>\lambda_{Q}<1</math>是稳定的网络，如果<math>\lambda_{Q}=1</math>是关键的网络，如果<math>\lambda_{Q}>1</math>，则不稳定。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*同质情况只是指网格，而网格只是对著名的[[Ising模型]]的简化。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2021-02-06T10:39:00Z

923397935：

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network '''由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间 ''t'' 的网络状态上的函数来确定整个网络在时间 ''t+1''的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 Regulatory Networks '''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有 ''2k'' 可能的规则中随机选择的，有 ''K'' 个输入。在 ''K=2'' 时，两类行为往往占主导地位。但对于 ''K>2'' ，人们看到的行为很快就会接近随机映射的典型特征，其中代表 ''N'' 个底层节点的 ''2k'' 种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

'''随机布尔网络 Random Boolean Network（RBN） '''是指从所有可能的特定大小的布尔网络 ''N'' 的集合中随机选取的网络。然后，人们可以从统计学上研究，这种网络的预期特性如何依赖于所有可能网络的集合的各种统计特性。例如，人们可以研究RBN行为如何随着''''''的改变而改变。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

1969年，Stuart A. Kauffman提出了第一个布尔网络，作为遗传调控网络的随机模型，但其数学理解在2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络只有 2N 种可能的状态，一个轨迹迟早会到达以前访问过的状态，因此，由于动力学是确定性的，轨迹将落入一个稳定状态或周期，称为吸引子（不过在更广泛的动力学系统领域，一个周期只有当来自它的扰动导致回到它时才是吸引子）。如果吸引子只有一个状态，则称为点吸引子，如果吸引子由一个以上的状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。只在轨迹开始时出现的状态（没有轨迹导致它们），称为'''伊甸园状态
garden-of-Eden states '''网络的动态从这些状态流向吸引子。到达吸引子所需的时间称为'''瞬时 transient time '''。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机能力的不断提高，对看似简单的模型的理解也越来越深刻，不同的作者对吸引子的平均数量和长度给出了不同的估计，这里简单总结一下主要的出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力系统理论中，网络的结构和吸引子的长度对应于网络的动态阶段。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定、临界或混乱的行为。这种现象受节点平均连接数的临界值（<math>K_{c}</math>）支配，可以用汉明距离作为距离度量。在非稳定体制下，两个初始接近状态之间的平均距离在时间上呈指数级增长，而在稳定体制下则呈指数级减小。在这其中，用 "最初接近的状态 "意味着汉明距离与网络中的节点数（<math>N</math>）相比是很小的。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于N-K模型，如果 <math>K<K_{c}</math> ，网络是稳定的；如果 <math>K=K_{c}</math> ，网络是临界的；如果 <math>K>K_{c}</math> ，网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

一个给定节点的状态 <math>n_{i}</math> 根据其真值表进行更新，真值表的输出是随机填充的。<math>p_{i}</math> 表示将关闭输出分配给给定系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果 <math>p_{i}=p=const.</math> 对于每一个节点，稳定和混沌范围之间的转换取决于 <math>p</math> 。根据伯纳德-德里达和伊夫-波莫的观点
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为 <math>K_{c}=1/[2p(1-p)]</math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果 <math>K</math> 不是常数，且内度和外度之间没有相关性，则稳定性的条件由 <math>\langle K^{in}\rangle</math> 决定，如果 <math>\langle K^{in}\rangle <K_{c}</math> ，网络是稳定的。如果 <math>\langle K^{in}\rangle =K_{c}</math> ，则为临界；如果<math>\langle K^{in}\rangle >K_{c}</math> ，则为不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于具有无标度拓扑的网络，其稳定性条件是相同的，其中进出度分布是幂律分布：<math> P(K) \propto K^{-\gamma} </math>和<math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>，因为节点的每个出度链接都是到另一个节点的入度链接。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度表示给定节点的布尔函数的输出更改（如果其输入更改）的概率。对于随机布尔网络，
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由矩阵<math> Q </math>的最大特征值<math> \lambda_{Q} </math>决定，其中<math> Q_{ij}=q_{i}A_{ij} </math>和<math> A </math>是网络的邻接矩阵。如果<math>\lambda_{Q}<1</math>是稳定的网络，如果<math>\lambda_{Q}=1</math>是关键的网络，如果<math>\lambda_{Q}>1</math>，则不稳定。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*同质情况只是指网格，而网格只是对著名的[[Ising模型]]的简化。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2021-02-06T10:35:06Z

923397935：

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network '''由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间 ''t'' 的网络状态上的函数来确定整个网络在时间 ''t+1''的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 Regulatory Networks '''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有 ''2k'' 可能的规则中随机选择的，有 ''K'' 个输入。在 ''K=2'' 时，两类行为往往占主导地位。但对于 ''K>2'' ，人们看到的行为很快就会接近随机映射的典型特征，其中代表 ''N'' 个底层节点的 ''2k'' 种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

'''随机布尔网络 Random Boolean Network（RBN） '''是指从所有可能的特定大小的布尔网络 ''N'' 的集合中随机选取的网络。然后，人们可以从统计学上研究，这种网络的预期特性如何依赖于所有可能网络的集合的各种统计特性。例如，人们可以研究RBN行为如何随着''''''的改变而改变。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

1969年，Stuart A. Kauffman提出了第一个布尔网络，作为遗传调控网络的随机模型，但其数学理解在2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络只有 2N 种可能的状态，一个轨迹迟早会到达以前访问过的状态，因此，由于动力学是确定性的，轨迹将落入一个稳定状态或周期，称为吸引子（不过在更广泛的动力学系统领域，一个周期只有当来自它的扰动导致回到它时才是吸引子）。如果吸引子只有一个状态，则称为点吸引子，如果吸引子由一个以上的状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。只在轨迹开始时出现的状态（没有轨迹导致它们），称为'''伊甸园状态
garden-of-Eden states '''网络的动态从这些状态流向吸引子。到达吸引子所需的时间称为'''瞬时 transient time '''。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机能力的不断提高，对看似简单的模型的理解也越来越深刻，不同的作者对吸引子的平均数量和长度给出了不同的估计，这里简单总结一下主要的出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力系统理论中，网络的结构和吸引子的长度对应于网络的动态阶段。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定、临界或混乱的行为。这种现象受节点平均连接数的临界值（<math>K_{c}</math>）支配，可以用汉明距离作为距离度量。在非稳定体制下，两个初始接近状态之间的平均距离在时间上呈指数级增长，而在稳定体制下则呈指数级减小。在这其中，用 "最初接近的状态 "意味着汉明距离与网络中的节点数（<math>N</math>）相比是很小的。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于N-K模型，如果 <math>K<K_{c}</math> ，网络是稳定的；如果 <math>K=K_{c}</math> ，网络是临界的；如果 <math>K>K_{c}</math> ，网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

一个给定节点的状态 <math>n_{i}</math> 根据其真值表进行更新，真值表的输出是随机填充的。<math>p_{i}</math> 表示将关闭输出分配给给定系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果 <math>p_{i}=p=const.</math> 对于每一个节点，稳定和混沌范围之间的转换取决于 <math>p</math> 。根据伯纳德-德里达和伊夫-波莫的观点
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为 <math>K_{c}=1/[2p(1-p)]</math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果 <math>K</math> 不是常数，且内度和外度之间没有相关性，则稳定性的条件由 <math>\langle K^{in}\rangle</math> 决定，如果 <math>\langle K^{in}/rangle <K_{c}</math> ，网络是稳定的。如果 <math>\langle K^{in}/rangle =K_{c}</math> ，则为临界；如果<math>\langle K^{in}/rangle >K_{c}</math> ，则为不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于具有无标度拓扑的网络，其稳定性条件是相同的，其中进出度分布是幂律分布：<math> P(K) \propto K^{-\gamma} </math>和<math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>，因为节点的每个出度链接都是到另一个节点的入度链接。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度表示给定节点的布尔函数的输出更改（如果其输入更改）的概率。对于随机布尔网络，
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由矩阵<math> Q </math>的最大特征值<math> \lambda_{Q} </math>决定，其中<math> Q_{ij}=q_{i}A_{ij} </math>和<math> A </math>是网络的邻接矩阵。如果<math>\lambda_{Q}<1</math>是稳定的网络，如果<math>\lambda_{Q}=1</math>是关键的网络，如果<math>\lambda_{Q}>1</math>，则不稳定。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*同质情况只是指网格，而网格只是对著名的[[Ising模型]]的简化。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2021-02-06T08:03:03Z

923397935：

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network '''由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间 ''t'' 的网络状态上的函数来确定整个网络在时间 ''t+1''的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 Regulatory Networks '''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有 ''2k'' 可能的规则中随机选择的，有 ''K'' 个输入。在 ''K=2'' 时，两类行为往往占主导地位。但对于 ''K>2'' ，人们看到的行为很快就会接近随机映射的典型特征，其中代表 ''N'' 个底层节点的 ''2k'' 种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

'''随机布尔网络 Random Boolean Network（RBN） '''是指从所有可能的特定大小的布尔网络 ''N'' 的集合中随机选取的网络。然后，人们可以从统计学上研究，这种网络的预期特性如何依赖于所有可能网络的集合的各种统计特性。例如，人们可以研究RBN行为如何随着''''''的改变而改变。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

1969年，Stuart A. Kauffman提出了第一个布尔网络，作为遗传调控网络的随机模型，但其数学理解在2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络仅具有2N个可能的状态，因此轨迹迟早会到达先前访问的状态，由于动力学是确定性的，因此轨迹会陷入稳态或称为一个吸引子（尽管在动力学系统的更广泛领域中，如果一个周期的扰动导致它返回，那么它只是一个吸引子）。如果吸引子只有一个状态，则称为点吸引子；如果吸引子由多个状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。仅在轨迹开始时出现的状态（没有轨迹导致它们出现）被称为伊甸园状态，网络的动力学从这些状态流向吸引子。到达吸引子所需的时间称为瞬态时间。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机功能的增强和对看似简单模型的了解的增加，不同的作者对吸引子的平均数量和长度给出了不同的估计，此处简要概述了主要出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力学系统理论中，网络吸引子的结构和长度与网络的动态相位相对应。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定，关键或混乱的行为。该现象由节点平均连接数的临界值（<math>K_{c}</math>）控制，并且可以通过'''海明距离 Hamming Distance '''作为距离度量来表征。在不稳定状态下，两个初始关闭状态之间的距离平均随时间呈指数增长，而在稳定状态下，其呈指数下降。在这种情况下，“初始关闭状态”表示与网络中的节点数（<math>N</math>）相比，海明距离较小。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于'''N-K模型 N-K-Model '''，如果<math>K<K_{c}</math>，则网络是稳定的；如果<math>K=K_{c}</math>，则网络是关键的；如果<math>K>K_{c}</math>，则网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

给定节点<math> n_{i} </math>的状态根据其真值表进行更新，该表的输出是随机填充的。 <math> p_{i} </math>表示将关闭输出分配给给定的一系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果<math> p_{i}=p=const. </math>,对于每个节点，稳定范围和混沌范围之间的过渡取决于<math> p </math>。根据Bernard Derrida和Yves Pomeau的说法
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为<math> K_{c}=1/[2p(1-p)] </math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果<math> K </math>不是常数，并且进度和出度之间没有相关性，则稳定性条件由<math> \langle K^{in}\rangle </math> 确定, 如果<math>\langle K^{in}\rangle <K_{c}</math>，则网络是稳定的；如果<math>\langle K^{in}\rangle =K_{c}</math>，则网络是稳定的 >，如果<math>\langle K^{in}\rangle >K_{c}</math>，则不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于具有无标度拓扑的网络，其稳定性条件是相同的，其中进出度分布是幂律分布：<math> P(K) \propto K^{-\gamma} </math>和<math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>，因为节点的每个出度链接都是到另一个节点的入度链接。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度表示给定节点的布尔函数的输出更改（如果其输入更改）的概率。对于随机布尔网络，
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由矩阵<math> Q </math>的最大特征值<math> \lambda_{Q} </math>决定，其中<math> Q_{ij}=q_{i}A_{ij} </math>和<math> A </math>是网络的邻接矩阵。如果<math>\lambda_{Q}<1</math>是稳定的网络，如果<math>\lambda_{Q}=1</math>是关键的网络，如果<math>\lambda_{Q}>1</math>，则不稳定。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*同质情况只是指网格，而网格只是对著名的[[Ising模型]]的简化。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2021-02-06T07:44:08Z

923397935：

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network '''由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间 ''t'' 的网络状态上的函数来确定整个网络在时间 ''t+1''的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 Regulatory Networks '''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有 ''2k'' 可能的规则中随机选择的，有 ''K'' 个输入。在 ''K=2'' 时，两类行为往往占主导地位。但对于 ''K>2'' ，人们看到的行为很快就会接近随机映射的典型特征，其中代表 ''N'' 个底层节点的 ''2k'' 种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

'''随机布尔网络 Random Boolean Network（RBN） '''是指从所有可能的特定大小的布尔网络 ''N'' 的集合中随机选取的网络。然后，人们可以从统计学上研究，这种网络的预期特性如何依赖于所有可能网络的集合的各种统计特性。例如，人们可以研究RBN行为如何随着'''平均连通性的改变而改变。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

第一个布尔网络是由 Stuart a. Kauffman 在1969年提出的，作为基因调控网络的随机模型，但是它们的数学理解直到2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络仅具有2N个可能的状态，因此轨迹迟早会到达先前访问的状态，由于动力学是确定性的，因此轨迹会陷入稳态或称为一个吸引子（尽管在动力学系统的更广泛领域中，如果一个周期的扰动导致它返回，那么它只是一个吸引子）。如果吸引子只有一个状态，则称为点吸引子；如果吸引子由多个状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。仅在轨迹开始时出现的状态（没有轨迹导致它们出现）被称为伊甸园状态，网络的动力学从这些状态流向吸引子。到达吸引子所需的时间称为瞬态时间。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机功能的增强和对看似简单模型的了解的增加，不同的作者对吸引子的平均数量和长度给出了不同的估计，此处简要概述了主要出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

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== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力学系统理论中，网络吸引子的结构和长度与网络的动态相位相对应。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定，关键或混乱的行为。该现象由节点平均连接数的临界值（<math>K_{c}</math>）控制，并且可以通过'''海明距离 Hamming Distance '''作为距离度量来表征。在不稳定状态下，两个初始关闭状态之间的距离平均随时间呈指数增长，而在稳定状态下，其呈指数下降。在这种情况下，“初始关闭状态”表示与网络中的节点数（<math>N</math>）相比，海明距离较小。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于'''N-K模型 N-K-Model '''，如果<math>K<K_{c}</math>，则网络是稳定的；如果<math>K=K_{c}</math>，则网络是关键的；如果<math>K>K_{c}</math>，则网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

给定节点<math> n_{i} </math>的状态根据其真值表进行更新，该表的输出是随机填充的。 <math> p_{i} </math>表示将关闭输出分配给给定的一系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果<math> p_{i}=p=const. </math>,对于每个节点，稳定范围和混沌范围之间的过渡取决于<math> p </math>。根据Bernard Derrida和Yves Pomeau的说法
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为<math> K_{c}=1/[2p(1-p)] </math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果<math> K </math>不是常数，并且进度和出度之间没有相关性，则稳定性条件由<math> \langle K^{in}\rangle </math> 确定, 如果<math>\langle K^{in}\rangle <K_{c}</math>，则网络是稳定的；如果<math>\langle K^{in}\rangle =K_{c}</math>，则网络是稳定的 >，如果<math>\langle K^{in}\rangle >K_{c}</math>，则不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于具有无标度拓扑的网络，其稳定性条件是相同的，其中进出度分布是幂律分布：<math> P(K) \propto K^{-\gamma} </math>和<math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>，因为节点的每个出度链接都是到另一个节点的入度链接。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度表示给定节点的布尔函数的输出更改（如果其输入更改）的概率。对于随机布尔网络，
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由矩阵<math> Q </math>的最大特征值<math> \lambda_{Q} </math>决定，其中<math> Q_{ij}=q_{i}A_{ij} </math>和<math> A </math>是网络的邻接矩阵。如果<math>\lambda_{Q}<1</math>是稳定的网络，如果<math>\lambda_{Q}=1</math>是关键的网络，如果<math>\lambda_{Q}>1</math>，则不稳定。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*同质情况只是指网格，而网格只是对著名的[[Ising模型]]的简化。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2021-02-06T07:35:33Z

923397935：

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network '''由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间t的网络状态上的函数来确定整个网络在时间t+1的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 regulatory networks '''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有''2k''可能的规则中随机选择的，有''K''个输入。在''K=2''时，两类行为往往占主导地位。但对于''K>2''，人们看到的行为很快就会接近随机映射的典型特征，其中代表''N''个底层节点的''2k''种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

随机布尔网络（RBN）是从所有可能的特定规模的布尔网络集合中随机选择的网络。然后，我们可以从统计学上研究此类网络的预期属性如何依赖于所有可能网络集合的各种统计特性。例如，可以研究RBN行为如何随着平均连接性的变化而变化。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

第一个布尔网络是由 Stuart a. Kauffman 在1969年提出的，作为基因调控网络的随机模型，但是它们的数学理解直到2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络仅具有2N个可能的状态，因此轨迹迟早会到达先前访问的状态，由于动力学是确定性的，因此轨迹会陷入稳态或称为一个吸引子（尽管在动力学系统的更广泛领域中，如果一个周期的扰动导致它返回，那么它只是一个吸引子）。如果吸引子只有一个状态，则称为点吸引子；如果吸引子由多个状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。仅在轨迹开始时出现的状态（没有轨迹导致它们出现）被称为伊甸园状态，网络的动力学从这些状态流向吸引子。到达吸引子所需的时间称为瞬态时间。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机功能的增强和对看似简单模型的了解的增加，不同的作者对吸引子的平均数量和长度给出了不同的估计，此处简要概述了主要出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力学系统理论中，网络吸引子的结构和长度与网络的动态相位相对应。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定，关键或混乱的行为。该现象由节点平均连接数的临界值（<math>K_{c}</math>）控制，并且可以通过'''海明距离 Hamming Distance '''作为距离度量来表征。在不稳定状态下，两个初始关闭状态之间的距离平均随时间呈指数增长，而在稳定状态下，其呈指数下降。在这种情况下，“初始关闭状态”表示与网络中的节点数（<math>N</math>）相比，海明距离较小。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于'''N-K模型 N-K-Model '''，如果<math>K<K_{c}</math>，则网络是稳定的；如果<math>K=K_{c}</math>，则网络是关键的；如果<math>K>K_{c}</math>，则网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

给定节点<math> n_{i} </math>的状态根据其真值表进行更新，该表的输出是随机填充的。 <math> p_{i} </math>表示将关闭输出分配给给定的一系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果<math> p_{i}=p=const. </math>,对于每个节点，稳定范围和混沌范围之间的过渡取决于<math> p </math>。根据Bernard Derrida和Yves Pomeau的说法
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为<math> K_{c}=1/[2p(1-p)] </math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果<math> K </math>不是常数，并且进度和出度之间没有相关性，则稳定性条件由<math> \langle K^{in}\rangle </math> 确定, 如果<math>\langle K^{in}\rangle <K_{c}</math>，则网络是稳定的；如果<math>\langle K^{in}\rangle =K_{c}</math>，则网络是稳定的 >，如果<math>\langle K^{in}\rangle >K_{c}</math>，则不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于具有无标度拓扑的网络，其稳定性条件是相同的，其中进出度分布是幂律分布：<math> P(K) \propto K^{-\gamma} </math>和<math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>，因为节点的每个出度链接都是到另一个节点的入度链接。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度表示给定节点的布尔函数的输出更改（如果其输入更改）的概率。对于随机布尔网络，
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由矩阵<math> Q </math>的最大特征值<math> \lambda_{Q} </math>决定，其中<math> Q_{ij}=q_{i}A_{ij} </math>和<math> A </math>是网络的邻接矩阵。如果<math>\lambda_{Q}<1</math>是稳定的网络，如果<math>\lambda_{Q}=1</math>是关键的网络，如果<math>\lambda_{Q}>1</math>，则不稳定。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*同质情况只是指网格，而网格只是对著名的[[Ising模型]]的简化。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2021-02-06T07:34:10Z

923397935：

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network '''由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间t的网络状态上的函数来确定整个网络在时间t+1的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 regulatory networks </fonts>'''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有''2k''可能的规则中随机选择的，有''K''个输入。在''K=2''时，两类行为往往占主导地位。但对于''K>2''，人们看到的行为很快就会接近随机映射的典型特征，其中代表''N''个底层节点的''2k''种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

随机布尔网络（RBN）是从所有可能的特定规模的布尔网络集合中随机选择的网络。然后，我们可以从统计学上研究此类网络的预期属性如何依赖于所有可能网络集合的各种统计特性。例如，可以研究RBN行为如何随着平均连接性的变化而变化。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

第一个布尔网络是由 Stuart a. Kauffman 在1969年提出的，作为基因调控网络的随机模型，但是它们的数学理解直到2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络仅具有2N个可能的状态，因此轨迹迟早会到达先前访问的状态，由于动力学是确定性的，因此轨迹会陷入稳态或称为一个吸引子（尽管在动力学系统的更广泛领域中，如果一个周期的扰动导致它返回，那么它只是一个吸引子）。如果吸引子只有一个状态，则称为点吸引子；如果吸引子由多个状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。仅在轨迹开始时出现的状态（没有轨迹导致它们出现）被称为伊甸园状态，网络的动力学从这些状态流向吸引子。到达吸引子所需的时间称为瞬态时间。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机功能的增强和对看似简单模型的了解的增加，不同的作者对吸引子的平均数量和长度给出了不同的估计，此处简要概述了主要出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力学系统理论中，网络吸引子的结构和长度与网络的动态相位相对应。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定，关键或混乱的行为。该现象由节点平均连接数的临界值（<math>K_{c}</math>）控制，并且可以通过'''海明距离 Hamming Distance '''作为距离度量来表征。在不稳定状态下，两个初始关闭状态之间的距离平均随时间呈指数增长，而在稳定状态下，其呈指数下降。在这种情况下，“初始关闭状态”表示与网络中的节点数（<math>N</math>）相比，海明距离较小。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于'''N-K模型 N-K-Model '''，如果<math>K<K_{c}</math>，则网络是稳定的；如果<math>K=K_{c}</math>，则网络是关键的；如果<math>K>K_{c}</math>，则网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

给定节点<math> n_{i} </math>的状态根据其真值表进行更新，该表的输出是随机填充的。 <math> p_{i} </math>表示将关闭输出分配给给定的一系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果<math> p_{i}=p=const. </math>,对于每个节点，稳定范围和混沌范围之间的过渡取决于<math> p </math>。根据Bernard Derrida和Yves Pomeau的说法
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为<math> K_{c}=1/[2p(1-p)] </math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果<math> K </math>不是常数，并且进度和出度之间没有相关性，则稳定性条件由<math> \langle K^{in}\rangle </math> 确定, 如果<math>\langle K^{in}\rangle <K_{c}</math>，则网络是稳定的；如果<math>\langle K^{in}\rangle =K_{c}</math>，则网络是稳定的 >，如果<math>\langle K^{in}\rangle >K_{c}</math>，则不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于具有无标度拓扑的网络，其稳定性条件是相同的，其中进出度分布是幂律分布：<math> P(K) \propto K^{-\gamma} </math>和<math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>，因为节点的每个出度链接都是到另一个节点的入度链接。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度表示给定节点的布尔函数的输出更改（如果其输入更改）的概率。对于随机布尔网络，
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由矩阵<math> Q </math>的最大特征值<math> \lambda_{Q} </math>决定，其中<math> Q_{ij}=q_{i}A_{ij} </math>和<math> A </math>是网络的邻接矩阵。如果<math>\lambda_{Q}<1</math>是稳定的网络，如果<math>\lambda_{Q}=1</math>是关键的网络，如果<math>\lambda_{Q}>1</math>，则不稳定。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*同质情况只是指网格，而网格只是对著名的[[Ising模型]]的简化。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2021-02-06T07:25:32Z

923397935：

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和每个节点K=1条链路。节点可以被打开（红色）或关闭（蓝色）。细（黑色）箭头象征着布尔函数的输入，布尔函数是每个节点的简单 "复制 "函数。粗（灰色）箭头表示同步更新的功能。总共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network 由一组离散的布尔变量组成，每个变量都被分配了一个布尔函数（可能每个变量都不同），它从这些变量的子集中获取输入，并输出决定其被分配的变量的状态。这一组函数实际上决定了变量集上的拓扑结构（连通性），这些变量就成为网络中的节点。通常，系统的动态是以离散时间序列的形式进行的，通过评估每个变量在时间t的网络状态上的函数来确定整个网络在时间t+1的状态，这可能是同步或异步完成的。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络在生物学中已被用于模拟'''调节网络 regulatory networks </fonts>'''。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

在2000年中期人们才完全理解看似数学上的简易（同步）模型。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态都是离散的，即时间序列中的变量集和状态集都各自有一个偏射到一个整数序列上。这样的系统就像网络上的细胞自动机一样，只是当它们被建立起来时，每个节点都有一个规则，这个规则是从所有2}}可能的规则中随机选择的，有K个输入。在K=2时，2类行为往往占主导地位。但对于K>2，人们看到的行为很快就会接近随机映射的典型特征，其中代表N个底层节点的2种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

随机布尔网络（RBN）是从所有可能的特定规模的布尔网络集合中随机选择的网络。然后，我们可以从统计学上研究此类网络的预期属性如何依赖于所有可能网络集合的各种统计特性。例如，可以研究RBN行为如何随着平均连接性的变化而变化。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

第一个布尔网络是由 Stuart a. Kauffman 在1969年提出的，作为基因调控网络的随机模型，但是它们的数学理解直到2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络仅具有2N个可能的状态，因此轨迹迟早会到达先前访问的状态，由于动力学是确定性的，因此轨迹会陷入稳态或称为一个吸引子（尽管在动力学系统的更广泛领域中，如果一个周期的扰动导致它返回，那么它只是一个吸引子）。如果吸引子只有一个状态，则称为点吸引子；如果吸引子由多个状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。仅在轨迹开始时出现的状态（没有轨迹导致它们出现）被称为伊甸园状态，网络的动力学从这些状态流向吸引子。到达吸引子所需的时间称为瞬态时间。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机功能的增强和对看似简单模型的了解的增加，不同的作者对吸引子的平均数量和长度给出了不同的估计，此处简要概述了主要出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力学系统理论中，网络吸引子的结构和长度与网络的动态相位相对应。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定，关键或混乱的行为。该现象由节点平均连接数的临界值（<math>K_{c}</math>）控制，并且可以通过'''海明距离 Hamming Distance '''作为距离度量来表征。在不稳定状态下，两个初始关闭状态之间的距离平均随时间呈指数增长，而在稳定状态下，其呈指数下降。在这种情况下，“初始关闭状态”表示与网络中的节点数（<math>N</math>）相比，海明距离较小。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于'''N-K模型 N-K-Model '''，如果<math>K<K_{c}</math>，则网络是稳定的；如果<math>K=K_{c}</math>，则网络是关键的；如果<math>K>K_{c}</math>，则网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

给定节点<math> n_{i} </math>的状态根据其真值表进行更新，该表的输出是随机填充的。 <math> p_{i} </math>表示将关闭输出分配给给定的一系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果<math> p_{i}=p=const. </math>,对于每个节点，稳定范围和混沌范围之间的过渡取决于<math> p </math>。根据Bernard Derrida和Yves Pomeau的说法
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为<math> K_{c}=1/[2p(1-p)] </math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果<math> K </math>不是常数，并且进度和出度之间没有相关性，则稳定性条件由<math> \langle K^{in}\rangle </math> 确定, 如果<math>\langle K^{in}\rangle <K_{c}</math>，则网络是稳定的；如果<math>\langle K^{in}\rangle =K_{c}</math>，则网络是稳定的 >，如果<math>\langle K^{in}\rangle >K_{c}</math>，则不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于具有无标度拓扑的网络，其稳定性条件是相同的，其中进出度分布是幂律分布：<math> P(K) \propto K^{-\gamma} </math>和<math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>，因为节点的每个出度链接都是到另一个节点的入度链接。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度表示给定节点的布尔函数的输出更改（如果其输入更改）的概率。对于随机布尔网络，
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由矩阵<math> Q </math>的最大特征值<math> \lambda_{Q} </math>决定，其中<math> Q_{ij}=q_{i}A_{ij} </math>和<math> A </math>是网络的邻接矩阵。如果<math>\lambda_{Q}<1</math>是稳定的网络，如果<math>\lambda_{Q}=1</math>是关键的网络，如果<math>\lambda_{Q}>1</math>，则不稳定。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*同质情况只是指网格，而网格只是对著名的[[Ising模型]]的简化。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

邻接表

2020-10-22T11:41:36Z

923397935：/* 实现 */

在'''图论和'''计算机科学'''中，'''邻接表 Adjacency List '''是用来表示'''有限图'''的无序表的集合。每个列表描述图中一个顶点的邻域集。这是计算机程序中常用的几种图形表示法之一。

[[文件:Simple cycle graph.svg.png|right|缩略图|
图：此''' 无向循环图'''可以用三个无序列表来描述:{a,b},{a,c},{b,c}]]

==实现==
{| class = "wikitable" align="right" style="width:18em;"
| colspan = "3" | 上图中的图表有这样的邻接表示法:
|-
| a || 毗邻 || b，c
|-
| b || 毗邻 || a，c
|-
| c || 毗邻 || a，b
|}

图的邻接表表示法将图中的每个顶点与其邻接顶点或边的集合关联起来。这个基本思想有许多变体，在如何实现顶点和集合之间的关联，如何实现集合，以及是否包括顶点和边还是只包括顶点作为第一类对象，以及什么类型的对象被用来表示顶点和边的细节上都有所不同。

* An implementation suggested by [[Guido van Rossum]] uses a [[hash table]] to associate each vertex in a graph with an [[array data structure|array]] of adjacent vertices. In this representation, a vertex may be represented by any hashable object. There is no explicit representation of edges as objects.<ref>{{cite web| author = Guido van Rossum| year = 1998| title = Python Patterns — Implementing Graphs| url = https://www.python.org/doc/essays/graphs/}}</ref>
由Guido van Rossum建议的一种实现方式使用哈希表将图中的每个顶点与相邻顶点的数组数据结构联系起来。在这种表示方式中，一个顶点可以由任何可哈希的对象表示，没有明确地将边表示为对象。<ref>{{cite web| author = Guido van Rossum| year = 1998| title = Python Patterns - Implementing Graphs| url = https://www.python.org/doc/essays/graphs/}}</ref>。

* Cormen et al. suggest an implementation in which the vertices are represented by index numbers.<ref>{{cite book| title = Introduction to Algorithms, Second Edition| publisher = MIT Press and McGraw-Hill | year = 2001|author1 = Thomas H. Cormen|author2 = Charles E. Leiserson| author3 = Ronald L. Rivest| author4 = Clifford Stein }}</ref> Their representation uses an array indexed by vertex number, in which the array cell for each vertex points to a [[singly linked list]] of the neighboring vertices of that vertex. In this representation, the nodes of the singly linked list may be interpreted as edge objects; however, they do not store the full information about each edge (they only store one of the two endpoints of the edge) and in undirected graphs there will be two different linked list nodes for each edge (one within the lists for each of the two endpoints of the edge).
Cormen等人提出了一种用索引号来表示顶点的实现方法。Leiserson|作者3 = Ronald L. Rivest|作者4 = Clifford Stein }}</ref>他们的表示方法使用了一个按顶点编号索引的数组，其中每个顶点的数组单元指向该顶点的相邻顶点的[[单链列表]]。在这种表示方式中，单链表的节点可以被解释为边缘对象；但是，它们并不存储每个边缘的全部信息（它们只存储边缘两个端点中的一个），在非定向图中，每个边缘将有两个不同的链表节点（边缘两个端点中的每一个在列表中）。

它们的表示使用一个按顶点数索引的数组，其中每个顶点的数组单元格指向该顶点的相邻顶点的单链表。在这种表示中，单链表的节点可以解释为边对象; 然而，它们并不存储关于每条边的完整信息(它们只存储边的两个端点中的一个) ，在'''无向图 Undirected Graph '''中，每条边有两个不同的链表节点(边的两个端点中的每个端点的列表中都有一个)。

*The [[object oriented]] incidence list structure suggested by Goodrich and Tamassia has special classes of vertex objects and edge objects. Each vertex object has an instance variable pointing to a collection object that lists the neighboring edge objects. In turn, each edge object points to the two vertex objects at its endpoints.<ref name="A">{{cite book| author = Michael T. Goodrich and Roberto Tamassia | title = Algorithm Design: Foundations, Analysis, and Internet Examples| publisher = John Wiley & Sons | year = 2002 | isbn = 0-471-38365-1}}</ref> This version of the adjacency list uses more memory than the version in which adjacent vertices are listed directly, but the existence of explicit edge objects allows it extra flexibility in storing additional information about edges.
Goodrich和Tamassia提出的[[面向对象]]入射列表结构有特殊的顶点对象和边缘对象类。每个顶点对象都有一个实例变量，指向一个集合对象，该集合对象列出了邻近的边缘对象。反过来，每个边缘对象又指向其端点的两个顶点对象。<ref name="A">{{cite book| author = Michael T. Goodrich and Roberto Tamassia | title = Algorithm Design:

这个版本的邻接表比直接列出相邻顶点的版本使用更多的内存，但是显示边对象的存在操作可以允许它在存储额外的关于边的信息方面有额外的灵活性。

==操作==

接表数据结构执行的主要操作是列出给定顶点的邻居列表。使用上面详细说明的任何表示方式，都可以在每个邻居的固定时间内执行。换句话说，列出一个顶点 ''v'' 所有邻居的总时间与 ''v'' 的'''度 Degree '''成正比。

也可以使用邻接列表来测试两个指定顶点之间是否存在边，但效率不高。在一个邻接表中，每个顶点的邻域都是不排序的，通过对该顶点的邻域进行顺序搜索，可以按照给定两个顶点的最小阶数按时间比例进行边的存在性测试。如果邻域被表示为一个排序数组，则可以使用二进制搜索，时间与次数的对数成正比。

==权衡==

'''邻接表'''的主要替代方法是'''邻接矩阵 Adjacency Matrix '''，该矩阵的行和列按顶点索引，其单元格包含一个布尔值，该值指示与单元格的行和列对应的顶点之间是否存在边。对于'''稀疏图 Sparse Graph '''（大多数顶点对不是由边连接的图），'''邻接表'''比邻接矩阵（存储为二维数组）更节省空间：'''邻接表'''的空间使用与图中的边和顶点的数量成正比，而对于以这种方式存储的邻接矩阵，其空间与顶点数的平方成正比。然而，通过使用由顶点对索引的'''哈希表'''而不是数组，可以更有效地存储邻接矩阵的空间，从而匹配'''邻接表'''的线性空间使用。

'''邻接表'''和'''邻接矩阵'''之间的另一个显著区别是它们执行操作的效率。在'''邻接表'''中，每个顶点的邻居可以有效地列出，在时间上与顶点的程度成正比。在'''邻接矩阵'''中，这个操作需要的时间与图中顶点的数量成正比，而顶点的数量可能明显高于度。另一方面，'''邻接矩阵'''允许测试两个顶点是否在固定的时间内彼此相邻; '''邻接表'''支持这一操作的速度较慢。

==数据结构==

'''作为一种数据结构，邻接表的主要替代方法是邻接矩阵。因为邻接矩阵中的每个条目只占一个比特(bit) ，所以它可以用非常紧凑的方式表示，只占用相邻空间的n2/8}}字节，其中n是图的顶点数。除了避免占用空间外，这种紧凑性还支持局部引用locality of reference。'''

用稀疏邻接矩阵表示邻接表时，将占用更少的空间。这是因为它能避免为不存在的边分配任何空间。在一台32位计算机上，如果使用原始的数组结构实现邻接表，那么对于一个无向图来说，它大约需要占用8e字节的存储空间，其中e表示边的个数。每条边都将会在两个邻接表中重复出现，并分别占用4字节空间。

注意到一个'''无向简单图'''最多可以有 n2边(允许循环)，我们可以让d=e/n2表示该图的密度。然后，8e>n2/8当d>1/64时，即邻接表表示比邻接矩阵表示占用更多的空间。因此，图必须足够稀疏，才适合使用邻接表表示。

除了空间上的权衡，不同的数据结构也有着不同的操作。在邻接表中找到与给定顶点相邻的所有顶点就像读取邻接表一样简单。对于邻接矩阵，必须扫描整行，这需要花费O(n)的时间。给定的两个顶点之间是否有边可以用邻接矩阵一次确定，并且运行时间与邻接列表中两个顶点的最小度数成正比。

==参考资料==

{{reflist}}

==延伸阅读==

* {{cite web | author = David Eppstein| year = 1996| title = ICS 161 Lecture Notes: Graph Algorithms| url = http://www.ics.uci.edu/~eppstein/161/960201.html}}

==其他链接==

{{commons category|Adjacency list}}

* The [[Boost Graph Library]] implements an efficient [http://www.boost.org/libs/graph adjacency list]

* [http://opendatastructures.org/versions/edition-0.1e/ods-java/12_2_AdjacencyLists_Graph_a.html Open Data Structures, Section 12.2, AdjacencyList: A Graph as a Collection of Lists], [[Pat Morin]]

{{Graph representations}}

[[Category:Graph data structures]]

Category:Graph data structures

类别: 图形数据结构

<noinclude>

This page was moved from [[wikipedia:en:Adjacency list]]. Its edit history can be viewed at [[邻接矩阵/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2020-09-27T12:54:26Z

923397935：/* See also */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

有多个节点，每个节点是K = 1个链接。节点可以打开（用红色表示）或关闭（用蓝色表示）。细（黑色）箭头表示布尔函数的输入，布尔函数是每个节点的简单“复制”函数。粗（灰色）箭头显示同步更新的作用。共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network 由一组离散的布尔变量组成，每个布尔变量都分配有一个布尔函数（每个变量可能不同），该布尔函数从这些变量的子集中获取输入，并确定变量所分配到的状态。这组功能实际上确定了变量集上的拓扑（连接性），这些变量随后成为网络中的节点。通常，系统的动力学被视为离散的时间序列，其中在时间t + 1时整个网络的状态是通过评估在时间t时网络状态的每个变量的功能来确定的。这可以同步或异步完成。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络已在生物学中用于建模监管网络。尽管布尔网络是遗传现实的粗略简化，其中基因不是简单的二进制开关，但在某些情况下，它们可以正确捕获表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

看似数学上简单的（同步）模型直到2000年代中期才被完全理解。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态是离散的，即时间序列中的变量集和状态集都具有对整数序列的双射。这样的系统就像网络上的蜂窝自动机一样，除了以下事实：建立它们时，每个节点都有一个规则，该规则是从所有2个具有K个输入的可能节点中随机选择的。在K = 2的情况下，第2类行为倾向于占主导地位。但是对于K> 2，人们看到的行为迅速接近了随机映射的典型行为，在随机映射中，代表N个基础节点的2个状态的演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

随机布尔网络（RBN）是从所有可能的特定规模的布尔网络集合中随机选择的网络。然后，我们可以从统计学上研究此类网络的预期属性如何依赖于所有可能网络集合的各种统计特性。例如，可以研究RBN行为如何随着平均连接性的变化而变化。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

第一个布尔网络是由 Stuart a. Kauffman 在1969年提出的，作为基因调控网络的随机模型，但是它们的数学理解直到2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络仅具有2N个可能的状态，因此轨迹迟早会到达先前访问的状态，由于动力学是确定性的，因此轨迹会陷入稳态或称为一个吸引子（尽管在动力学系统的更广泛领域中，如果一个周期的扰动导致它返回，那么它只是一个吸引子）。如果吸引子只有一个状态，则称为点吸引子；如果吸引子由多个状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。仅在轨迹开始时出现的状态（没有轨迹导致它们出现）被称为伊甸园状态，网络的动力学从这些状态流向吸引子。到达吸引子所需的时间称为瞬态时间。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机功能的增强和对看似简单模型的了解的增加，不同的作者对吸引子的平均数量和长度给出了不同的估计，此处简要概述了主要出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力学系统理论中，网络吸引子的结构和长度与网络的动态相位相对应。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定，关键或混乱的行为。该现象由节点平均连接数的临界值（<math>K_{c}</math>）控制，并且可以通过'''海明距离 Hamming Distance '''作为距离度量来表征。在不稳定状态下，两个初始关闭状态之间的距离平均随时间呈指数增长，而在稳定状态下，其呈指数下降。在这种情况下，“初始关闭状态”表示与网络中的节点数（<math>N</math>）相比，海明距离较小。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于'''N-K模型 N-K-Model '''，如果<math>K<K_{c}</math>，则网络是稳定的；如果<math>K=K_{c}</math>，则网络是关键的；如果<math>K>K_{c}</math>，则网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

给定节点<math> n_{i} </math>的状态根据其真值表进行更新，该表的输出是随机填充的。 <math> p_{i} </math>表示将关闭输出分配给给定的一系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果<math> p_{i}=p=const. </math>,对于每个节点，稳定范围和混沌范围之间的过渡取决于<math> p </math>。根据Bernard Derrida和Yves Pomeau的说法
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为<math> K_{c}=1/[2p(1-p)] </math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果<math> K </math>不是常数，并且进度和出度之间没有相关性，则稳定性条件由<math> \langle K^{in}\rangle </math> 确定, 如果<math>\langle K^{in}\rangle <K_{c}</math>，则网络是稳定的；如果<math>\langle K^{in}\rangle =K_{c}</math>，则网络是稳定的 >，如果<math>\langle K^{in}\rangle >K_{c}</math>，则不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于具有无标度拓扑的网络，其稳定性条件是相同的，其中进出度分布是幂律分布：<math> P(K) \propto K^{-\gamma} </math>和<math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>，因为节点的每个出度链接都是到另一个节点的入度链接。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度表示给定节点的布尔函数的输出更改（如果其输入更改）的概率。对于随机布尔网络，
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由矩阵<math> Q </math>的最大特征值<math> \lambda_{Q} </math>决定，其中<math> Q_{ij}=q_{i}A_{ij} </math>和<math> A </math>是网络的邻接矩阵。如果<math>\lambda_{Q}<1</math>是稳定的网络，如果<math>\lambda_{Q}=1</math>是关键的网络，如果<math>\lambda_{Q}>1</math>，则不稳定。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*同质情况只是指网格，而网格只是对著名的[[Ising模型]]的简化。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 
'''NK模型 NK Model '''

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2020-09-27T12:51:32Z

923397935：/* Classical model */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

有多个节点，每个节点是K = 1个链接。节点可以打开（用红色表示）或关闭（用蓝色表示）。细（黑色）箭头表示布尔函数的输入，布尔函数是每个节点的简单“复制”函数。粗（灰色）箭头显示同步更新的作用。共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network 由一组离散的布尔变量组成，每个布尔变量都分配有一个布尔函数（每个变量可能不同），该布尔函数从这些变量的子集中获取输入，并确定变量所分配到的状态。这组功能实际上确定了变量集上的拓扑（连接性），这些变量随后成为网络中的节点。通常，系统的动力学被视为离散的时间序列，其中在时间t + 1时整个网络的状态是通过评估在时间t时网络状态的每个变量的功能来确定的。这可以同步或异步完成。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络已在生物学中用于建模监管网络。尽管布尔网络是遗传现实的粗略简化，其中基因不是简单的二进制开关，但在某些情况下，它们可以正确捕获表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

看似数学上简单的（同步）模型直到2000年代中期才被完全理解。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态是离散的，即时间序列中的变量集和状态集都具有对整数序列的双射。这样的系统就像网络上的蜂窝自动机一样，除了以下事实：建立它们时，每个节点都有一个规则，该规则是从所有2个具有K个输入的可能节点中随机选择的。在K = 2的情况下，第2类行为倾向于占主导地位。但是对于K> 2，人们看到的行为迅速接近了随机映射的典型行为，在随机映射中，代表N个基础节点的2个状态的演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

随机布尔网络（RBN）是从所有可能的特定规模的布尔网络集合中随机选择的网络。然后，我们可以从统计学上研究此类网络的预期属性如何依赖于所有可能网络集合的各种统计特性。例如，可以研究RBN行为如何随着平均连接性的变化而变化。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

第一个布尔网络是由 Stuart a. Kauffman 在1969年提出的，作为基因调控网络的随机模型，但是它们的数学理解直到2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络仅具有2N个可能的状态，因此轨迹迟早会到达先前访问的状态，由于动力学是确定性的，因此轨迹会陷入稳态或称为一个吸引子（尽管在动力学系统的更广泛领域中，如果一个周期的扰动导致它返回，那么它只是一个吸引子）。如果吸引子只有一个状态，则称为点吸引子；如果吸引子由多个状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。仅在轨迹开始时出现的状态（没有轨迹导致它们出现）被称为伊甸园状态，网络的动力学从这些状态流向吸引子。到达吸引子所需的时间称为瞬态时间。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机功能的增强和对看似简单模型的了解的增加，不同的作者对吸引子的平均数量和长度给出了不同的估计，此处简要概述了主要出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力学系统理论中，网络吸引子的结构和长度与网络的动态相位相对应。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定，关键或混乱的行为。该现象由节点平均连接数的临界值（<math>K_{c}</math>）控制，并且可以通过'''海明距离 Hamming Distance '''作为距离度量来表征。在不稳定状态下，两个初始关闭状态之间的距离平均随时间呈指数增长，而在稳定状态下，其呈指数下降。在这种情况下，“初始关闭状态”表示与网络中的节点数（<math>N</math>）相比，海明距离较小。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于'''N-K模型 N-K-Model '''，如果<math>K<K_{c}</math>，则网络是稳定的；如果<math>K=K_{c}</math>，则网络是关键的；如果<math>K>K_{c}</math>，则网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

给定节点<math> n_{i} </math>的状态根据其真值表进行更新，该表的输出是随机填充的。 <math> p_{i} </math>表示将关闭输出分配给给定的一系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果<math> p_{i}=p=const. </math>,对于每个节点，稳定范围和混沌范围之间的过渡取决于<math> p </math>。根据Bernard Derrida和Yves Pomeau的说法
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为<math> K_{c}=1/[2p(1-p)] </math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果<math> K </math>不是常数，并且进度和出度之间没有相关性，则稳定性条件由<math> \langle K^{in}\rangle </math> 确定, 如果<math>\langle K^{in}\rangle <K_{c}</math>，则网络是稳定的；如果<math>\langle K^{in}\rangle =K_{c}</math>，则网络是稳定的 >，如果<math>\langle K^{in}\rangle >K_{c}</math>，则不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于具有无标度拓扑的网络，其稳定性条件是相同的，其中进出度分布是幂律分布：<math> P(K) \propto K^{-\gamma} </math>和<math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>，因为节点的每个出度链接都是到另一个节点的入度链接。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度表示给定节点的布尔函数的输出更改（如果其输入更改）的概率。对于随机布尔网络，
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由矩阵<math> Q </math>的最大特征值<math> \lambda_{Q} </math>决定，其中<math> Q_{ij}=q_{i}A_{ij} </math>和<math> A </math>是网络的邻接矩阵。如果<math>\lambda_{Q}<1</math>是稳定的网络，如果<math>\lambda_{Q}=1</math>是关键的网络，如果<math>\lambda_{Q}>1</math>，则不稳定。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*同质情况只是指网格，而网格只是对著名的[[Ising模型]]的简化。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2020-09-27T12:50:42Z

923397935：

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

有多个节点，每个节点是K = 1个链接。节点可以打开（用红色表示）或关闭（用蓝色表示）。细（黑色）箭头表示布尔函数的输入，布尔函数是每个节点的简单“复制”函数。粗（灰色）箭头显示同步更新的作用。共有6个（橙色）吸引子，其中4个是固定点。

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

'''布尔网络 Boolean Network 由一组离散的布尔变量组成，每个布尔变量都分配有一个布尔函数（每个变量可能不同），该布尔函数从这些变量的子集中获取输入，并确定变量所分配到的状态。这组功能实际上确定了变量集上的拓扑（连接性），这些变量随后成为网络中的节点。通常，系统的动力学被视为离散的时间序列，其中在时间t + 1时整个网络的状态是通过评估在时间t时网络状态的每个变量的功能来确定的。这可以同步或异步完成。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络已在生物学中用于建模监管网络。尽管布尔网络是遗传现实的粗略简化，其中基因不是简单的二进制开关，但在某些情况下，它们可以正确捕获表达和抑制基因的正确模式。
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

看似数学上简单的（同步）模型直到2000年代中期才被完全理解。

== Classical model ==
经典模型 

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力学系统，其中时间和状态是离散的，即时间序列中的变量集和状态集都具有对整数序列的双射。这样的系统就像网络上的蜂窝自动机一样，除了以下事实：建立它们时，每个节点都有一个规则，该规则是从所有2个具有K个输入的可能节点中随机选择的。在K = 2的情况下，第2类行为倾向于占主导地位。但是对于K> 2，人们看到的行为迅速接近了随机映射的典型行为，在随机映射中，代表N个基础节点的2个状态的演化的网络本身基本上是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

随机布尔网络（RBN）是从所有可能的特定规模的布尔网络集合中随机选择的网络。然后，我们可以从统计学上研究此类网络的预期属性如何依赖于所有可能网络集合的各种统计特性。例如，可以研究RBN行为如何随着平均连接性的变化而变化。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

第一个布尔网络是由 Stuart a. Kauffman 在1969年提出的，作为基因调控网络的随机模型，但是他们的数学理解直到2000年才开始。

== Attractors ==
'''吸引子 Attractors ''' 

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于布尔网络仅具有2N个可能的状态，因此轨迹迟早会到达先前访问的状态，由于动力学是确定性的，因此轨迹会陷入稳态或称为一个吸引子（尽管在动力学系统的更广泛领域中，如果一个周期的扰动导致它返回，那么它只是一个吸引子）。如果吸引子只有一个状态，则称为点吸引子；如果吸引子由多个状态组成，则称为周期吸引子。导致吸引子的状态集称为吸引子的盆地。仅在轨迹开始时出现的状态（没有轨迹导致它们出现）被称为伊甸园状态，网络的动力学从这些状态流向吸引子。到达吸引子所需的时间称为瞬态时间。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机功能的增强和对看似简单模型的了解的增加，不同的作者对吸引子的平均数量和长度给出了不同的估计，此处简要概述了主要出版物。
{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==
'''稳定性 Stability ''' 

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在动力学系统理论中，网络吸引子的结构和长度与网络的动态相位相对应。布尔网络的稳定性取决于其节点的连接。布尔网络可以表现出稳定，关键或混乱的行为。该现象由节点平均连接数的临界值（<math>K_{c}</math>）控制，并且可以通过'''海明距离 Hamming Distance '''作为距离度量来表征。在不稳定状态下，两个初始关闭状态之间的距离平均随时间呈指数增长，而在稳定状态下，其呈指数下降。在这种情况下，“初始关闭状态”表示与网络中的节点数（<math>N</math>）相比，海明距离较小。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于'''N-K模型 N-K-Model '''，如果<math>K<K_{c}</math>，则网络是稳定的；如果<math>K=K_{c}</math>，则网络是关键的；如果<math>K>K_{c}</math>，则网络是不稳定的。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

给定节点<math> n_{i} </math>的状态根据其真值表进行更新，该表的输出是随机填充的。 <math> p_{i} </math>表示将关闭输出分配给给定的一系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果<math> p_{i}=p=const. </math>,对于每个节点，稳定范围和混沌范围之间的过渡取决于<math> p </math>。根据Bernard Derrida和Yves Pomeau的说法
, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值为<math> K_{c}=1/[2p(1-p)] </math>。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果<math> K </math>不是常数，并且进度和出度之间没有相关性，则稳定性条件由<math> \langle K^{in}\rangle </math> 确定, 如果<math>\langle K^{in}\rangle <K_{c}</math>，则网络是稳定的；如果<math>\langle K^{in}\rangle =K_{c}</math>，则网络是稳定的 >，如果<math>\langle K^{in}\rangle >K_{c}</math>，则不稳定。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于具有无标度拓扑的网络，其稳定性条件是相同的，其中进出度分布是幂律分布：<math> P(K) \propto K^{-\gamma} </math>和<math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>，因为节点的每个出度链接都是到另一个节点的入度链接。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度表示给定节点的布尔函数的输出更改（如果其输入更改）的概率。对于随机布尔网络，
<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>。在一般情况下，网络的稳定性由矩阵<math> Q </math>的最大特征值<math> \lambda_{Q} </math>决定，其中<math> Q_{ij}=q_{i}A_{ij} </math>和<math> A </math>是网络的邻接矩阵。如果<math>\lambda_{Q}<1</math>是稳定的网络，如果<math>\lambda_{Q}=1</math>是关键的网络，如果<math>\lambda_{Q}>1</math>，则不稳定。

== Variations of the model ==
模型的变化 

== Other topologies ==
其他拓扑性质 

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑。
* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].
*同质情况只是指网格，而网格只是对著名的[[Ising模型]]的简化。
* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

== Other updating schemes ==
其他更新方案 

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络（有时称为CRBN，即经典随机布尔网络）。由于基因通常不会同时改变状态，因此引入了不同的选择。常见的分类如下：
* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>
确定性异步更新的布尔网络'（'DRBN'）不会同步更新，但确定性解决方案仍然存在。当''t ≡ Qi (''mod'' Pi)''其中''t''是节点时，将更新节点''i'' 时间步长。
* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
*最一般的情况是完全随机更新（'''GARBN'''，一般的异步随机布尔网络）。在此，在每个计算步骤中选择一个（或多个）节点进行更新。
* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==
布尔网络的应用 

== Classification ==
分类 

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

'''可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification ''' <ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E..可伸缩的最佳贝叶斯算法管制模型不确定性下的单细胞弹道分类。 ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref>开发了考虑潜在模型不确定性的轨迹的最佳分类，并提出了基于粒子的轨迹分类，该分类对于具有许多功能的大型网络具有高度可扩展性低于最佳解决方案的复杂性。

== See also ==

* [[NK model]] 

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2020-09-27T12:21:54Z

923397935：

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

个节点，每个节点K = 1个链接。节点可以打开（红色）或关闭（蓝色）。细（黑色）箭头表示布尔函数的输入，布尔函数是每个节点的简单“复制”函数。粗（灰色）箭头显示同步更新的作用。共有6个（橙色）吸引子，其中4个是固定点。]]

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

一个布尔网络由一组离散的布尔变量组成，每个布尔变量都有一个布尔函数值(每个变量可能不同) ，这个值从这些变量的一个子集中获取输入，然后输出决定它被分配给的变量的状态。这组函数实际上决定了变量集上的拓扑结构(连通性) ，这些变量随后成为网络中的节点。通常，系统的动力学被看作是一个离散的时间序列，其中整个网络在时间 t + 1时的状态是通过计算每个变量在时间 t 时的状态上的函数来确定的。这可以同步或异步地完成。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络已经在生物学中被用来为调控网络建模。虽然布尔网络是基因现实的粗略简化，其中基因不是简单的二进制开关，但在一些情况下，它们正确地捕获了表达和抑制基因的正确模式。

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

看似简单(同步)的数学模型直到2000年代中期才被完全理解。

== Classical model ==

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力系统，其中时间和状态是离散的，例如。时间序列中的变量集和状态集都对整数序列有一个双射。这样的系统就像网络上的细胞自动机，只不过当它们被设置好的时候，每个节点都有一个规则，这个规则是从所有2个有 k 输入的可能节点中随机选择的。当 k = 2时，2类行为倾向于占主导地位。但是对于 k > 2，这种行为很快就接近于随机映射的典型行为，在这种映射中，代表 n 个底层节点两种状态演化的网络本身是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

一个随机布尔网络(RBN)是从一个特定大小的所有可能的布尔网络(n。然后我们可以从统计学的角度来研究，这些网络的预期特性是如何依赖于所有可能网络集合的各种统计特性的。例如，可以研究随着平均连接性的改变，RBN 行为是如何变化的。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

第一个布尔网络是由 Stuart a. Kauffman 在1969年提出的，作为基因调控网络的随机模型，但是他们的数学理解直到2000年才开始。

=== Attractors ===

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于一个布尔网络只有2个可能的状态，一个轨迹迟早会达到一个先前访问过的状态，因此，由于动力学是确定性的，轨迹将陷入一个稳定的状态或周期，称为吸引子(尽管在更广泛的动力系统领域中，一个周期只是一个吸引子，如果从它的扰动导致它)。如果吸引子只有一个状态，称为点吸引子; 如果吸引子由多个状态组成，称为循环吸引子。引出吸引子的一组状态称为吸引子的盆。这种只发生在轨道开始时的状态(没有轨道指向它们)被称为伊甸园状态，网络的动力学从这些状态流向吸引子。到达吸引子所需的时间称为瞬态时间。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机能力的提高和对这个看似简单的模型的理解，不同的作者对吸引子的平均数和长度给出了不同的估计，这里简要地总结了一些关键的发表。

{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在21动态系统理论，网络吸引子的结构和长度对应于网络的动态阶段。布尔网络的稳定性取决于节点之间的连接。布尔网络可以表现出稳定的、临界的或混沌的行为。这种现象是由平均节点连接数的临界值控制的(< math > k { c } </math >) ，可以用拥有属性汉明距离作为距离度量。在不稳定区域，两个初始闭合态之间的距离随时间呈指数增长，而在稳定区域则呈指数减小。在这里，“初始关闭状态”意味着与网络中的节点数相比，汉明距离是小的(< math > n </math >)。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于 n-k 模型，网络是稳定的，如果 < math > k < k { c } </math > ，临界的，如果 < math > k = k { c } </math > ，网络是不稳定的，如果 < math > k > k { c } </math > 。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

给定节点的状态 < math > n _ { i } </math > 根据其真值表更新，该真值表的输出是随机填充的。< math > p _ { i } </math > 表示将输出分配给给定系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果 < math > p _ { i } = p = const。对于每一个节点，稳定范围和混沌范围之间的过渡取决于。根据伯纳德 · 德里达和伊夫 · 奥博美的研究

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值是 < math > k _ { c } = 1/[2p (1-p)] </math > 。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果 < math > k </math > 不是常数，并且 in-degrees 和 out-degrees 之间没有相关性，则稳定的条件由 < math > langle k ^ { in } rangle </math > 网络是稳定的，如果 < math > langle k ^ { in } rangle < k { c } </math > ，临界如果 < math > langle k ^ { in rangle = k { c } </math > ，而不稳定如果 math < langle k ^ { in rangle > k { c } </math > 。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于无标度拓扑的网络，稳定性的条件是相同的，其中输入和输出度分布是幂律分布: < math > p (k) propto k ^ {-gamma </math > ，和 < math > langle k ^ { in } rangle = langle k ^ { out } rangle </math > ，因为每个节点的输出链路都是一个输入链路到另一个节点。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度表示给定节点的布尔函数输出在其输入发生变化时发生变化的概率。对于随机布尔网络,

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

2p { i }(1-p { i }).在一般情况下，网络的稳定性是由矩阵的最大特征值 > lambda { q } </math > 决定的，其中 < math > q { ij } = q { i } a { ij } </math > ，< math > a </math > 是网络的邻接矩阵。如果 < math > lambda { q } < 1 </math > ，网络是稳定的; 如果 < math > lambda > lambda { q } = 1 </math > ，网络是不稳定的; 如果 < math > lambda { q } > 1 </math > 。

== Variations of the model ==

=== Other topologies ===

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的底层图拓扑。

* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].

* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

=== Other updating schemes ===

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络(有时称为 CRBN，即。经典随机布尔网络)同步更新。由于基因通常不会同时改变它们的状态，因此引入了不同的选择。常见的分类如下:

* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>

* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.

* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==

=== Classification ===

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

== See also ==

* [[NK model]] 

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

布尔网络

2020-09-27T12:20:46Z

923397935：

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 
本词条由信白初步翻译 

{{Cleanup|date=August 2011}}

{{Network Science}}

[[File:Hou710 BooleanNetwork.svg|thumb|State space of a Boolean Network with ''N=4'' [[Vertex (graph theory)|nodes]] and ''K=1'' [[Glossary of graph theory#Basics|links]] per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the [[Boolean function]] which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) [[attractor]]s, 4 of them are [[Fixed point (mathematics)|fixed points]].]]

nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.]]

节点和 k = 每个节点1个链接。节点可以打开(红色)或关闭(蓝色)。细(黑色)箭头表示布尔函数的输入，这是一个简单的“复制”函数，用于每个节点。粗(灰色)箭头显示同步更新的作用。共有6个(橙色)吸引子，其中4个是不动点。]

A '''Boolean network''' consists of a discrete set of [[boolean variable]]s each of which has a [[Boolean function]] (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a [[network (mathematics)|network]]. Usually, the dynamics of the system is taken as a discrete [[time series]] where the state of the entire network at time ''t''+1 is determined by evaluating each variable's function on the state of the network at time ''t''. This may be done [[synchronous]]ly or [[wikt:asynchronous|asynchronous]]ly.<ref>{{cite journal|last1=Naldi|first1=A.|last2=Monteiro|first2=P. T.|last3=Mussel|first3=C.|last4=Kestler|first4=H. A.|last5=Thieffry|first5=D.|last6=Xenarios|first6=I.|last7=Saez-Rodriguez|first7=J.|last8=Helikar|first8=T.|last9=Chaouiya|first9=C.|title=Cooperative development of logical modelling standards and tools with CoLoMoTo|journal=Bioinformatics|date=25 January 2015|volume=31|issue=7|pages=1154–1159|doi=10.1093/bioinformatics/btv013|pmid=25619997|doi-access=free}}</ref>

A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

一个布尔网络由一组离散的布尔变量组成，每个布尔变量都有一个布尔函数值(每个变量可能不同) ，这个值从这些变量的一个子集中获取输入，然后输出决定它被分配给的变量的状态。这组函数实际上决定了变量集上的拓扑结构(连通性) ，这些变量随后成为网络中的节点。通常，系统的动力学被看作是一个离散的时间序列，其中整个网络在时间 t + 1时的状态是通过计算每个变量在时间 t 时的状态上的函数来确定的。这可以同步或异步地完成。

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.<ref>{{cite journal|last1=Albert|first1=Réka|last2=Othmer|first2=Hans G|title=The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster|journal=Journal of Theoretical Biology|date=July 2003|volume=223|issue=1|pages=1–18|doi=10.1016/S0022-5193(03)00035-3|pmid=12782112|pmc=6388622|citeseerx=10.1.1.13.3370}}</ref><ref>{{cite journal|last1=Li|first1=J.|last2=Bench|first2=A. J.|last3=Vassiliou|first3=G. S.|last4=Fourouclas|first4=N.|last5=Ferguson-Smith|first5=A. C.|last6=Green|first6=A. R.|title=Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies |journal=Proceedings of the National Academy of Sciences|date=30 April 2004 |volume=101|issue=19 |pages=7341–7346 |doi=10.1073/pnas.0308195101|pmid=15123827 |pmc=409920|bibcode = 2004PNAS..101.7341L }}</ref>

Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.

布尔网络已经在生物学中被用来为调控网络建模。虽然布尔网络是基因现实的粗略简化，其中基因不是简单的二进制开关，但在一些情况下，它们正确地捕获了表达和抑制基因的正确模式。

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.<ref name=DrosselRbn>{{cite book|last1=Drossel|first1=Barbara|editor1-last=Schuster|editor1-first=Heinz Georg|title=Chapter 3. Random Boolean Networks|date=December 2009|doi=10.1002/9783527626359.ch3|arxiv=0706.3351|series=Reviews of Nonlinear Dynamics and Complexity|publisher=Wiley|pages=69–110|isbn=9783527626359|chapter=Random Boolean Networks}}</ref>

The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

看似简单(同步)的数学模型直到2000年代中期才被完全理解。

== Classical model ==

A Boolean network is a particular kind of [[sequential dynamical system]], where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a [[bijection]] onto an integer series. Such systems are like [[cellular automata]] on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all ''2{{sup|2{{sup|K}}}}'' possible ones with ''K'' inputs. With ''K=2'' class 2 behavior tends to dominate. But for ''K>2'', the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the ''2{{sup|N}}'' states of the ''N'' underlying nodes is itself connected essentially randomly.<ref>{{cite book|last1=Wolfram|first1=Stephen|title=A New Kind of Science|date=2002|publisher=Wolfram Media, Inc.|location=Champaign, Illinois|isbn=978-1579550080|page=[https://archive.org/details/newkindofscience00wolf/page/936 936]|url=https://archive.org/details/newkindofscience00wolf/page/936|accessdate=15 March 2018|url-access=registration}}</ref>

A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series. Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2}} possible ones with K inputs. With K=2 class 2 behavior tends to dominate. But for K>2, the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 states of the N underlying nodes is itself connected essentially randomly.

布尔网络是一种特殊的顺序动力系统，其中时间和状态是离散的，例如。时间序列中的变量集和状态集都对整数序列有一个双射。这样的系统就像网络上的细胞自动机，只不过当它们被设置好的时候，每个节点都有一个规则，这个规则是从所有2个有 k 输入的可能节点中随机选择的。当 k = 2时，2类行为倾向于占主导地位。但是对于 k > 2，这种行为很快就接近于随机映射的典型行为，在这种映射中，代表 n 个底层节点两种状态演化的网络本身是随机连接的。

A '''random Boolean network''' (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, ''N''. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.

一个随机布尔网络(RBN)是从一个特定大小的所有可能的布尔网络(n。然后我们可以从统计学的角度来研究，这些网络的预期特性是如何依赖于所有可能网络集合的各种统计特性的。例如，可以研究随着平均连接性的改变，RBN 行为是如何变化的。

The first Boolean networks were proposed by [[Stuart A. Kauffman]] in 1969, as [[random]] models of [[genetic regulatory network]]s<ref name=KauffmanOriginal>{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K }}</ref> but their mathematical understanding only started in the 2000s.<ref name=AldanaCoppersmithKadanoff>{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Boolean Dynamics with Random Couplings|journal=Perspectives and Problems in Nonlinear Sciences|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9}}</ref><ref>{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}</ref>

The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks but their mathematical understanding only started in the 2000s.

第一个布尔网络是由 Stuart a. Kauffman 在1969年提出的，作为基因调控网络的随机模型，但是他们的数学理解直到2000年才开始。

=== Attractors ===

Since a Boolean network has only 2''N'' possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an [[attractor]] (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a ''point attractor'', and if the attractor consists of more than one state it is called a ''cycle attractor''. The set of states that lead to an attractor is called the ''basin'' of the attractor. States which occur only at the beginning of trajectories (no trajectories lead ''to'' them), are called ''garden-of-Eden'' states<ref name=WuenscheBook>{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16|accessdate=12 January 2016}}</ref> and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called ''transient time''.<ref name=DrosselRbn />

Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

由于一个布尔网络只有2个可能的状态，一个轨迹迟早会达到一个先前访问过的状态，因此，由于动力学是确定性的，轨迹将陷入一个稳定的状态或周期，称为吸引子(尽管在更广泛的动力系统领域中，一个周期只是一个吸引子，如果从它的扰动导致它)。如果吸引子只有一个状态，称为点吸引子; 如果吸引子由多个状态组成，称为循环吸引子。引出吸引子的一组状态称为吸引子的盆。这种只发生在轨道开始时的状态(没有轨道指向它们)被称为伊甸园状态，网络的动力学从这些状态流向吸引子。到达吸引子所需的时间称为瞬态时间。

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.<ref name=GreilReview>{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389}}</ref>

With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.

随着计算机能力的提高和对这个看似简单的模型的理解，不同的作者对吸引子的平均数和长度给出了不同的估计，这里简要地总结了一些关键的发表。

{| class="wikitable sortable"

{| class="wikitable sortable"

{ | class = “ wikitable sortable”

|-

|-

|-

! Author

! Author

!作者

! Year

! Year

!年份

! Mean attractor length

! Mean attractor length

!平均吸引长度

! Mean attractor number

! Mean attractor number

!平均吸引子数

! comment

! comment

!评论

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

| Kauffmann

考夫曼

| 1969

| 1969

| 1969

| <math>\langle A\rangle\sim \sqrt{N}</math>

| <math>\langle A\rangle\sim \sqrt{N}</math>

| < math > langle a rangle sim sqrt { n } </math >

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| <math>\langle\nu\rangle\sim \sqrt{N}</math>

| < math > langle nu rangle sim sqrt { n } </math >

|

|

|

|-

|-

|-

| Bastolla/ Parisi<ref name=BastollaParisi1998>{{cite journal|last1=Bastolla|first1=U.|last2=Parisi|first2=G.|title=The modular structure of Kauffman networks|journal=Physica D: Nonlinear Phenomena|date=May 1998|volume=115|issue=3–4|pages=219–233|doi=10.1016/S0167-2789(97)00242-X|arxiv = cond-mat/9708214 |bibcode = 1998PhyD..115..219B }}</ref>

| Bastolla/ Parisi

| Bastolla/ Parisi

| 1998

| 1998

| 1998

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

| first numerical evidences

| first numerical evidences

第一个数字证据

|-

|-

|-

| Bilke/ Sjunnesson<ref>{{cite journal|last1=Bilke|first1=Sven|last2=Sjunnesson|first2=Fredrik|title=Stability of the Kauffman model|journal=Physical Review E|date=December 2001|volume=65|issue=1|pages=016129|doi=10.1103/PhysRevE.65.016129|pmid=11800758|arxiv = cond-mat/0107035 |bibcode = 2002PhRvE..65a6129B }}</ref>

| Bilke/ Sjunnesson

| Bilke/Sjunnesson

| 2002

| 2002

| 2002

|

|

|

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| linear with system size, <math>\langle\nu\rangle \sim N</math>

| 与系统大小成线性关系

|

|

|

|-

|-

|-

| Socolar/Kauffman<ref>{{cite journal|last1=Socolar|first1=J.|last2=Kauffman|first2=S.|title=Scaling in Ordered and Critical Random Boolean Networks|journal=Physical Review Letters|date=February 2003|volume=90|issue=6|pages=068702|doi=10.1103/PhysRevLett.90.068702|pmid=12633339|bibcode=2003PhRvL..90f8702S|arxiv = cond-mat/0212306 }}</ref>

| Socolar/Kauffman

| Socolar/Kauffman

| 2003

| 2003

| 2003

|

|

|

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

| faster than linear, <math>\langle\nu\rangle > N^x</math> with <math>x > 1</math>

快于线性，< math > langle nu rangle > n ^ x </math > with < math > x > 1 </math >

|

|

|

|-

|-

|-

| Samuelsson/Troein<ref>{{cite journal|last1=Samuelsson|first1=Björn|last2=Troein|first2=Carl|title=Superpolynomial Growth in the Number of Attractors in Kauffman Networks|journal=Physical Review Letters|date=March 2003|volume=90|issue=9|doi=10.1103/PhysRevLett.90.098701|bibcode=2003PhRvL..90i8701S|pmid=12689263|page=098701}}</ref>

| Samuelsson/Troein

| Samuelsson/Troein

| 2003

| 2003

| 2003

|

|

|

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| superpolynomial growth, <math>\langle\nu\rangle > N^x \forall x</math>

| 超多项式生长，< math > langle nu rangle > n ^ x for all x </math >

| mathematical proof

| mathematical proof

数学证明

|-

|-

|-

| Mihaljev/Drossel<ref>{{cite journal|last1=Mihaljev|first1=Tamara|last2=Drossel|first2=Barbara|title=Scaling in a general class of critical random Boolean networks|journal=Physical Review E|date=October 2006|volume=74|issue=4|pages=046101|doi=10.1103/PhysRevE.74.046101|pmid=17155127|arxiv = cond-mat/0606612 |bibcode = 2006PhRvE..74d6101M }}</ref>

| Mihaljev/Drossel

| Mihaljev/Drossel

| 2005

| 2005

| 2005

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle A\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle a rangle > n ^ x for all x </math >

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| faster than a power law, <math>\langle\nu\rangle > N^x \forall x</math>

| 比幂定律快，< math > langle nu rangle > n ^ x for all x </math >

|

|

|

|}

|}

|}

== Stability ==

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The '''stability of Boolean networks''' depends on the connections of their [[Node (graph theory)|node]]s. A Boolean network can exhibit stable, critical or [[chaotic behavior]]. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the [[Hamming distance]] as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (<math>K_{c}</math>), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes (<math>N</math>) in the network.

在21动态系统理论，网络吸引子的结构和长度对应于网络的动态阶段。布尔网络的稳定性取决于节点之间的连接。布尔网络可以表现出稳定的、临界的或混沌的行为。这种现象是由平均节点连接数的临界值控制的(< math > k { c } </math >) ，可以用拥有属性汉明距离作为距离度量。在不稳定区域，两个初始闭合态之间的距离随时间呈指数增长，而在稳定区域则呈指数减小。在这里，“初始关闭状态”意味着与网络中的节点数相比，汉明距离是小的(< math > n </math >)。

For '''N-K-model'''<ref>{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 }}</ref> the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

For N-K-model the network is stable if <math>K<K_{c}</math>, critical if <math>K=K_{c}</math>, and unstable if <math>K>K_{c}</math>.

对于 n-k 模型，网络是稳定的，如果 < math > k < k { c } </math > ，临界的，如果 < math > k = k { c } </math > ，网络是不稳定的，如果 < math > k > k { c } </math > 。

The state of a given node <math> n_{i} </math> is updated according to its [[truth table]], whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

The state of a given node <math> n_{i} </math> is updated according to its truth table, whose outputs are randomly populated. <math> p_{i} </math> denotes the probability of assigning an off output to a given series of input signals.

给定节点的状态 < math > n _ { i } </math > 根据其真值表更新，该真值表的输出是随机填充的。< math > p _ { i } </math > 表示将输出分配给给定系列输入信号的概率。

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to [[Bernard Derrida]] and [[Yves Pomeau]]<ref>{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters (EPL)|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first = B|last = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D}}</ref>

If <math> p_{i}=p=const. </math> for every node, the transition between the stable and chaotic range depends on <math> p </math>. According to Bernard Derrida and Yves Pomeau

如果 < math > p _ { i } = p = const。对于每一个节点，稳定范围和混沌范围之间的过渡取决于。根据伯纳德 · 德里达和伊夫 · 奥博美的研究

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

, the critical value of the average number of connections is <math> K_{c}=1/[2p(1-p)] </math>.

，平均连接数的临界值是 < math > k _ { c } = 1/[2p (1-p)] </math > 。

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math><ref>{{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first = Ricard V.|last = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}</ref><ref>{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first = Bartolo|last = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}</ref><ref>{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first = Jeffrey J.|last = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F}}</ref> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

If <math> K </math> is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by <math> \langle K^{in}\rangle </math> The network is stable if <math>\langle K^{in}\rangle <K_{c}</math>, critical if <math>\langle K^{in}\rangle =K_{c}</math>, and unstable if <math>\langle K^{in}\rangle >K_{c}</math>.

如果 < math > k </math > 不是常数，并且 in-degrees 和 out-degrees 之间没有相关性，则稳定的条件由 < math > langle k ^ { in } rangle </math > 网络是稳定的，如果 < math > langle k ^ { in } rangle < k { c } </math > ，临界如果 < math > langle k ^ { in rangle = k { c } </math > ，而不稳定如果 math < langle k ^ { in rangle > k { c } </math > 。

The conditions of stability are the same in the case of networks with [[Scale-free network|scale-free]] [[network topology|topology]] where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.<ref>{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first = Maximino|last = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A}}</ref>

The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: <math> P(K) \propto K^{-\gamma} </math>, and <math>\langle K^{in} \rangle=\langle K^{out} \rangle </math>, since every out-link from a node is an in-link to another.

对于无标度拓扑的网络，稳定性的条件是相同的，其中输入和输出度分布是幂律分布: < math > p (k) propto k ^ {-gamma </math > ，和 < math > langle k ^ { in } rangle = langle k ^ { out } rangle </math > ，因为每个节点的输出链路都是一个输入链路到另一个节点。

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

灵敏度表示给定节点的布尔函数输出在其输入发生变化时发生变化的概率。对于随机布尔网络,

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest [[Eigenvalues and eigenvectors|eigenvalue]] <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the [[adjacency matrix]] of the network.<ref>{{Cite journal|title = The effect of network topology on the stability of discrete state models of genetic control|journal = Proceedings of the National Academy of Sciences|date = 2009-05-19|issn = 0027-8424|pmc = 2688895|pmid = 19416903|pages = 8209–8214|volume = 106|issue = 20|doi = 10.1073/pnas.0900142106|first = Andrew|last = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P}}</ref> The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

<math> q_{i}=2p_{i}(1-p_{i}) </math>. In the general case, stability of the network is governed by the largest eigenvalue <math> \lambda_{Q} </math> of matrix <math> Q </math>, where <math> Q_{ij}=q_{i}A_{ij} </math>, and <math> A </math> is the adjacency matrix of the network. The network is stable if <math>\lambda_{Q}<1</math>, critical if <math>\lambda_{Q}=1</math>, unstable if <math>\lambda_{Q}>1</math>.

2p { i }(1-p { i }).在一般情况下，网络的稳定性是由矩阵的最大特征值 > lambda { q } </math > 决定的，其中 < math > q { ij } = q { i } a { ij } </math > ，< math > a </math > 是网络的邻接矩阵。如果 < math > lambda { q } < 1 </math > ，网络是稳定的; 如果 < math > lambda > lambda { q } = 1 </math > ，网络是不稳定的; 如果 < math > lambda { q } > 1 </math > 。

== Variations of the model ==

=== Other topologies ===

One theme is to study '''different underlying [[Graph topology|graph topologies]]'''.

One theme is to study different underlying graph topologies.

一个主题是研究不同的底层图拓扑。

* The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].

* [[Scale-free network|Scale-free]] topologies may be chosen for Boolean networks.<ref name=AldanaScaleFree>{{cite journal|last1=Aldana|first1=Maximino|title=Boolean dynamics of networks with scale-free topology|journal=Physica D: Nonlinear Phenomena|date=October 2003|volume=185|issue=1|pages=45–66|doi=10.1016/s0167-2789(03)00174-x|arxiv=cond-mat/0209571|bibcode=2003PhyD..185...45A}}</ref> One can distinguish the case where only in-degree distribution in power-law distributed,<ref name=ScaleFreeInDegree>{{cite journal|last1=Drossel|first1=Barbara|last2=Greil|first2=Florian|title=Critical Boolean networks with scale-free in-degree distribution|journal=Physical Review E|date=4 August 2009|volume=80|issue=2|pages=026102|doi=10.1103/PhysRevE.80.026102|pmid=19792195|arxiv=0901.0387|bibcode=2009PhRvE..80b6102D}}</ref> or only the out-degree-distribution or both.

=== Other updating schemes ===

Classical Boolean networks (sometimes called '''CRBN''', i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,<ref name=HarveyBossomaier1997>{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|title=Time out of joint: Attractors in asynchronous random Boolean networks|journal=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578}}</ref> different alternatives have been introduced. A common classification<ref name=Gershenson2004>{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|title=Classification of Random Boolean Networks|journal=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1|accessdate=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G}}</ref> is the following:

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

经典布尔网络(有时称为 CRBN，即。经典随机布尔网络)同步更新。由于基因通常不会同时改变它们的状态，因此引入了不同的选择。常见的分类如下:

* '''Deterministic asynchronous updated Boolean networks''' ('''DRBN'''s) are not synchronously updated but a deterministic solution still exists. A node ''i'' will be updated when ''t ≡ Qi (''mod'' Pi)'' where ''t'' is the time step.<ref name=GershensonDrbn>{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Contextual Random Boolean Networks|journal=Advances in Artificial Life|date=14 September 2003|volume=2801|pages=615–624|doi=10.1007/978-3-540-39432-7_66|arxiv=nlin/0303021|series=Lecture Notes in Computer Science|trans-title=7th European Conference, ECAL 2003|location=Dortmund, Germany|isbn=978-3-540-39432-7}}</ref>

* The most general case is full stochastic updating ('''GARBN''', general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.

* The '''Partially-Observed Boolean Dynamical System (POBDS)'''<ref>{{Cite journal|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|date=2017-01-01|title=Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems|journal=IEEE Transactions on Signal Processing|volume=65|issue=2|pages=359–371|doi=10.1109/TSP.2016.2614798|issn=1053-587X|arxiv=1702.07269|bibcode=2017ITSP...65..359I}}</ref><ref>{{Cite book|pages=972–976|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/GlobalSIP.2015.7418342|chapter=Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother|year=2015|isbn=978-1-4799-7591-4|title=2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP)}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U. M.|language=en-US|doi=10.1109/ACC.2016.7524920|title=2016 American Control Conference (ACC)|pages=227–232|year=2016|isbn=978-1-4673-8682-1}}</ref><ref>{{Cite book|last=Imani|first=M.|last2=Braga-Neto|first2=U.|date=2016-12-01|title=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space|journal=2016 IEEE 55th Conference on Decision and Control (CDC)|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6}}</ref> signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.

== Application of Boolean Networks ==

=== Classification ===

* The '''Scalable Optimal Bayesian Classification'''<ref name=":bmdl">Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689</ref> developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

== See also ==

* [[NK model]] 

== References ==

{{Reflist|30em}}

* Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484. 

== External links ==

*[http://www.ddlab.com/ DDLab]

*[https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]

*[https://sourceforge.net/projects/rbn/ RBNLab]

*[https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]

*[https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]

*[http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]

*[https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]

*[https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

*[http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]

{{Stochastic processes}}

[[Category:Bioinformatics]]

Category:Bioinformatics

类别: 生物信息学

[[Category:Logic]]

Category:Logic

分类: 逻辑

[[Category:Spin models]]

Category:Spin models

分类: 旋转模型

[[Category:Exactly solvable models]]

Category:Exactly solvable models

类别: 完全可解模型

[[Category:Statistical mechanics]]

Category:Statistical mechanics

类别: 统计力学

<noinclude>

This page was moved from [[wikipedia:en:Boolean network]]. Its edit history can be viewed at [[布尔网络/edithistory]]</noinclude>

[[Category:待整理页面]]

同质性

2020-09-25T12:37:40Z

923397935：/* Causes */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

{{distinguish|Homophile}}

{{Short description|Process by which people befriend similar people}}{{expert needed|sociology|reason=statements about humans are unqualified|date=November 2011}}

{{Network science}}

'''Homophily''' (from [[Ancient Greek]]: ''[[:wiktionary:ὁμοῦ#Ancient Greek|homoû]]'', 'together' + ''philíē'', 'friendship, love') is the tendency of individuals to associate and [[human bonding|bond]] with similar others, as in the [[proverb]] "[[wikt:birds of a feather flock together|birds of a feather flock together]]."<ref>{{cite magazine |magazine=[[Foreign Affairs]] |first=Niall |last=Ferguson |quote=At the same time, birds of a feather flock together. Because of the phenomenon known as “homophily”, or attraction to similarity, social networks tend to form clusters of nodes with similar properties or attitudes. |url=https://www.foreignaffairs.com/articles/2017-08-15/false-prophecy-hyperconnection |title=The False Prophecy of Hyperconnection |date=August 15, 2017 |access-date=October 1, 2017}}</ref> The presence of homophily has been discovered in a vast array of [[social network|network]] studies: over {{nowrap|100 studies}} have observed homophily in some form or another, and they establish that similarity is associated with connection.<ref name=birds>{{cite journal | last1 = McPherson | first1 = M. | last2 = Smith-Lovin | first2 = L. | last3 = Cook | first3 = J. M. | year = 2001 | title = Birds of a Feather: Homophily in Social Networks | journal = Annual Review of Sociology | volume = 27 | issue = | pages = 415–444 | doi=10.1146/annurev.soc.27.1.415}}</ref> The categories on which homophily occurs include [[Age group|age]], [[gender]], [[social class|class]], and organizational role.<ref name=aaron>{{cite news|last=Retica|first=Aaron|title=Homophily|url=https://www.nytimes.com/2006/12/10/magazine/10Section2a.t-4.html|newspaper=New York Times|date=10 December 2006}}</ref>

Homophily (from Ancient Greek: homoû, 'together' + philíē, 'friendship, love') is the tendency of individuals to associate and bond with similar others, as in the proverb "birds of a feather flock together." The presence of homophily has been discovered in a vast array of network studies: over have observed homophily in some form or another, and they establish that similarity is associated with connection. The categories on which homophily occurs include age, gender, class, and organizational role.

'''同质性 Homophily '''（源自古希腊语：homoû，“ together” +philíē，“ friendship，love”）是个体趋向于与相似的其他人交往和联系的一种倾向，就像谚语“羽毛鸟聚在一起”。在各种各样的网络研究中都发现了同质性的存在：过去以某种形式或其他形式观察到同质性，并且他们确定相似性与连接有关。同形发生的类别包括年龄，性别，阶级和组织角色。

The opposite of homophily is [[heterophily]] or [[intermingling]]. Individuals in homophilic relationships share common characteristics (beliefs, [[values]], education, etc.) that make communication and relationship formation easier. Homophily between [[Breeding pair|mated pairs]] in animals has been extensively studied in the field of [[evolutionary biology]], where it is known as [[assortative mating|''assortative mating'']]. Homophily between mated pairs is common within natural animal mating populations.<ref name="Assortative Mating in Animals">{{cite journal |last1=Jiang |first1=Yuexin |last2=Bolnick |first2=Daniel I. |last3=Kirkpatrick |first3=Mark |title=Assortative Mating in Animals |journal=The American Naturalist |date=June 2013 |volume=181 |issue=6 |pages=E125–38 |doi=10.1086/670160 |pmid=23669548 |jstor=10.1086/670160 |ref=Assortative Mating in Animals|hdl=2152/31270 |hdl-access=free }}</ref>

The opposite of homophily is heterophily or intermingling. Individuals in homophilic relationships share common characteristics (beliefs, values, education, etc.) that make communication and relationship formation easier. Homophily between mated pairs in animals has been extensively studied in the field of evolutionary biology, where it is known as assortative mating. Homophily between mated pairs is common within natural animal mating populations.

同质性的相反是'''异构 heterophily '''或'''混合 Intermingling '''。处于亲密关系中的个人具有共同的特征（信念，价值观，教育程度等），这使沟通和关系的形成变得更加容易。动物的交配对之间的同质性已在进化生物学领域进行了广泛的研究，这被称为'''分类交配 Assortative Mating '''。在自然动物的交配种群中，交配对之间的同质性很普遍。

Homophily has a variety of consequences for social and economic outcomes, ranging from facilitating cooperation<ref name=":0" /> to slowing processes of [[Consensus decision-making|consensus-formation]],<ref name=":1">{{Cite journal|last=Golub and Jackson|first=|date=2012z|title=How homophily affects the speed of learning and best-response dynamics|url=http://www.bengolub.net/papers/homophily.pdf|journal=Quarterly Journal of Economics|volume=127(3)|pages=1287–1338|via=}}</ref> among others.

Homophily has a variety of consequences for social and economic outcomes, ranging from facilitating cooperation among others.

同质性对社会和经济成果有多种影响，包括促进彼此之间的合作。

==Types and dimensions==
类型和尺寸 

==Baseline vs. inbreeding==
基线与近交 

To test the relevance of homophily, researchers have distinguished between two types:<ref name=birds />

To test the relevance of homophily, researchers have distinguished between two types:

为了测试同质性的相关性，研究人员区分了两种类型：

* '''Baseline homophily''': simply the amount of homophily that would be expected by chance given an existing uneven distribution of people with varying characteristics; and
*'''基线同质性 Baseline Homophily '''：简单地讲，由于存在具有不同特征的人的不均匀分布，偶然地可以预期的同质性；
* '''Inbreeding homophily''': the amount of homophily over and above this expected value, typically due to personal preferences and choices.

*'''近亲繁殖 Inbreeding Homophily '''：通常超出个人期望和选择，超出期望值的同构数量。
==Status vs. value==
状态与价值 

In their original formulation of homophily, [[Paul Lazarsfeld]] and [[Robert K. Merton]] (1954) distinguished between ''status homophily'' and ''value homophily'', find that individuals with similar [[social status]] characteristics are more likely to associate with each other than by chance:<ref name="lazar">[[Paul Lazarsfeld|Lazarsfeld, Paul F.]], and [[Robert K. Merton|Merton, Robert K.]] 1954. "Friendship as a Social Process: A Substantive and Methodological Analysis." Pp. 18–66 in ''Freedom and Control in Modern Society'', edited by M. Berger, T. Abel, and C. H. Page. New York: [[Van Nostrand Reinhold|Van Nostrand]]. </ref><ref name="birds" />

In their original formulation of homophily, Paul Lazarsfeld and Robert K. Merton (1954) distinguished between status homophily and value homophily, find that individuals with similar social status characteristics are more likely to associate with each other than by chance:

保罗·拉扎斯菲尔德（Paul Lazarsfeld）和罗伯特·K·默顿（Robert K.Merton）（1954）在最初的同构表述中，区分了地位同质和价值同质，发现具有相似社会地位特征的个体更可能彼此交往而不是偶然：

* '''Status homophily''': includes both [[ascribed characteristics]] (e.g. race, ethnicity, sex, and age) and [[Acquired characteristic|acquired characteristics]] (e.g., religion and education).
*'''同质状态 Status Homophily '''：既包括指定特征（例如种族，种族，性别和年龄），也包括获得的特征（例如宗教和教育）。
* '''Value homophily''': involves association with others who think in similar ways, regardless of differences in status characteristics.
*'''价值同质 Value Homophily '''：涉及与以相似方式思考的其他人的联系，而不论身份特征如何。

== Dimensions ==
外型尺寸 

==Race and ethnicity==
种族与民族

[[Social network]]s in the United States today are strongly divided by [[Race (human categorization)|race]] and [[ethnicity]],<ref>{{Cite journal|last=Moody|first=James|date=November 2001|title=Race, School Integration, and Friendship Segregation in America|journal=American Journal of Sociology|language=en|volume=107|issue=3|pages=679–716|doi=10.1086/338954|issn=0002-9602}}</ref> which account for the greatest proportion of ''inbreeding'' homophily (though classification by these criteria can be problematic in [[sociology]] due to fuzzy boundaries and different definitions of race).

Social networks in the United States today are strongly divided by race and ethnicity, which account for the greatest proportion of inbreeding homophily (though classification by these criteria can be problematic in sociology due to fuzzy boundaries and different definitions of race).

今天，在美国，社交网络在种族和族裔之间有很强的区分，种族和族裔在近亲近亲中占最大比例（尽管由于模糊的边界和种族的不同定义，根据这些标准进行分类在社会学上可能会出现问题）。

Smaller groups have lower [[Racial diversity|diversity]] simply due to the number of members. This tends to give racial and ethnic minority groups a higher ''baseline'' homophily. Race and ethnicity also correlates with [[educational attainment]] and occupation, which further increase baseline homophily.<ref name="birds" />

Smaller groups have lower diversity simply due to the number of members. This tends to give racial and ethnic minority groups a higher baseline homophily. Race and ethnicity also correlates with educational attainment and occupation, which further increase baseline homophily.

较小的组仅由于成员数而具有较低的多样性。这倾向于使种族和少数民族群体具有更高的基线同质性。种族和种族也与受教育程度和职业相关，这进一步增加了基线的同质性。

==Sex and gender==
性别 

In regard to [[sex]] and [[gender]], ''baseline'' homophily of networks is relatively low compared to race and ethnicity. Men and women frequently live together, and are both large and equally-sized populations. Most sex homophily is of the inbreeding type.<ref name="birds" /> Especially in schools, students tend to have a high gender homophily.<ref>{{Cite journal|last=Shrum|first=Wesley|last2=Cheek|first2=Neil H.|last3=Hunter|first3=Saundra MacD.|date=October 1988|title=Friendship in School: Gender and Racial Homophily|journal=Sociology of Education|volume=61|issue=4|pages=227|doi=10.2307/2112441|issn=0038-0407|jstor=2112441}}</ref>

In regard to sex and gender, baseline homophily of networks is relatively low compared to race and ethnicity. Men and women frequently live together, and are both large and equally-sized populations. Most sex homophily is of the inbreeding type.

关于性别，与种族和种族相比，网络的同质性相对较低。男女经常生活在一起，人口众多且大小相等。大多数同性是近交类型。

==Age==
年龄

Most age homophily is of the ''baseline'' type. An interesting pattern of inbreeding age homophily for groups of different ages was found by Marsden (1988).<ref>Marsden, P. V. 1988. "Homogeneity in confiding relations." ''[[Social Networks (journal)|Social Networks]]'' 10:57–76.</ref> It indicated a strong relationship between someone's age and the social distance to other people with regard to confiding in someone. For example, the larger age gap someone had, the smaller chances that they were confided by others with lower ages to "discuss important matters."<ref name="birds" />

Most age homophily is of the baseline type. An interesting pattern of inbreeding age homophily for groups of different ages was found by Marsden (1988). It indicated a strong relationship between someone's age and the social distance to other people with regard to confiding in someone. For example, the larger age gap someone had, the smaller chances that they were confided by others with lower ages to "discuss important matters."

大多数同龄人属于基线类型。 Marsden（1988）发现了不同年龄组的近亲同质的有趣模式。它表明某人的年龄与与他人交往方面的社交距离之间有很强的关系。例如，某人的年龄差距越大，他们被较低年龄的其他人“讨论重要问题”的机率就越小。

==Religion==
宗教 

Homophily based on religion is due to both ''baseline'' and ''inbreeding'' homophily.<ref name="birds" />

Homophily based on religion is due to both baseline and inbreeding homophily.

基于宗教的同质性是由于基线和近亲同性造成的。

==Education, occupation and social class==
教育，职业和社会阶层教育，职业和社会阶层 

Family of birth accounts for considerable baseline homophily with respect to education, occupation, and [[social class]].<ref name="birds" />

Family of birth accounts for considerable baseline homophily with respect to education, occupation, and social class.

出生家庭在教育，职业和社会阶层方面占相当大的基线同质性。

== Social media ==
社交媒体 

As [[social network]]s are largely divided by race, [[Social networking service|social-networking websites]] like [[Facebook]] also foster homophilic atmospheres.<ref>{{Cite journal|last=Aiello|first=Luca Maria|last2=Barrat|first2=Alain|last3=Schifanella|first3=Rossano|last4=Cattuto|first4=Ciro|last5=Markines|first5=Benjamin|last6=Menczer|first6=Filippo|date=2012-05-01|title=Friendship prediction and homophily in social media|journal=ACM Transactions on the Web|volume=6|issue=2|pages=1–33|doi=10.1145/2180861.2180866|issn=1559-1131}}</ref> When a Facebook user '[[Facebook like button|likes]]' or interacts with an article or post of a certain [[ideology]], Facebook continues to show that user posts of that similar ideology (which Facebook believes they will be drawn to). In a research article, McPherson, Smith-Lovin, and Cook (2003) write that [[Homogeneity and heterogeneity|homogeneous]] [[Personal network|personal networks]] result in limited "social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience."<ref>{{Cite journal |last=McPherson |first=Miller |last2=Smith-Lovin |first2=Lynn |last3=Cook |first3=James M. |date=2003-11-28 |title=Birds of a Feather: Homophily in Social Networks |journal=Annual Review of Sociology |language=en |volume=27 |issue=1 |pages=415–444 |doi=10.1146/annurev.soc.27.1.415}}</ref> This homophily can foster divides and [[Echo chamber (media)|echo chambers]] on social networking sites, where people of similar ideologies only interact with each other.

As social networks are largely divided by race, social-networking websites like Facebook also foster homophilic atmospheres. When a Facebook user 'likes' or interacts with an article or post of a certain ideology, Facebook continues to show that user posts of that similar ideology (which Facebook believes they will be drawn to). In a research article, McPherson, Smith-Lovin, and Cook (2003) write that homogeneous personal networks result in limited "social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience." This homophily can foster divides and echo chambers on social networking sites, where people of similar ideologies only interact with each other.

由于社交网络在很大程度上因种族而异，因此诸如Facebook之类的社交网站也营造了同情的氛围。当Facebook用户“喜欢”某种特定意识形态的文章或帖子或与之互动时，Facebook继续显示该相似意识形态的用户帖子（Facebook相信会吸引他们）。 McPherson，Smith-Lovin和Cook（2003）在一篇研究文章中写道，同质的个人网络会导致有限的“社会世界”，从而对其所接收的信息，所形成的态度以及所经历的互动产生强大的影响。。” 这种同质性可以促进社交网站上的分歧和回声室，在这里社交网络上意识形态相似的人只能相互交流。

==Causes and effects==
因果 

== Causes ==
原因 

{{Unreferenced section|date=July 2020}}

'''Geography''': Baseline homophily often arises when the people who are located nearby also have similar characteristics. People are more likely to have contact with those who are geographically closer than those who are distant. Technology such as the telephone, e-mail, and social networks have reduced but not eliminated this effect.

Geography: Baseline homophily often arises when the people who are located nearby also have similar characteristics. People are more likely to have contact with those who are geographically closer than those who are distant. Technology such as the telephone, e-mail, and social networks have reduced but not eliminated this effect.

'''地理环境 Geography '''：当附近的人也具有相似的特征时，通常会产生基线同质性。人们更可能与地理位置较远的人接触。电话，电子邮件和社交网络等技术已经减少但并未消除这种影响。

'''Family ties''': [[Family relationships]] often produce relatively close, frequent contact among those who are at great geographic distance. These ties tend to decay slowly, but can be dramatically restructured when new marriages occur.

Family ties: Family relationships often produce relatively close, frequent contact among those who are at great geographic distance. These ties tend to decay slowly, but can be dramatically restructured when new marriages occur.

'''家庭关系 Family Ties '''：家庭关系通常会在相距遥远的人之间产生相对紧密的频繁联系。这些纽带趋向于缓慢衰减，但是当新婚发生时，它们可以显着地重组。

'''Organizations''': School, work, and volunteer activities provide the great majority of non-family ties. Many friendships, confiding relations, and social support ties are formed within voluntary groups. The social homogeneity of most organizations creates a strong baseline homophily in networks that are formed there.

Organizations: School, work, and volunteer activities provide the great majority of non-family ties. Many friendships, confiding relations, and social support ties are formed within voluntary groups. The social homogeneity of most organizations creates a strong baseline homophily in networks that are formed there.

'''组织 Organizations'''：学校，工作和志愿者活动提供了绝大多数的非家庭纽带。志愿团体内部建立了许多友谊，信任关系和社会支持纽带。大多数组织的社会同质性在形成于其中的网络中建立了强大的基线同质性。

'''Isomorphic sources''': The connections between people who occupy equivalent roles will induce homophily in the system of network ties. This is common in three domains: [[workplace]] (e.g., all heads of [[HR department|HR departments]] will tend to associate with other HR heads), family (e.g., mothers tend to associate with other mothers), and informal networks.

Isomorphic sources: The connections between people who occupy equivalent roles will induce homophily in the system of network ties. This is common in three domains: workplace (e.g., all heads of HR departments will tend to associate with other HR heads), family (e.g., mothers tend to associate with other mothers), and informal networks.

'''同构信息源 Isomorphic Sources '''：扮演同等角色的人们之间的联系将导致网络联系系统中的同质。这在三个领域很常见：工作场所（例如，人力资源部门的所有负责人都倾向于与其他人力资源负责人联系），家庭（例如，母亲倾向于与其他母亲联系）以及非正式网络。

'''Cognitive processes''': People who have demographic similarity tend to own shared knowledge, and therefore they have a greater ease of communication and share cultural tastes, which can also generate homophily.

Cognitive processes: People who have demographic similarity tend to own shared knowledge, and therefore they have a greater ease of communication and share cultural tastes, which can also generate homophily.

'''认知过程 Cognitive Processes'''：具有人口统计学相似性的人倾向于拥有共同的知识，因此他们更容易交流和分享文化品味，也可能产生同质性。

==Effects==
结果 

According to one study, [[perception]] of [[Interpersonal attraction|interpersonal]] similarity improves coordination and increase the expected payoff of interactions, above and beyond the effect of merely "liking others."<ref>{{cite journal|last1=Chierchia|first1=Gabriele|last2=Coricelli|first2=Giorgio|title=The impact of perceived similarity on tacit coordination: propensity for matching and aversion to decoupling choices|journal=Frontiers in Behavioral Neuroscience|doi=10.3389/fnbeh.2015.00202|pmid=26283940|pmc=4516978|volume=9|pages=202|year=2015}}</ref> Another study claims that homophily produces [[Social tolerance|tolerance]] and cooperation in [[Social space|social spaces]].<ref name=":0">{{cite journal|last1=Mark|first1=N. P.|title=Culture and competition: Homophily and distancing explanations for cultural niches|journal=American Sociological Review|date=2003|volume=68|issue=3|pages=319–345|jstor=1519727|doi=10.2307/1519727}}</ref> However, homophilic patterns can also restrict access to information or inclusion for minorities.<ref>{{Cite journal|last=Karimi|first=Fariba|last2=Génois|first2=Mathieu|last3=Wagner|first3=Claudia|last4=Singer|first4=Philipp|last5=Strohmaier|first5=Markus|date=2018-07-23|title=Homophily influences ranking of minorities in social networks|journal=Scientific Reports|volume=8|issue=1|pages=11077|doi=10.1038/s41598-018-29405-7|pmid=30038426|bibcode=2018NatSR...811077K|issn=2045-2322|doi-access=free}}</ref>

According to one study, perception of interpersonal similarity improves coordination and increase the expected payoff of interactions, above and beyond the effect of merely "liking others." Another study claims that homophily produces tolerance and cooperation in social spaces. However, homophilic patterns can also restrict access to information or inclusion for minorities.

根据一项研究，对人际相似性的感知不仅可以“喜欢他人”，还可以提高协调能力并增加预期的互动收益。另一项研究声称同质在社会空间中产生宽容与合作。但是，同质模式也可能限制少数群体获得信息或将其纳入。

The effects of homophily on the diffusion of information and behaviors are also complex. Some studies have claimed that homophily facilitates [[Access to information|access information]],<ref>{{cite arXiv|last1=Choudhury|first1=M. D|title="Birds of a feather": Does user homophily impact information diffusion in social media |date=2010 |eprint=1006.1702|class=cs.CY}}</ref> the diffusion of innovations and behaviors,<ref>{{cite journal|last1=Christakis|first1=N. A|last2=Fowler|first2=J. H.|title=The spread of obesity in a large social network over 32 years|journal=The New England Journal of Medicine|date=2007|volume=357|issue=4|pages=370–379|url=http://media.timesfreepress.com/docs/2010/01/Obesity_studies_0104.pdf|doi=10.1056/nejmsa066082|pmid=17652652|citeseerx=10.1.1.581.4893}}</ref> and the formation of [[Social norm|social norms]].<ref>{{cite journal|last1=Centola|first1=D|last2=R|first2=Willer|last3=M|first3=Macy|title=The emperor's dilemma: A computational model of self-enforcing norms. |journal=American Journal of Sociology|date=2005|volume=110|issue=4|pages=1009–1040|jstor=10.1086/427321|doi=10.1086/427321|url=https://repository.upenn.edu/cgi/viewcontent.cgi?article=1604&context=asc_papers}}</ref> Other studies, however, highlight mechanisms through which homophily can maintain disagreement, exacerbate [[Social polarization|polarization]] of opinions, and slow the formation of an overall [[Consensus decision-making|consensus]].<ref name=":1" /><ref>{{Cite journal|last=Centola D, Gonzalez-Avella JC, Eguiluz VM, San Miguel M.|first=|date=2007|title=Homophily, cultural drift, and the co-evolution of cultural groups|url=|journal=Journal of Conflict Resolution|volume=51(6)|pages=905–29|via=}}</ref>

The effects of homophily on the diffusion of information and behaviors are also complex. Some studies have claimed that homophily facilitates access information, the diffusion of innovations and behaviors, and the formation of social norms. Other studies, however, highlight mechanisms through which homophily can maintain disagreement, exacerbate polarization of opinions, and slow the formation of an overall consensus.

同质性对信息和行为传播的影响也很复杂。一些研究声称，同质促进了信息的获取，创新和行为的传播以及社会规范的形成。但是，其他研究强调了同质可以维持分歧，加剧意见分歧和减缓整体共识形成的机制。

Homophily is a cause of [[Homogamy (sociology)|homogamy]]—marriage between people with similar characteristics.<ref name="fiore">Fiore, A. T. and Donath, J. S. (2005). [http://smg.media.mit.edu/papers/Fiore/fiore_donath_chi2005_short.pdf "Homophily in Online Dating: When Do You Like Someone Like Yourself?"]. ''MIT Media Lab''.</ref> Homophily is a [[fertility factor (demography)|fertility factor]]; an increased [[fertility]] is seen in people with a tendency to seek acquaintance among those with common characteristics.<ref name="fent2013" /> Governmental family policies have a decreased influence on [[Total fertility rate|fertility rates]] in such populations.<ref name="fent2013">{{cite journal|title=Family policies in the context of low fertility and social structure|url=http://www.demographic-research.org/Volumes/Vol29/37/|journal=Demographic Research|volume=29|issue=37|author1=Thomas Fent |author2=Belinda Aparicio Diaz |author3=Alexia Prskawetz |year=2013}}</ref>

Homophily is a cause of homogamy—marriage between people with similar characteristics. Homophily is a fertility factor; an increased fertility is seen in people with a tendency to seek acquaintance among those with common characteristics.

同性恋是同性婚姻的原因，即具有相似特征的人之间的婚姻。同性恋是一个生育因素。在具有共同特征的人中倾向于结识的人会发现生育力增加。

==See also==

*[[Groupthink]]

==References==

{{reflist|30em}}

{{World view}}

[[Category:Interpersonal relationships]]

Category:Interpersonal relationships

类别: 人际关系

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Homophily]]. Its edit history can be viewed at [[同质性/edithistory]]</noinclude>

[[Category:待整理页面]]

同质性

2020-09-25T12:33:52Z

923397935：/* Causes and effects */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

{{distinguish|Homophile}}

{{Short description|Process by which people befriend similar people}}{{expert needed|sociology|reason=statements about humans are unqualified|date=November 2011}}

{{Network science}}

'''Homophily''' (from [[Ancient Greek]]: ''[[:wiktionary:ὁμοῦ#Ancient Greek|homoû]]'', 'together' + ''philíē'', 'friendship, love') is the tendency of individuals to associate and [[human bonding|bond]] with similar others, as in the [[proverb]] "[[wikt:birds of a feather flock together|birds of a feather flock together]]."<ref>{{cite magazine |magazine=[[Foreign Affairs]] |first=Niall |last=Ferguson |quote=At the same time, birds of a feather flock together. Because of the phenomenon known as “homophily”, or attraction to similarity, social networks tend to form clusters of nodes with similar properties or attitudes. |url=https://www.foreignaffairs.com/articles/2017-08-15/false-prophecy-hyperconnection |title=The False Prophecy of Hyperconnection |date=August 15, 2017 |access-date=October 1, 2017}}</ref> The presence of homophily has been discovered in a vast array of [[social network|network]] studies: over {{nowrap|100 studies}} have observed homophily in some form or another, and they establish that similarity is associated with connection.<ref name=birds>{{cite journal | last1 = McPherson | first1 = M. | last2 = Smith-Lovin | first2 = L. | last3 = Cook | first3 = J. M. | year = 2001 | title = Birds of a Feather: Homophily in Social Networks | journal = Annual Review of Sociology | volume = 27 | issue = | pages = 415–444 | doi=10.1146/annurev.soc.27.1.415}}</ref> The categories on which homophily occurs include [[Age group|age]], [[gender]], [[social class|class]], and organizational role.<ref name=aaron>{{cite news|last=Retica|first=Aaron|title=Homophily|url=https://www.nytimes.com/2006/12/10/magazine/10Section2a.t-4.html|newspaper=New York Times|date=10 December 2006}}</ref>

Homophily (from Ancient Greek: homoû, 'together' + philíē, 'friendship, love') is the tendency of individuals to associate and bond with similar others, as in the proverb "birds of a feather flock together." The presence of homophily has been discovered in a vast array of network studies: over have observed homophily in some form or another, and they establish that similarity is associated with connection. The categories on which homophily occurs include age, gender, class, and organizational role.

'''同质性 Homophily '''（源自古希腊语：homoû，“ together” +philíē，“ friendship，love”）是个体趋向于与相似的其他人交往和联系的一种倾向，就像谚语“羽毛鸟聚在一起”。在各种各样的网络研究中都发现了同质性的存在：过去以某种形式或其他形式观察到同质性，并且他们确定相似性与连接有关。同形发生的类别包括年龄，性别，阶级和组织角色。

The opposite of homophily is [[heterophily]] or [[intermingling]]. Individuals in homophilic relationships share common characteristics (beliefs, [[values]], education, etc.) that make communication and relationship formation easier. Homophily between [[Breeding pair|mated pairs]] in animals has been extensively studied in the field of [[evolutionary biology]], where it is known as [[assortative mating|''assortative mating'']]. Homophily between mated pairs is common within natural animal mating populations.<ref name="Assortative Mating in Animals">{{cite journal |last1=Jiang |first1=Yuexin |last2=Bolnick |first2=Daniel I. |last3=Kirkpatrick |first3=Mark |title=Assortative Mating in Animals |journal=The American Naturalist |date=June 2013 |volume=181 |issue=6 |pages=E125–38 |doi=10.1086/670160 |pmid=23669548 |jstor=10.1086/670160 |ref=Assortative Mating in Animals|hdl=2152/31270 |hdl-access=free }}</ref>

The opposite of homophily is heterophily or intermingling. Individuals in homophilic relationships share common characteristics (beliefs, values, education, etc.) that make communication and relationship formation easier. Homophily between mated pairs in animals has been extensively studied in the field of evolutionary biology, where it is known as assortative mating. Homophily between mated pairs is common within natural animal mating populations.

同质性的相反是'''异构 heterophily '''或'''混合 Intermingling '''。处于亲密关系中的个人具有共同的特征（信念，价值观，教育程度等），这使沟通和关系的形成变得更加容易。动物的交配对之间的同质性已在进化生物学领域进行了广泛的研究，这被称为'''分类交配 Assortative Mating '''。在自然动物的交配种群中，交配对之间的同质性很普遍。

Homophily has a variety of consequences for social and economic outcomes, ranging from facilitating cooperation<ref name=":0" /> to slowing processes of [[Consensus decision-making|consensus-formation]],<ref name=":1">{{Cite journal|last=Golub and Jackson|first=|date=2012z|title=How homophily affects the speed of learning and best-response dynamics|url=http://www.bengolub.net/papers/homophily.pdf|journal=Quarterly Journal of Economics|volume=127(3)|pages=1287–1338|via=}}</ref> among others.

Homophily has a variety of consequences for social and economic outcomes, ranging from facilitating cooperation among others.

同质性对社会和经济成果有多种影响，包括促进彼此之间的合作。

==Types and dimensions==
类型和尺寸 

==Baseline vs. inbreeding==
基线与近交 

To test the relevance of homophily, researchers have distinguished between two types:<ref name=birds />

To test the relevance of homophily, researchers have distinguished between two types:

为了测试同质性的相关性，研究人员区分了两种类型：

* '''Baseline homophily''': simply the amount of homophily that would be expected by chance given an existing uneven distribution of people with varying characteristics; and
*'''基线同质性 Baseline Homophily '''：简单地讲，由于存在具有不同特征的人的不均匀分布，偶然地可以预期的同质性；
* '''Inbreeding homophily''': the amount of homophily over and above this expected value, typically due to personal preferences and choices.

*'''近亲繁殖 Inbreeding Homophily '''：通常超出个人期望和选择，超出期望值的同构数量。
==Status vs. value==
状态与价值 

In their original formulation of homophily, [[Paul Lazarsfeld]] and [[Robert K. Merton]] (1954) distinguished between ''status homophily'' and ''value homophily'', find that individuals with similar [[social status]] characteristics are more likely to associate with each other than by chance:<ref name="lazar">[[Paul Lazarsfeld|Lazarsfeld, Paul F.]], and [[Robert K. Merton|Merton, Robert K.]] 1954. "Friendship as a Social Process: A Substantive and Methodological Analysis." Pp. 18–66 in ''Freedom and Control in Modern Society'', edited by M. Berger, T. Abel, and C. H. Page. New York: [[Van Nostrand Reinhold|Van Nostrand]]. </ref><ref name="birds" />

In their original formulation of homophily, Paul Lazarsfeld and Robert K. Merton (1954) distinguished between status homophily and value homophily, find that individuals with similar social status characteristics are more likely to associate with each other than by chance:

保罗·拉扎斯菲尔德（Paul Lazarsfeld）和罗伯特·K·默顿（Robert K.Merton）（1954）在最初的同构表述中，区分了地位同质和价值同质，发现具有相似社会地位特征的个体更可能彼此交往而不是偶然：

* '''Status homophily''': includes both [[ascribed characteristics]] (e.g. race, ethnicity, sex, and age) and [[Acquired characteristic|acquired characteristics]] (e.g., religion and education).
*'''同质状态 Status Homophily '''：既包括指定特征（例如种族，种族，性别和年龄），也包括获得的特征（例如宗教和教育）。
* '''Value homophily''': involves association with others who think in similar ways, regardless of differences in status characteristics.
*'''价值同质 Value Homophily '''：涉及与以相似方式思考的其他人的联系，而不论身份特征如何。

== Dimensions ==
外型尺寸 

==Race and ethnicity==
种族与民族

[[Social network]]s in the United States today are strongly divided by [[Race (human categorization)|race]] and [[ethnicity]],<ref>{{Cite journal|last=Moody|first=James|date=November 2001|title=Race, School Integration, and Friendship Segregation in America|journal=American Journal of Sociology|language=en|volume=107|issue=3|pages=679–716|doi=10.1086/338954|issn=0002-9602}}</ref> which account for the greatest proportion of ''inbreeding'' homophily (though classification by these criteria can be problematic in [[sociology]] due to fuzzy boundaries and different definitions of race).

Social networks in the United States today are strongly divided by race and ethnicity, which account for the greatest proportion of inbreeding homophily (though classification by these criteria can be problematic in sociology due to fuzzy boundaries and different definitions of race).

今天，在美国，社交网络在种族和族裔之间有很强的区分，种族和族裔在近亲近亲中占最大比例（尽管由于模糊的边界和种族的不同定义，根据这些标准进行分类在社会学上可能会出现问题）。

Smaller groups have lower [[Racial diversity|diversity]] simply due to the number of members. This tends to give racial and ethnic minority groups a higher ''baseline'' homophily. Race and ethnicity also correlates with [[educational attainment]] and occupation, which further increase baseline homophily.<ref name="birds" />

Smaller groups have lower diversity simply due to the number of members. This tends to give racial and ethnic minority groups a higher baseline homophily. Race and ethnicity also correlates with educational attainment and occupation, which further increase baseline homophily.

较小的组仅由于成员数而具有较低的多样性。这倾向于使种族和少数民族群体具有更高的基线同质性。种族和种族也与受教育程度和职业相关，这进一步增加了基线的同质性。

==Sex and gender==
性别 

In regard to [[sex]] and [[gender]], ''baseline'' homophily of networks is relatively low compared to race and ethnicity. Men and women frequently live together, and are both large and equally-sized populations. Most sex homophily is of the inbreeding type.<ref name="birds" /> Especially in schools, students tend to have a high gender homophily.<ref>{{Cite journal|last=Shrum|first=Wesley|last2=Cheek|first2=Neil H.|last3=Hunter|first3=Saundra MacD.|date=October 1988|title=Friendship in School: Gender and Racial Homophily|journal=Sociology of Education|volume=61|issue=4|pages=227|doi=10.2307/2112441|issn=0038-0407|jstor=2112441}}</ref>

In regard to sex and gender, baseline homophily of networks is relatively low compared to race and ethnicity. Men and women frequently live together, and are both large and equally-sized populations. Most sex homophily is of the inbreeding type.

关于性别，与种族和种族相比，网络的同质性相对较低。男女经常生活在一起，人口众多且大小相等。大多数同性是近交类型。

==Age==
年龄

Most age homophily is of the ''baseline'' type. An interesting pattern of inbreeding age homophily for groups of different ages was found by Marsden (1988).<ref>Marsden, P. V. 1988. "Homogeneity in confiding relations." ''[[Social Networks (journal)|Social Networks]]'' 10:57–76.</ref> It indicated a strong relationship between someone's age and the social distance to other people with regard to confiding in someone. For example, the larger age gap someone had, the smaller chances that they were confided by others with lower ages to "discuss important matters."<ref name="birds" />

Most age homophily is of the baseline type. An interesting pattern of inbreeding age homophily for groups of different ages was found by Marsden (1988). It indicated a strong relationship between someone's age and the social distance to other people with regard to confiding in someone. For example, the larger age gap someone had, the smaller chances that they were confided by others with lower ages to "discuss important matters."

大多数同龄人属于基线类型。 Marsden（1988）发现了不同年龄组的近亲同质的有趣模式。它表明某人的年龄与与他人交往方面的社交距离之间有很强的关系。例如，某人的年龄差距越大，他们被较低年龄的其他人“讨论重要问题”的机率就越小。

==Religion==
宗教 

Homophily based on religion is due to both ''baseline'' and ''inbreeding'' homophily.<ref name="birds" />

Homophily based on religion is due to both baseline and inbreeding homophily.

基于宗教的同质性是由于基线和近亲同性造成的。

==Education, occupation and social class==
教育，职业和社会阶层教育，职业和社会阶层 

Family of birth accounts for considerable baseline homophily with respect to education, occupation, and [[social class]].<ref name="birds" />

Family of birth accounts for considerable baseline homophily with respect to education, occupation, and social class.

出生家庭在教育，职业和社会阶层方面占相当大的基线同质性。

== Social media ==
社交媒体 

As [[social network]]s are largely divided by race, [[Social networking service|social-networking websites]] like [[Facebook]] also foster homophilic atmospheres.<ref>{{Cite journal|last=Aiello|first=Luca Maria|last2=Barrat|first2=Alain|last3=Schifanella|first3=Rossano|last4=Cattuto|first4=Ciro|last5=Markines|first5=Benjamin|last6=Menczer|first6=Filippo|date=2012-05-01|title=Friendship prediction and homophily in social media|journal=ACM Transactions on the Web|volume=6|issue=2|pages=1–33|doi=10.1145/2180861.2180866|issn=1559-1131}}</ref> When a Facebook user '[[Facebook like button|likes]]' or interacts with an article or post of a certain [[ideology]], Facebook continues to show that user posts of that similar ideology (which Facebook believes they will be drawn to). In a research article, McPherson, Smith-Lovin, and Cook (2003) write that [[Homogeneity and heterogeneity|homogeneous]] [[Personal network|personal networks]] result in limited "social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience."<ref>{{Cite journal |last=McPherson |first=Miller |last2=Smith-Lovin |first2=Lynn |last3=Cook |first3=James M. |date=2003-11-28 |title=Birds of a Feather: Homophily in Social Networks |journal=Annual Review of Sociology |language=en |volume=27 |issue=1 |pages=415–444 |doi=10.1146/annurev.soc.27.1.415}}</ref> This homophily can foster divides and [[Echo chamber (media)|echo chambers]] on social networking sites, where people of similar ideologies only interact with each other.

As social networks are largely divided by race, social-networking websites like Facebook also foster homophilic atmospheres. When a Facebook user 'likes' or interacts with an article or post of a certain ideology, Facebook continues to show that user posts of that similar ideology (which Facebook believes they will be drawn to). In a research article, McPherson, Smith-Lovin, and Cook (2003) write that homogeneous personal networks result in limited "social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience." This homophily can foster divides and echo chambers on social networking sites, where people of similar ideologies only interact with each other.

由于社交网络在很大程度上因种族而异，因此诸如Facebook之类的社交网站也营造了同情的氛围。当Facebook用户“喜欢”某种特定意识形态的文章或帖子或与之互动时，Facebook继续显示该相似意识形态的用户帖子（Facebook相信会吸引他们）。 McPherson，Smith-Lovin和Cook（2003）在一篇研究文章中写道，同质的个人网络会导致有限的“社会世界”，从而对其所接收的信息，所形成的态度以及所经历的互动产生强大的影响。。” 这种同质性可以促进社交网站上的分歧和回声室，在这里社交网络上意识形态相似的人只能相互交流。

==Causes and effects==
因果 

== Causes ==
原因 

{{Unreferenced section|date=July 2020}}

'''Geography''': Baseline homophily often arises when the people who are located nearby also have similar characteristics. People are more likely to have contact with those who are geographically closer than those who are distant. Technology such as the telephone, e-mail, and social networks have reduced but not eliminated this effect.

Geography: Baseline homophily often arises when the people who are located nearby also have similar characteristics. People are more likely to have contact with those who are geographically closer than those who are distant. Technology such as the telephone, e-mail, and social networks have reduced but not eliminated this effect.

地理环境：当附近的人也具有相似的特征时，通常会产生基线同质性。人们更可能与地理位置较远的人接触。电话，电子邮件和社交网络等技术已经减少但并未消除这种影响。

'''Family ties''': [[Family relationships]] often produce relatively close, frequent contact among those who are at great geographic distance. These ties tend to decay slowly, but can be dramatically restructured when new marriages occur.

Family ties: Family relationships often produce relatively close, frequent contact among those who are at great geographic distance. These ties tend to decay slowly, but can be dramatically restructured when new marriages occur.

'''家庭关系 Family Ties '''：家庭关系通常会在相距遥远的人之间产生相对紧密的频繁联系。这些纽带趋向于缓慢衰减，但是当新婚发生时，它们可以显着地重组。

'''Organizations''': School, work, and volunteer activities provide the great majority of non-family ties. Many friendships, confiding relations, and social support ties are formed within voluntary groups. The social homogeneity of most organizations creates a strong baseline homophily in networks that are formed there.

Organizations: School, work, and volunteer activities provide the great majority of non-family ties. Many friendships, confiding relations, and social support ties are formed within voluntary groups. The social homogeneity of most organizations creates a strong baseline homophily in networks that are formed there.

组织：学校，工作和志愿者活动提供了绝大多数的非家庭纽带。志愿团体内部建立了许多友谊，信任关系和社会支持纽带。大多数组织的社会同质性在形成于其中的网络中建立了强大的基线同质性。

'''Isomorphic sources''': The connections between people who occupy equivalent roles will induce homophily in the system of network ties. This is common in three domains: [[workplace]] (e.g., all heads of [[HR department|HR departments]] will tend to associate with other HR heads), family (e.g., mothers tend to associate with other mothers), and informal networks.

Isomorphic sources: The connections between people who occupy equivalent roles will induce homophily in the system of network ties. This is common in three domains: workplace (e.g., all heads of HR departments will tend to associate with other HR heads), family (e.g., mothers tend to associate with other mothers), and informal networks.

同构信息源：扮演同等角色的人们之间的联系将导致网络联系系统中的同质。这在三个领域很常见：工作场所（例如，人力资源部门的所有负责人都倾向于与其他人力资源负责人联系），家庭（例如，母亲倾向于与其他母亲联系）以及非正式网络。

'''Cognitive processes''': People who have demographic similarity tend to own shared knowledge, and therefore they have a greater ease of communication and share cultural tastes, which can also generate homophily.

Cognitive processes: People who have demographic similarity tend to own shared knowledge, and therefore they have a greater ease of communication and share cultural tastes, which can also generate homophily.

认知过程：具有人口统计学相似性的人倾向于拥有共同的知识，因此他们更容易交流和分享文化品味，也可能产生同质性。

==Effects==
结果 

According to one study, [[perception]] of [[Interpersonal attraction|interpersonal]] similarity improves coordination and increase the expected payoff of interactions, above and beyond the effect of merely "liking others."<ref>{{cite journal|last1=Chierchia|first1=Gabriele|last2=Coricelli|first2=Giorgio|title=The impact of perceived similarity on tacit coordination: propensity for matching and aversion to decoupling choices|journal=Frontiers in Behavioral Neuroscience|doi=10.3389/fnbeh.2015.00202|pmid=26283940|pmc=4516978|volume=9|pages=202|year=2015}}</ref> Another study claims that homophily produces [[Social tolerance|tolerance]] and cooperation in [[Social space|social spaces]].<ref name=":0">{{cite journal|last1=Mark|first1=N. P.|title=Culture and competition: Homophily and distancing explanations for cultural niches|journal=American Sociological Review|date=2003|volume=68|issue=3|pages=319–345|jstor=1519727|doi=10.2307/1519727}}</ref> However, homophilic patterns can also restrict access to information or inclusion for minorities.<ref>{{Cite journal|last=Karimi|first=Fariba|last2=Génois|first2=Mathieu|last3=Wagner|first3=Claudia|last4=Singer|first4=Philipp|last5=Strohmaier|first5=Markus|date=2018-07-23|title=Homophily influences ranking of minorities in social networks|journal=Scientific Reports|volume=8|issue=1|pages=11077|doi=10.1038/s41598-018-29405-7|pmid=30038426|bibcode=2018NatSR...811077K|issn=2045-2322|doi-access=free}}</ref>

According to one study, perception of interpersonal similarity improves coordination and increase the expected payoff of interactions, above and beyond the effect of merely "liking others." Another study claims that homophily produces tolerance and cooperation in social spaces. However, homophilic patterns can also restrict access to information or inclusion for minorities.

根据一项研究，对人际相似性的感知不仅可以“喜欢他人”，还可以提高协调能力并增加预期的互动收益。另一项研究声称同质在社会空间中产生宽容与合作。但是，同质模式也可能限制少数群体获得信息或将其纳入。

The effects of homophily on the diffusion of information and behaviors are also complex. Some studies have claimed that homophily facilitates [[Access to information|access information]],<ref>{{cite arXiv|last1=Choudhury|first1=M. D|title="Birds of a feather": Does user homophily impact information diffusion in social media |date=2010 |eprint=1006.1702|class=cs.CY}}</ref> the diffusion of innovations and behaviors,<ref>{{cite journal|last1=Christakis|first1=N. A|last2=Fowler|first2=J. H.|title=The spread of obesity in a large social network over 32 years|journal=The New England Journal of Medicine|date=2007|volume=357|issue=4|pages=370–379|url=http://media.timesfreepress.com/docs/2010/01/Obesity_studies_0104.pdf|doi=10.1056/nejmsa066082|pmid=17652652|citeseerx=10.1.1.581.4893}}</ref> and the formation of [[Social norm|social norms]].<ref>{{cite journal|last1=Centola|first1=D|last2=R|first2=Willer|last3=M|first3=Macy|title=The emperor's dilemma: A computational model of self-enforcing norms. |journal=American Journal of Sociology|date=2005|volume=110|issue=4|pages=1009–1040|jstor=10.1086/427321|doi=10.1086/427321|url=https://repository.upenn.edu/cgi/viewcontent.cgi?article=1604&context=asc_papers}}</ref> Other studies, however, highlight mechanisms through which homophily can maintain disagreement, exacerbate [[Social polarization|polarization]] of opinions, and slow the formation of an overall [[Consensus decision-making|consensus]].<ref name=":1" /><ref>{{Cite journal|last=Centola D, Gonzalez-Avella JC, Eguiluz VM, San Miguel M.|first=|date=2007|title=Homophily, cultural drift, and the co-evolution of cultural groups|url=|journal=Journal of Conflict Resolution|volume=51(6)|pages=905–29|via=}}</ref>

The effects of homophily on the diffusion of information and behaviors are also complex. Some studies have claimed that homophily facilitates access information, the diffusion of innovations and behaviors, and the formation of social norms. Other studies, however, highlight mechanisms through which homophily can maintain disagreement, exacerbate polarization of opinions, and slow the formation of an overall consensus.

同质性对信息和行为传播的影响也很复杂。一些研究声称，同质促进了信息的获取，创新和行为的传播以及社会规范的形成。但是，其他研究强调了同质可以维持分歧，加剧意见分歧和减缓整体共识形成的机制。

Homophily is a cause of [[Homogamy (sociology)|homogamy]]—marriage between people with similar characteristics.<ref name="fiore">Fiore, A. T. and Donath, J. S. (2005). [http://smg.media.mit.edu/papers/Fiore/fiore_donath_chi2005_short.pdf "Homophily in Online Dating: When Do You Like Someone Like Yourself?"]. ''MIT Media Lab''.</ref> Homophily is a [[fertility factor (demography)|fertility factor]]; an increased [[fertility]] is seen in people with a tendency to seek acquaintance among those with common characteristics.<ref name="fent2013" /> Governmental family policies have a decreased influence on [[Total fertility rate|fertility rates]] in such populations.<ref name="fent2013">{{cite journal|title=Family policies in the context of low fertility and social structure|url=http://www.demographic-research.org/Volumes/Vol29/37/|journal=Demographic Research|volume=29|issue=37|author1=Thomas Fent |author2=Belinda Aparicio Diaz |author3=Alexia Prskawetz |year=2013}}</ref>

Homophily is a cause of homogamy—marriage between people with similar characteristics. Homophily is a fertility factor; an increased fertility is seen in people with a tendency to seek acquaintance among those with common characteristics.

同性恋是同性婚姻的原因，即具有相似特征的人之间的婚姻。同性恋是一个生育因素。在具有共同特征的人中倾向于结识的人会发现生育力增加。

==See also==

*[[Groupthink]]

==References==

{{reflist|30em}}

{{World view}}

[[Category:Interpersonal relationships]]

Category:Interpersonal relationships

类别: 人际关系

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Homophily]]. Its edit history can be viewed at [[同质性/edithistory]]</noinclude>

[[Category:待整理页面]]

同质性

2020-09-25T12:18:19Z

923397935：/* Education, occupation and social class== */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

{{distinguish|Homophile}}

{{Short description|Process by which people befriend similar people}}{{expert needed|sociology|reason=statements about humans are unqualified|date=November 2011}}

{{Network science}}

'''Homophily''' (from [[Ancient Greek]]: ''[[:wiktionary:ὁμοῦ#Ancient Greek|homoû]]'', 'together' + ''philíē'', 'friendship, love') is the tendency of individuals to associate and [[human bonding|bond]] with similar others, as in the [[proverb]] "[[wikt:birds of a feather flock together|birds of a feather flock together]]."<ref>{{cite magazine |magazine=[[Foreign Affairs]] |first=Niall |last=Ferguson |quote=At the same time, birds of a feather flock together. Because of the phenomenon known as “homophily”, or attraction to similarity, social networks tend to form clusters of nodes with similar properties or attitudes. |url=https://www.foreignaffairs.com/articles/2017-08-15/false-prophecy-hyperconnection |title=The False Prophecy of Hyperconnection |date=August 15, 2017 |access-date=October 1, 2017}}</ref> The presence of homophily has been discovered in a vast array of [[social network|network]] studies: over {{nowrap|100 studies}} have observed homophily in some form or another, and they establish that similarity is associated with connection.<ref name=birds>{{cite journal | last1 = McPherson | first1 = M. | last2 = Smith-Lovin | first2 = L. | last3 = Cook | first3 = J. M. | year = 2001 | title = Birds of a Feather: Homophily in Social Networks | journal = Annual Review of Sociology | volume = 27 | issue = | pages = 415–444 | doi=10.1146/annurev.soc.27.1.415}}</ref> The categories on which homophily occurs include [[Age group|age]], [[gender]], [[social class|class]], and organizational role.<ref name=aaron>{{cite news|last=Retica|first=Aaron|title=Homophily|url=https://www.nytimes.com/2006/12/10/magazine/10Section2a.t-4.html|newspaper=New York Times|date=10 December 2006}}</ref>

Homophily (from Ancient Greek: homoû, 'together' + philíē, 'friendship, love') is the tendency of individuals to associate and bond with similar others, as in the proverb "birds of a feather flock together." The presence of homophily has been discovered in a vast array of network studies: over have observed homophily in some form or another, and they establish that similarity is associated with connection. The categories on which homophily occurs include age, gender, class, and organizational role.

'''同质性 Homophily '''（源自古希腊语：homoû，“ together” +philíē，“ friendship，love”）是个体趋向于与相似的其他人交往和联系的一种倾向，就像谚语“羽毛鸟聚在一起”。在各种各样的网络研究中都发现了同质性的存在：过去以某种形式或其他形式观察到同质性，并且他们确定相似性与连接有关。同形发生的类别包括年龄，性别，阶级和组织角色。

The opposite of homophily is [[heterophily]] or [[intermingling]]. Individuals in homophilic relationships share common characteristics (beliefs, [[values]], education, etc.) that make communication and relationship formation easier. Homophily between [[Breeding pair|mated pairs]] in animals has been extensively studied in the field of [[evolutionary biology]], where it is known as [[assortative mating|''assortative mating'']]. Homophily between mated pairs is common within natural animal mating populations.<ref name="Assortative Mating in Animals">{{cite journal |last1=Jiang |first1=Yuexin |last2=Bolnick |first2=Daniel I. |last3=Kirkpatrick |first3=Mark |title=Assortative Mating in Animals |journal=The American Naturalist |date=June 2013 |volume=181 |issue=6 |pages=E125–38 |doi=10.1086/670160 |pmid=23669548 |jstor=10.1086/670160 |ref=Assortative Mating in Animals|hdl=2152/31270 |hdl-access=free }}</ref>

The opposite of homophily is heterophily or intermingling. Individuals in homophilic relationships share common characteristics (beliefs, values, education, etc.) that make communication and relationship formation easier. Homophily between mated pairs in animals has been extensively studied in the field of evolutionary biology, where it is known as assortative mating. Homophily between mated pairs is common within natural animal mating populations.

同质性的相反是'''异构 heterophily '''或'''混合 Intermingling '''。处于亲密关系中的个人具有共同的特征（信念，价值观，教育程度等），这使沟通和关系的形成变得更加容易。动物的交配对之间的同质性已在进化生物学领域进行了广泛的研究，这被称为'''分类交配 Assortative Mating '''。在自然动物的交配种群中，交配对之间的同质性很普遍。

Homophily has a variety of consequences for social and economic outcomes, ranging from facilitating cooperation<ref name=":0" /> to slowing processes of [[Consensus decision-making|consensus-formation]],<ref name=":1">{{Cite journal|last=Golub and Jackson|first=|date=2012z|title=How homophily affects the speed of learning and best-response dynamics|url=http://www.bengolub.net/papers/homophily.pdf|journal=Quarterly Journal of Economics|volume=127(3)|pages=1287–1338|via=}}</ref> among others.

Homophily has a variety of consequences for social and economic outcomes, ranging from facilitating cooperation among others.

同质性对社会和经济成果有多种影响，包括促进彼此之间的合作。

==Types and dimensions==
类型和尺寸 

==Baseline vs. inbreeding==
基线与近交 

To test the relevance of homophily, researchers have distinguished between two types:<ref name=birds />

To test the relevance of homophily, researchers have distinguished between two types:

为了测试同质性的相关性，研究人员区分了两种类型：

* '''Baseline homophily''': simply the amount of homophily that would be expected by chance given an existing uneven distribution of people with varying characteristics; and
*'''基线同质性 Baseline Homophily '''：简单地讲，由于存在具有不同特征的人的不均匀分布，偶然地可以预期的同质性；
* '''Inbreeding homophily''': the amount of homophily over and above this expected value, typically due to personal preferences and choices.

*'''近亲繁殖 Inbreeding Homophily '''：通常超出个人期望和选择，超出期望值的同构数量。
==Status vs. value==
状态与价值 

In their original formulation of homophily, [[Paul Lazarsfeld]] and [[Robert K. Merton]] (1954) distinguished between ''status homophily'' and ''value homophily'', find that individuals with similar [[social status]] characteristics are more likely to associate with each other than by chance:<ref name="lazar">[[Paul Lazarsfeld|Lazarsfeld, Paul F.]], and [[Robert K. Merton|Merton, Robert K.]] 1954. "Friendship as a Social Process: A Substantive and Methodological Analysis." Pp. 18–66 in ''Freedom and Control in Modern Society'', edited by M. Berger, T. Abel, and C. H. Page. New York: [[Van Nostrand Reinhold|Van Nostrand]]. </ref><ref name="birds" />

In their original formulation of homophily, Paul Lazarsfeld and Robert K. Merton (1954) distinguished between status homophily and value homophily, find that individuals with similar social status characteristics are more likely to associate with each other than by chance:

保罗·拉扎斯菲尔德（Paul Lazarsfeld）和罗伯特·K·默顿（Robert K.Merton）（1954）在最初的同构表述中，区分了地位同质和价值同质，发现具有相似社会地位特征的个体更可能彼此交往而不是偶然：

* '''Status homophily''': includes both [[ascribed characteristics]] (e.g. race, ethnicity, sex, and age) and [[Acquired characteristic|acquired characteristics]] (e.g., religion and education).
*'''同质状态 Status Homophily '''：既包括指定特征（例如种族，种族，性别和年龄），也包括获得的特征（例如宗教和教育）。
* '''Value homophily''': involves association with others who think in similar ways, regardless of differences in status characteristics.
*'''价值同质 Value Homophily '''：涉及与以相似方式思考的其他人的联系，而不论身份特征如何。

== Dimensions ==
外型尺寸 

==Race and ethnicity==
种族与民族

[[Social network]]s in the United States today are strongly divided by [[Race (human categorization)|race]] and [[ethnicity]],<ref>{{Cite journal|last=Moody|first=James|date=November 2001|title=Race, School Integration, and Friendship Segregation in America|journal=American Journal of Sociology|language=en|volume=107|issue=3|pages=679–716|doi=10.1086/338954|issn=0002-9602}}</ref> which account for the greatest proportion of ''inbreeding'' homophily (though classification by these criteria can be problematic in [[sociology]] due to fuzzy boundaries and different definitions of race).

Social networks in the United States today are strongly divided by race and ethnicity, which account for the greatest proportion of inbreeding homophily (though classification by these criteria can be problematic in sociology due to fuzzy boundaries and different definitions of race).

今天，在美国，社交网络在种族和族裔之间有很强的区分，种族和族裔在近亲近亲中占最大比例（尽管由于模糊的边界和种族的不同定义，根据这些标准进行分类在社会学上可能会出现问题）。

Smaller groups have lower [[Racial diversity|diversity]] simply due to the number of members. This tends to give racial and ethnic minority groups a higher ''baseline'' homophily. Race and ethnicity also correlates with [[educational attainment]] and occupation, which further increase baseline homophily.<ref name="birds" />

Smaller groups have lower diversity simply due to the number of members. This tends to give racial and ethnic minority groups a higher baseline homophily. Race and ethnicity also correlates with educational attainment and occupation, which further increase baseline homophily.

较小的组仅由于成员数而具有较低的多样性。这倾向于使种族和少数民族群体具有更高的基线同质性。种族和种族也与受教育程度和职业相关，这进一步增加了基线的同质性。

==Sex and gender==
性别 

In regard to [[sex]] and [[gender]], ''baseline'' homophily of networks is relatively low compared to race and ethnicity. Men and women frequently live together, and are both large and equally-sized populations. Most sex homophily is of the inbreeding type.<ref name="birds" /> Especially in schools, students tend to have a high gender homophily.<ref>{{Cite journal|last=Shrum|first=Wesley|last2=Cheek|first2=Neil H.|last3=Hunter|first3=Saundra MacD.|date=October 1988|title=Friendship in School: Gender and Racial Homophily|journal=Sociology of Education|volume=61|issue=4|pages=227|doi=10.2307/2112441|issn=0038-0407|jstor=2112441}}</ref>

In regard to sex and gender, baseline homophily of networks is relatively low compared to race and ethnicity. Men and women frequently live together, and are both large and equally-sized populations. Most sex homophily is of the inbreeding type.

关于性别，与种族和种族相比，网络的同质性相对较低。男女经常生活在一起，人口众多且大小相等。大多数同性是近交类型。

==Age==
年龄

Most age homophily is of the ''baseline'' type. An interesting pattern of inbreeding age homophily for groups of different ages was found by Marsden (1988).<ref>Marsden, P. V. 1988. "Homogeneity in confiding relations." ''[[Social Networks (journal)|Social Networks]]'' 10:57–76.</ref> It indicated a strong relationship between someone's age and the social distance to other people with regard to confiding in someone. For example, the larger age gap someone had, the smaller chances that they were confided by others with lower ages to "discuss important matters."<ref name="birds" />

Most age homophily is of the baseline type. An interesting pattern of inbreeding age homophily for groups of different ages was found by Marsden (1988). It indicated a strong relationship between someone's age and the social distance to other people with regard to confiding in someone. For example, the larger age gap someone had, the smaller chances that they were confided by others with lower ages to "discuss important matters."

大多数同龄人属于基线类型。 Marsden（1988）发现了不同年龄组的近亲同质的有趣模式。它表明某人的年龄与与他人交往方面的社交距离之间有很强的关系。例如，某人的年龄差距越大，他们被较低年龄的其他人“讨论重要问题”的机率就越小。

==Religion==
宗教 

Homophily based on religion is due to both ''baseline'' and ''inbreeding'' homophily.<ref name="birds" />

Homophily based on religion is due to both baseline and inbreeding homophily.

基于宗教的同质性是由于基线和近亲同性造成的。

==Education, occupation and social class==
教育，职业和社会阶层教育，职业和社会阶层 

Family of birth accounts for considerable baseline homophily with respect to education, occupation, and [[social class]].<ref name="birds" />

Family of birth accounts for considerable baseline homophily with respect to education, occupation, and social class.

出生家庭在教育，职业和社会阶层方面占相当大的基线同质性。

== Social media ==
社交媒体 

As [[social network]]s are largely divided by race, [[Social networking service|social-networking websites]] like [[Facebook]] also foster homophilic atmospheres.<ref>{{Cite journal|last=Aiello|first=Luca Maria|last2=Barrat|first2=Alain|last3=Schifanella|first3=Rossano|last4=Cattuto|first4=Ciro|last5=Markines|first5=Benjamin|last6=Menczer|first6=Filippo|date=2012-05-01|title=Friendship prediction and homophily in social media|journal=ACM Transactions on the Web|volume=6|issue=2|pages=1–33|doi=10.1145/2180861.2180866|issn=1559-1131}}</ref> When a Facebook user '[[Facebook like button|likes]]' or interacts with an article or post of a certain [[ideology]], Facebook continues to show that user posts of that similar ideology (which Facebook believes they will be drawn to). In a research article, McPherson, Smith-Lovin, and Cook (2003) write that [[Homogeneity and heterogeneity|homogeneous]] [[Personal network|personal networks]] result in limited "social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience."<ref>{{Cite journal |last=McPherson |first=Miller |last2=Smith-Lovin |first2=Lynn |last3=Cook |first3=James M. |date=2003-11-28 |title=Birds of a Feather: Homophily in Social Networks |journal=Annual Review of Sociology |language=en |volume=27 |issue=1 |pages=415–444 |doi=10.1146/annurev.soc.27.1.415}}</ref> This homophily can foster divides and [[Echo chamber (media)|echo chambers]] on social networking sites, where people of similar ideologies only interact with each other.

As social networks are largely divided by race, social-networking websites like Facebook also foster homophilic atmospheres. When a Facebook user 'likes' or interacts with an article or post of a certain ideology, Facebook continues to show that user posts of that similar ideology (which Facebook believes they will be drawn to). In a research article, McPherson, Smith-Lovin, and Cook (2003) write that homogeneous personal networks result in limited "social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience." This homophily can foster divides and echo chambers on social networking sites, where people of similar ideologies only interact with each other.

由于社交网络在很大程度上因种族而异，因此诸如Facebook之类的社交网站也营造了同情的氛围。当Facebook用户“喜欢”某种特定意识形态的文章或帖子或与之互动时，Facebook继续显示该相似意识形态的用户帖子（Facebook相信会吸引他们）。 McPherson，Smith-Lovin和Cook（2003）在一篇研究文章中写道，同质的个人网络会导致有限的“社会世界”，从而对其所接收的信息，所形成的态度以及所经历的互动产生强大的影响。。” 这种同质性可以促进社交网站上的分歧和回声室，在这里社交网络上意识形态相似的人只能相互交流。

==Causes and effects==

=== Causes ===

{{Unreferenced section|date=July 2020}}

'''Geography''': Baseline homophily often arises when the people who are located nearby also have similar characteristics. People are more likely to have contact with those who are geographically closer than those who are distant. Technology such as the telephone, e-mail, and social networks have reduced but not eliminated this effect.

Geography: Baseline homophily often arises when the people who are located nearby also have similar characteristics. People are more likely to have contact with those who are geographically closer than those who are distant. Technology such as the telephone, e-mail, and social networks have reduced but not eliminated this effect.

地理位置: 当周围的人也有相似的特征时，基线同质性常常出现。人们更有可能与那些地理上比较近的人接触，而不是那些距离较远的人。电话、电子邮件和社交网络等技术已经减少但并未消除这种影响。

'''Family ties''': [[Family relationships]] often produce relatively close, frequent contact among those who are at great geographic distance. These ties tend to decay slowly, but can be dramatically restructured when new marriages occur.

Family ties: Family relationships often produce relatively close, frequent contact among those who are at great geographic distance. These ties tend to decay slowly, but can be dramatically restructured when new marriages occur.

家庭关系: 家庭关系往往产生相对密切，频繁的接触之间的地理距离很远。这些关系往往会缓慢衰退，但当新的婚姻出现时，它们可以戏剧性地重组。

'''Organizations''': School, work, and volunteer activities provide the great majority of non-family ties. Many friendships, confiding relations, and social support ties are formed within voluntary groups. The social homogeneity of most organizations creates a strong baseline homophily in networks that are formed there.

Organizations: School, work, and volunteer activities provide the great majority of non-family ties. Many friendships, confiding relations, and social support ties are formed within voluntary groups. The social homogeneity of most organizations creates a strong baseline homophily in networks that are formed there.

组织: 学校、工作和志愿者活动提供了绝大多数的非家庭纽带。许多友谊、信任关系和社会支持关系都是在志愿者团体中形成的。大多数组织的社会同质性在形成的网络中创造了一个强大的基线同质性。

'''Isomorphic sources''': The connections between people who occupy equivalent roles will induce homophily in the system of network ties. This is common in three domains: [[workplace]] (e.g., all heads of [[HR department|HR departments]] will tend to associate with other HR heads), family (e.g., mothers tend to associate with other mothers), and informal networks.

Isomorphic sources: The connections between people who occupy equivalent roles will induce homophily in the system of network ties. This is common in three domains: workplace (e.g., all heads of HR departments will tend to associate with other HR heads), family (e.g., mothers tend to associate with other mothers), and informal networks.

同构性来源: 在网络关系系统中，身处同等角色的人之间的联系会导致同构性。这在三个领域很常见: 工作场所(例如，所有人力资源部门的负责人都倾向于与其他人力资源负责人联系)、家庭(例如，母亲倾向于与其他母亲联系)和非正式网络。

'''Cognitive processes''': People who have demographic similarity tend to own shared knowledge, and therefore they have a greater ease of communication and share cultural tastes, which can also generate homophily.

Cognitive processes: People who have demographic similarity tend to own shared knowledge, and therefore they have a greater ease of communication and share cultural tastes, which can also generate homophily.

认知过程: 人口统计学上相似的人倾向于拥有共享的知识，因此他们有更大的交流和共享的文化品味，这也可以产生同质性。

===Effects===

According to one study, [[perception]] of [[Interpersonal attraction|interpersonal]] similarity improves coordination and increase the expected payoff of interactions, above and beyond the effect of merely "liking others."<ref>{{cite journal|last1=Chierchia|first1=Gabriele|last2=Coricelli|first2=Giorgio|title=The impact of perceived similarity on tacit coordination: propensity for matching and aversion to decoupling choices|journal=Frontiers in Behavioral Neuroscience|doi=10.3389/fnbeh.2015.00202|pmid=26283940|pmc=4516978|volume=9|pages=202|year=2015}}</ref> Another study claims that homophily produces [[Social tolerance|tolerance]] and cooperation in [[Social space|social spaces]].<ref name=":0">{{cite journal|last1=Mark|first1=N. P.|title=Culture and competition: Homophily and distancing explanations for cultural niches|journal=American Sociological Review|date=2003|volume=68|issue=3|pages=319–345|jstor=1519727|doi=10.2307/1519727}}</ref> However, homophilic patterns can also restrict access to information or inclusion for minorities.<ref>{{Cite journal|last=Karimi|first=Fariba|last2=Génois|first2=Mathieu|last3=Wagner|first3=Claudia|last4=Singer|first4=Philipp|last5=Strohmaier|first5=Markus|date=2018-07-23|title=Homophily influences ranking of minorities in social networks|journal=Scientific Reports|volume=8|issue=1|pages=11077|doi=10.1038/s41598-018-29405-7|pmid=30038426|bibcode=2018NatSR...811077K|issn=2045-2322|doi-access=free}}</ref>

According to one study, perception of interpersonal similarity improves coordination and increase the expected payoff of interactions, above and beyond the effect of merely "liking others." Another study claims that homophily produces tolerance and cooperation in social spaces. However, homophilic patterns can also restrict access to information or inclusion for minorities.

根据一项研究，人际相似性感知可以改善协调性，提高互动的预期回报，远远超过仅仅“喜欢他人”的效果另一项研究声称，同质性在社会空间产生宽容与合作。然而，同源模式也会限制少数族裔获取信息或包容性。

The effects of homophily on the diffusion of information and behaviors are also complex. Some studies have claimed that homophily facilitates [[Access to information|access information]],<ref>{{cite arXiv|last1=Choudhury|first1=M. D|title="Birds of a feather": Does user homophily impact information diffusion in social media |date=2010 |eprint=1006.1702|class=cs.CY}}</ref> the diffusion of innovations and behaviors,<ref>{{cite journal|last1=Christakis|first1=N. A|last2=Fowler|first2=J. H.|title=The spread of obesity in a large social network over 32 years|journal=The New England Journal of Medicine|date=2007|volume=357|issue=4|pages=370–379|url=http://media.timesfreepress.com/docs/2010/01/Obesity_studies_0104.pdf|doi=10.1056/nejmsa066082|pmid=17652652|citeseerx=10.1.1.581.4893}}</ref> and the formation of [[Social norm|social norms]].<ref>{{cite journal|last1=Centola|first1=D|last2=R|first2=Willer|last3=M|first3=Macy|title=The emperor's dilemma: A computational model of self-enforcing norms. |journal=American Journal of Sociology|date=2005|volume=110|issue=4|pages=1009–1040|jstor=10.1086/427321|doi=10.1086/427321|url=https://repository.upenn.edu/cgi/viewcontent.cgi?article=1604&context=asc_papers}}</ref> Other studies, however, highlight mechanisms through which homophily can maintain disagreement, exacerbate [[Social polarization|polarization]] of opinions, and slow the formation of an overall [[Consensus decision-making|consensus]].<ref name=":1" /><ref>{{Cite journal|last=Centola D, Gonzalez-Avella JC, Eguiluz VM, San Miguel M.|first=|date=2007|title=Homophily, cultural drift, and the co-evolution of cultural groups|url=|journal=Journal of Conflict Resolution|volume=51(6)|pages=905–29|via=}}</ref>

The effects of homophily on the diffusion of information and behaviors are also complex. Some studies have claimed that homophily facilitates access information, the diffusion of innovations and behaviors, and the formation of social norms. Other studies, however, highlight mechanisms through which homophily can maintain disagreement, exacerbate polarization of opinions, and slow the formation of an overall consensus.

同质性对信息和行为扩散的影响也是复杂的。一些研究声称，同质性促进了信息的获取，创新产品渗透理论和行为，以及社会规范的形成。然而，其他的研究强调了同质性可以维持分歧，加剧意见的两极分化，并减缓整体共识形成的机制。

Homophily is a cause of [[Homogamy (sociology)|homogamy]]—marriage between people with similar characteristics.<ref name="fiore">Fiore, A. T. and Donath, J. S. (2005). [http://smg.media.mit.edu/papers/Fiore/fiore_donath_chi2005_short.pdf "Homophily in Online Dating: When Do You Like Someone Like Yourself?"]. ''MIT Media Lab''.</ref> Homophily is a [[fertility factor (demography)|fertility factor]]; an increased [[fertility]] is seen in people with a tendency to seek acquaintance among those with common characteristics.<ref name="fent2013" /> Governmental family policies have a decreased influence on [[Total fertility rate|fertility rates]] in such populations.<ref name="fent2013">{{cite journal|title=Family policies in the context of low fertility and social structure|url=http://www.demographic-research.org/Volumes/Vol29/37/|journal=Demographic Research|volume=29|issue=37|author1=Thomas Fent |author2=Belinda Aparicio Diaz |author3=Alexia Prskawetz |year=2013}}</ref>

Homophily is a cause of homogamy—marriage between people with similar characteristics. Homophily is a fertility factor; an increased fertility is seen in people with a tendency to seek acquaintance among those with common characteristics.

同质性是同质婚姻的一个原因ーー具有相似特征的人之间的婚姻。同质性是一种 F质粒，人们倾向于在具有共同特征的人群中寻求熟人，这种倾向会导致生育力的提高。

==See also==

*[[Groupthink]]

==References==

{{reflist|30em}}

{{World view}}

[[Category:Interpersonal relationships]]

Category:Interpersonal relationships

类别: 人际关系

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Homophily]]. Its edit history can be viewed at [[同质性/edithistory]]</noinclude>

[[Category:待整理页面]]

同质性

2020-09-25T12:17:45Z

923397935：/* Baseline vs. inbreeding */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

{{distinguish|Homophile}}

{{Short description|Process by which people befriend similar people}}{{expert needed|sociology|reason=statements about humans are unqualified|date=November 2011}}

{{Network science}}

'''Homophily''' (from [[Ancient Greek]]: ''[[:wiktionary:ὁμοῦ#Ancient Greek|homoû]]'', 'together' + ''philíē'', 'friendship, love') is the tendency of individuals to associate and [[human bonding|bond]] with similar others, as in the [[proverb]] "[[wikt:birds of a feather flock together|birds of a feather flock together]]."<ref>{{cite magazine |magazine=[[Foreign Affairs]] |first=Niall |last=Ferguson |quote=At the same time, birds of a feather flock together. Because of the phenomenon known as “homophily”, or attraction to similarity, social networks tend to form clusters of nodes with similar properties or attitudes. |url=https://www.foreignaffairs.com/articles/2017-08-15/false-prophecy-hyperconnection |title=The False Prophecy of Hyperconnection |date=August 15, 2017 |access-date=October 1, 2017}}</ref> The presence of homophily has been discovered in a vast array of [[social network|network]] studies: over {{nowrap|100 studies}} have observed homophily in some form or another, and they establish that similarity is associated with connection.<ref name=birds>{{cite journal | last1 = McPherson | first1 = M. | last2 = Smith-Lovin | first2 = L. | last3 = Cook | first3 = J. M. | year = 2001 | title = Birds of a Feather: Homophily in Social Networks | journal = Annual Review of Sociology | volume = 27 | issue = | pages = 415–444 | doi=10.1146/annurev.soc.27.1.415}}</ref> The categories on which homophily occurs include [[Age group|age]], [[gender]], [[social class|class]], and organizational role.<ref name=aaron>{{cite news|last=Retica|first=Aaron|title=Homophily|url=https://www.nytimes.com/2006/12/10/magazine/10Section2a.t-4.html|newspaper=New York Times|date=10 December 2006}}</ref>

Homophily (from Ancient Greek: homoû, 'together' + philíē, 'friendship, love') is the tendency of individuals to associate and bond with similar others, as in the proverb "birds of a feather flock together." The presence of homophily has been discovered in a vast array of network studies: over have observed homophily in some form or another, and they establish that similarity is associated with connection. The categories on which homophily occurs include age, gender, class, and organizational role.

'''同质性 Homophily '''（源自古希腊语：homoû，“ together” +philíē，“ friendship，love”）是个体趋向于与相似的其他人交往和联系的一种倾向，就像谚语“羽毛鸟聚在一起”。在各种各样的网络研究中都发现了同质性的存在：过去以某种形式或其他形式观察到同质性，并且他们确定相似性与连接有关。同形发生的类别包括年龄，性别，阶级和组织角色。

The opposite of homophily is [[heterophily]] or [[intermingling]]. Individuals in homophilic relationships share common characteristics (beliefs, [[values]], education, etc.) that make communication and relationship formation easier. Homophily between [[Breeding pair|mated pairs]] in animals has been extensively studied in the field of [[evolutionary biology]], where it is known as [[assortative mating|''assortative mating'']]. Homophily between mated pairs is common within natural animal mating populations.<ref name="Assortative Mating in Animals">{{cite journal |last1=Jiang |first1=Yuexin |last2=Bolnick |first2=Daniel I. |last3=Kirkpatrick |first3=Mark |title=Assortative Mating in Animals |journal=The American Naturalist |date=June 2013 |volume=181 |issue=6 |pages=E125–38 |doi=10.1086/670160 |pmid=23669548 |jstor=10.1086/670160 |ref=Assortative Mating in Animals|hdl=2152/31270 |hdl-access=free }}</ref>

The opposite of homophily is heterophily or intermingling. Individuals in homophilic relationships share common characteristics (beliefs, values, education, etc.) that make communication and relationship formation easier. Homophily between mated pairs in animals has been extensively studied in the field of evolutionary biology, where it is known as assortative mating. Homophily between mated pairs is common within natural animal mating populations.

同质性的相反是'''异构 heterophily '''或'''混合 Intermingling '''。处于亲密关系中的个人具有共同的特征（信念，价值观，教育程度等），这使沟通和关系的形成变得更加容易。动物的交配对之间的同质性已在进化生物学领域进行了广泛的研究，这被称为'''分类交配 Assortative Mating '''。在自然动物的交配种群中，交配对之间的同质性很普遍。

Homophily has a variety of consequences for social and economic outcomes, ranging from facilitating cooperation<ref name=":0" /> to slowing processes of [[Consensus decision-making|consensus-formation]],<ref name=":1">{{Cite journal|last=Golub and Jackson|first=|date=2012z|title=How homophily affects the speed of learning and best-response dynamics|url=http://www.bengolub.net/papers/homophily.pdf|journal=Quarterly Journal of Economics|volume=127(3)|pages=1287–1338|via=}}</ref> among others.

Homophily has a variety of consequences for social and economic outcomes, ranging from facilitating cooperation among others.

同质性对社会和经济成果有多种影响，包括促进彼此之间的合作。

==Types and dimensions==
类型和尺寸 

==Baseline vs. inbreeding==
基线与近交 

To test the relevance of homophily, researchers have distinguished between two types:<ref name=birds />

To test the relevance of homophily, researchers have distinguished between two types:

为了测试同质性的相关性，研究人员区分了两种类型：

* '''Baseline homophily''': simply the amount of homophily that would be expected by chance given an existing uneven distribution of people with varying characteristics; and
*'''基线同质性 Baseline Homophily '''：简单地讲，由于存在具有不同特征的人的不均匀分布，偶然地可以预期的同质性；
* '''Inbreeding homophily''': the amount of homophily over and above this expected value, typically due to personal preferences and choices.

*'''近亲繁殖 Inbreeding Homophily '''：通常超出个人期望和选择，超出期望值的同构数量。
==Status vs. value==
状态与价值 

In their original formulation of homophily, [[Paul Lazarsfeld]] and [[Robert K. Merton]] (1954) distinguished between ''status homophily'' and ''value homophily'', find that individuals with similar [[social status]] characteristics are more likely to associate with each other than by chance:<ref name="lazar">[[Paul Lazarsfeld|Lazarsfeld, Paul F.]], and [[Robert K. Merton|Merton, Robert K.]] 1954. "Friendship as a Social Process: A Substantive and Methodological Analysis." Pp. 18–66 in ''Freedom and Control in Modern Society'', edited by M. Berger, T. Abel, and C. H. Page. New York: [[Van Nostrand Reinhold|Van Nostrand]]. </ref><ref name="birds" />

In their original formulation of homophily, Paul Lazarsfeld and Robert K. Merton (1954) distinguished between status homophily and value homophily, find that individuals with similar social status characteristics are more likely to associate with each other than by chance:

保罗·拉扎斯菲尔德（Paul Lazarsfeld）和罗伯特·K·默顿（Robert K.Merton）（1954）在最初的同构表述中，区分了地位同质和价值同质，发现具有相似社会地位特征的个体更可能彼此交往而不是偶然：

* '''Status homophily''': includes both [[ascribed characteristics]] (e.g. race, ethnicity, sex, and age) and [[Acquired characteristic|acquired characteristics]] (e.g., religion and education).
*'''同质状态 Status Homophily '''：既包括指定特征（例如种族，种族，性别和年龄），也包括获得的特征（例如宗教和教育）。
* '''Value homophily''': involves association with others who think in similar ways, regardless of differences in status characteristics.
*'''价值同质 Value Homophily '''：涉及与以相似方式思考的其他人的联系，而不论身份特征如何。

== Dimensions ==
外型尺寸 

==Race and ethnicity==
种族与民族

[[Social network]]s in the United States today are strongly divided by [[Race (human categorization)|race]] and [[ethnicity]],<ref>{{Cite journal|last=Moody|first=James|date=November 2001|title=Race, School Integration, and Friendship Segregation in America|journal=American Journal of Sociology|language=en|volume=107|issue=3|pages=679–716|doi=10.1086/338954|issn=0002-9602}}</ref> which account for the greatest proportion of ''inbreeding'' homophily (though classification by these criteria can be problematic in [[sociology]] due to fuzzy boundaries and different definitions of race).

Social networks in the United States today are strongly divided by race and ethnicity, which account for the greatest proportion of inbreeding homophily (though classification by these criteria can be problematic in sociology due to fuzzy boundaries and different definitions of race).

今天，在美国，社交网络在种族和族裔之间有很强的区分，种族和族裔在近亲近亲中占最大比例（尽管由于模糊的边界和种族的不同定义，根据这些标准进行分类在社会学上可能会出现问题）。

Smaller groups have lower [[Racial diversity|diversity]] simply due to the number of members. This tends to give racial and ethnic minority groups a higher ''baseline'' homophily. Race and ethnicity also correlates with [[educational attainment]] and occupation, which further increase baseline homophily.<ref name="birds" />

Smaller groups have lower diversity simply due to the number of members. This tends to give racial and ethnic minority groups a higher baseline homophily. Race and ethnicity also correlates with educational attainment and occupation, which further increase baseline homophily.

较小的组仅由于成员数而具有较低的多样性。这倾向于使种族和少数民族群体具有更高的基线同质性。种族和种族也与受教育程度和职业相关，这进一步增加了基线的同质性。

==Sex and gender==
性别 

In regard to [[sex]] and [[gender]], ''baseline'' homophily of networks is relatively low compared to race and ethnicity. Men and women frequently live together, and are both large and equally-sized populations. Most sex homophily is of the inbreeding type.<ref name="birds" /> Especially in schools, students tend to have a high gender homophily.<ref>{{Cite journal|last=Shrum|first=Wesley|last2=Cheek|first2=Neil H.|last3=Hunter|first3=Saundra MacD.|date=October 1988|title=Friendship in School: Gender and Racial Homophily|journal=Sociology of Education|volume=61|issue=4|pages=227|doi=10.2307/2112441|issn=0038-0407|jstor=2112441}}</ref>

In regard to sex and gender, baseline homophily of networks is relatively low compared to race and ethnicity. Men and women frequently live together, and are both large and equally-sized populations. Most sex homophily is of the inbreeding type.

关于性别，与种族和种族相比，网络的同质性相对较低。男女经常生活在一起，人口众多且大小相等。大多数同性是近交类型。

==Age==
年龄

Most age homophily is of the ''baseline'' type. An interesting pattern of inbreeding age homophily for groups of different ages was found by Marsden (1988).<ref>Marsden, P. V. 1988. "Homogeneity in confiding relations." ''[[Social Networks (journal)|Social Networks]]'' 10:57–76.</ref> It indicated a strong relationship between someone's age and the social distance to other people with regard to confiding in someone. For example, the larger age gap someone had, the smaller chances that they were confided by others with lower ages to "discuss important matters."<ref name="birds" />

Most age homophily is of the baseline type. An interesting pattern of inbreeding age homophily for groups of different ages was found by Marsden (1988). It indicated a strong relationship between someone's age and the social distance to other people with regard to confiding in someone. For example, the larger age gap someone had, the smaller chances that they were confided by others with lower ages to "discuss important matters."

大多数同龄人属于基线类型。 Marsden（1988）发现了不同年龄组的近亲同质的有趣模式。它表明某人的年龄与与他人交往方面的社交距离之间有很强的关系。例如，某人的年龄差距越大，他们被较低年龄的其他人“讨论重要问题”的机率就越小。

==Religion==
宗教 

Homophily based on religion is due to both ''baseline'' and ''inbreeding'' homophily.<ref name="birds" />

Homophily based on religion is due to both baseline and inbreeding homophily.

基于宗教的同质性是由于基线和近亲同性造成的。

==Education, occupation and social class====
教育，职业和社会阶层教育，职业和社会阶层 

Family of birth accounts for considerable baseline homophily with respect to education, occupation, and [[social class]].<ref name="birds" />

Family of birth accounts for considerable baseline homophily with respect to education, occupation, and social class.

出生家庭在教育，职业和社会阶层方面占相当大的基线同质性。

== Social media ==
社交媒体 

As [[social network]]s are largely divided by race, [[Social networking service|social-networking websites]] like [[Facebook]] also foster homophilic atmospheres.<ref>{{Cite journal|last=Aiello|first=Luca Maria|last2=Barrat|first2=Alain|last3=Schifanella|first3=Rossano|last4=Cattuto|first4=Ciro|last5=Markines|first5=Benjamin|last6=Menczer|first6=Filippo|date=2012-05-01|title=Friendship prediction and homophily in social media|journal=ACM Transactions on the Web|volume=6|issue=2|pages=1–33|doi=10.1145/2180861.2180866|issn=1559-1131}}</ref> When a Facebook user '[[Facebook like button|likes]]' or interacts with an article or post of a certain [[ideology]], Facebook continues to show that user posts of that similar ideology (which Facebook believes they will be drawn to). In a research article, McPherson, Smith-Lovin, and Cook (2003) write that [[Homogeneity and heterogeneity|homogeneous]] [[Personal network|personal networks]] result in limited "social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience."<ref>{{Cite journal |last=McPherson |first=Miller |last2=Smith-Lovin |first2=Lynn |last3=Cook |first3=James M. |date=2003-11-28 |title=Birds of a Feather: Homophily in Social Networks |journal=Annual Review of Sociology |language=en |volume=27 |issue=1 |pages=415–444 |doi=10.1146/annurev.soc.27.1.415}}</ref> This homophily can foster divides and [[Echo chamber (media)|echo chambers]] on social networking sites, where people of similar ideologies only interact with each other.

As social networks are largely divided by race, social-networking websites like Facebook also foster homophilic atmospheres. When a Facebook user 'likes' or interacts with an article or post of a certain ideology, Facebook continues to show that user posts of that similar ideology (which Facebook believes they will be drawn to). In a research article, McPherson, Smith-Lovin, and Cook (2003) write that homogeneous personal networks result in limited "social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience." This homophily can foster divides and echo chambers on social networking sites, where people of similar ideologies only interact with each other.

由于社交网络在很大程度上因种族而异，因此诸如Facebook之类的社交网站也营造了同情的氛围。当Facebook用户“喜欢”某种特定意识形态的文章或帖子或与之互动时，Facebook继续显示该相似意识形态的用户帖子（Facebook相信会吸引他们）。 McPherson，Smith-Lovin和Cook（2003）在一篇研究文章中写道，同质的个人网络会导致有限的“社会世界”，从而对其所接收的信息，所形成的态度以及所经历的互动产生强大的影响。。” 这种同质性可以促进社交网站上的分歧和回声室，在这里社交网络上意识形态相似的人只能相互交流。

==Causes and effects==

=== Causes ===

{{Unreferenced section|date=July 2020}}

'''Geography''': Baseline homophily often arises when the people who are located nearby also have similar characteristics. People are more likely to have contact with those who are geographically closer than those who are distant. Technology such as the telephone, e-mail, and social networks have reduced but not eliminated this effect.

Geography: Baseline homophily often arises when the people who are located nearby also have similar characteristics. People are more likely to have contact with those who are geographically closer than those who are distant. Technology such as the telephone, e-mail, and social networks have reduced but not eliminated this effect.

地理位置: 当周围的人也有相似的特征时，基线同质性常常出现。人们更有可能与那些地理上比较近的人接触，而不是那些距离较远的人。电话、电子邮件和社交网络等技术已经减少但并未消除这种影响。

'''Family ties''': [[Family relationships]] often produce relatively close, frequent contact among those who are at great geographic distance. These ties tend to decay slowly, but can be dramatically restructured when new marriages occur.

Family ties: Family relationships often produce relatively close, frequent contact among those who are at great geographic distance. These ties tend to decay slowly, but can be dramatically restructured when new marriages occur.

家庭关系: 家庭关系往往产生相对密切，频繁的接触之间的地理距离很远。这些关系往往会缓慢衰退，但当新的婚姻出现时，它们可以戏剧性地重组。

'''Organizations''': School, work, and volunteer activities provide the great majority of non-family ties. Many friendships, confiding relations, and social support ties are formed within voluntary groups. The social homogeneity of most organizations creates a strong baseline homophily in networks that are formed there.

Organizations: School, work, and volunteer activities provide the great majority of non-family ties. Many friendships, confiding relations, and social support ties are formed within voluntary groups. The social homogeneity of most organizations creates a strong baseline homophily in networks that are formed there.

组织: 学校、工作和志愿者活动提供了绝大多数的非家庭纽带。许多友谊、信任关系和社会支持关系都是在志愿者团体中形成的。大多数组织的社会同质性在形成的网络中创造了一个强大的基线同质性。

'''Isomorphic sources''': The connections between people who occupy equivalent roles will induce homophily in the system of network ties. This is common in three domains: [[workplace]] (e.g., all heads of [[HR department|HR departments]] will tend to associate with other HR heads), family (e.g., mothers tend to associate with other mothers), and informal networks.

Isomorphic sources: The connections between people who occupy equivalent roles will induce homophily in the system of network ties. This is common in three domains: workplace (e.g., all heads of HR departments will tend to associate with other HR heads), family (e.g., mothers tend to associate with other mothers), and informal networks.

同构性来源: 在网络关系系统中，身处同等角色的人之间的联系会导致同构性。这在三个领域很常见: 工作场所(例如，所有人力资源部门的负责人都倾向于与其他人力资源负责人联系)、家庭(例如，母亲倾向于与其他母亲联系)和非正式网络。

'''Cognitive processes''': People who have demographic similarity tend to own shared knowledge, and therefore they have a greater ease of communication and share cultural tastes, which can also generate homophily.

Cognitive processes: People who have demographic similarity tend to own shared knowledge, and therefore they have a greater ease of communication and share cultural tastes, which can also generate homophily.

认知过程: 人口统计学上相似的人倾向于拥有共享的知识，因此他们有更大的交流和共享的文化品味，这也可以产生同质性。

===Effects===

According to one study, [[perception]] of [[Interpersonal attraction|interpersonal]] similarity improves coordination and increase the expected payoff of interactions, above and beyond the effect of merely "liking others."<ref>{{cite journal|last1=Chierchia|first1=Gabriele|last2=Coricelli|first2=Giorgio|title=The impact of perceived similarity on tacit coordination: propensity for matching and aversion to decoupling choices|journal=Frontiers in Behavioral Neuroscience|doi=10.3389/fnbeh.2015.00202|pmid=26283940|pmc=4516978|volume=9|pages=202|year=2015}}</ref> Another study claims that homophily produces [[Social tolerance|tolerance]] and cooperation in [[Social space|social spaces]].<ref name=":0">{{cite journal|last1=Mark|first1=N. P.|title=Culture and competition: Homophily and distancing explanations for cultural niches|journal=American Sociological Review|date=2003|volume=68|issue=3|pages=319–345|jstor=1519727|doi=10.2307/1519727}}</ref> However, homophilic patterns can also restrict access to information or inclusion for minorities.<ref>{{Cite journal|last=Karimi|first=Fariba|last2=Génois|first2=Mathieu|last3=Wagner|first3=Claudia|last4=Singer|first4=Philipp|last5=Strohmaier|first5=Markus|date=2018-07-23|title=Homophily influences ranking of minorities in social networks|journal=Scientific Reports|volume=8|issue=1|pages=11077|doi=10.1038/s41598-018-29405-7|pmid=30038426|bibcode=2018NatSR...811077K|issn=2045-2322|doi-access=free}}</ref>

According to one study, perception of interpersonal similarity improves coordination and increase the expected payoff of interactions, above and beyond the effect of merely "liking others." Another study claims that homophily produces tolerance and cooperation in social spaces. However, homophilic patterns can also restrict access to information or inclusion for minorities.

根据一项研究，人际相似性感知可以改善协调性，提高互动的预期回报，远远超过仅仅“喜欢他人”的效果另一项研究声称，同质性在社会空间产生宽容与合作。然而，同源模式也会限制少数族裔获取信息或包容性。

The effects of homophily on the diffusion of information and behaviors are also complex. Some studies have claimed that homophily facilitates [[Access to information|access information]],<ref>{{cite arXiv|last1=Choudhury|first1=M. D|title="Birds of a feather": Does user homophily impact information diffusion in social media |date=2010 |eprint=1006.1702|class=cs.CY}}</ref> the diffusion of innovations and behaviors,<ref>{{cite journal|last1=Christakis|first1=N. A|last2=Fowler|first2=J. H.|title=The spread of obesity in a large social network over 32 years|journal=The New England Journal of Medicine|date=2007|volume=357|issue=4|pages=370–379|url=http://media.timesfreepress.com/docs/2010/01/Obesity_studies_0104.pdf|doi=10.1056/nejmsa066082|pmid=17652652|citeseerx=10.1.1.581.4893}}</ref> and the formation of [[Social norm|social norms]].<ref>{{cite journal|last1=Centola|first1=D|last2=R|first2=Willer|last3=M|first3=Macy|title=The emperor's dilemma: A computational model of self-enforcing norms. |journal=American Journal of Sociology|date=2005|volume=110|issue=4|pages=1009–1040|jstor=10.1086/427321|doi=10.1086/427321|url=https://repository.upenn.edu/cgi/viewcontent.cgi?article=1604&context=asc_papers}}</ref> Other studies, however, highlight mechanisms through which homophily can maintain disagreement, exacerbate [[Social polarization|polarization]] of opinions, and slow the formation of an overall [[Consensus decision-making|consensus]].<ref name=":1" /><ref>{{Cite journal|last=Centola D, Gonzalez-Avella JC, Eguiluz VM, San Miguel M.|first=|date=2007|title=Homophily, cultural drift, and the co-evolution of cultural groups|url=|journal=Journal of Conflict Resolution|volume=51(6)|pages=905–29|via=}}</ref>

The effects of homophily on the diffusion of information and behaviors are also complex. Some studies have claimed that homophily facilitates access information, the diffusion of innovations and behaviors, and the formation of social norms. Other studies, however, highlight mechanisms through which homophily can maintain disagreement, exacerbate polarization of opinions, and slow the formation of an overall consensus.

同质性对信息和行为扩散的影响也是复杂的。一些研究声称，同质性促进了信息的获取，创新产品渗透理论和行为，以及社会规范的形成。然而，其他的研究强调了同质性可以维持分歧，加剧意见的两极分化，并减缓整体共识形成的机制。

Homophily is a cause of [[Homogamy (sociology)|homogamy]]—marriage between people with similar characteristics.<ref name="fiore">Fiore, A. T. and Donath, J. S. (2005). [http://smg.media.mit.edu/papers/Fiore/fiore_donath_chi2005_short.pdf "Homophily in Online Dating: When Do You Like Someone Like Yourself?"]. ''MIT Media Lab''.</ref> Homophily is a [[fertility factor (demography)|fertility factor]]; an increased [[fertility]] is seen in people with a tendency to seek acquaintance among those with common characteristics.<ref name="fent2013" /> Governmental family policies have a decreased influence on [[Total fertility rate|fertility rates]] in such populations.<ref name="fent2013">{{cite journal|title=Family policies in the context of low fertility and social structure|url=http://www.demographic-research.org/Volumes/Vol29/37/|journal=Demographic Research|volume=29|issue=37|author1=Thomas Fent |author2=Belinda Aparicio Diaz |author3=Alexia Prskawetz |year=2013}}</ref>

Homophily is a cause of homogamy—marriage between people with similar characteristics. Homophily is a fertility factor; an increased fertility is seen in people with a tendency to seek acquaintance among those with common characteristics.

同质性是同质婚姻的一个原因ーー具有相似特征的人之间的婚姻。同质性是一种 F质粒，人们倾向于在具有共同特征的人群中寻求熟人，这种倾向会导致生育力的提高。

==See also==

*[[Groupthink]]

==References==

{{reflist|30em}}

{{World view}}

[[Category:Interpersonal relationships]]

Category:Interpersonal relationships

类别: 人际关系

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Homophily]]. Its edit history can be viewed at [[同质性/edithistory]]</noinclude>

[[Category:待整理页面]]

同质性

2020-09-25T11:51:08Z

923397935：/* Baseline vs. inbreeding */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

{{distinguish|Homophile}}

{{Short description|Process by which people befriend similar people}}{{expert needed|sociology|reason=statements about humans are unqualified|date=November 2011}}

{{Network science}}

'''Homophily''' (from [[Ancient Greek]]: ''[[:wiktionary:ὁμοῦ#Ancient Greek|homoû]]'', 'together' + ''philíē'', 'friendship, love') is the tendency of individuals to associate and [[human bonding|bond]] with similar others, as in the [[proverb]] "[[wikt:birds of a feather flock together|birds of a feather flock together]]."<ref>{{cite magazine |magazine=[[Foreign Affairs]] |first=Niall |last=Ferguson |quote=At the same time, birds of a feather flock together. Because of the phenomenon known as “homophily”, or attraction to similarity, social networks tend to form clusters of nodes with similar properties or attitudes. |url=https://www.foreignaffairs.com/articles/2017-08-15/false-prophecy-hyperconnection |title=The False Prophecy of Hyperconnection |date=August 15, 2017 |access-date=October 1, 2017}}</ref> The presence of homophily has been discovered in a vast array of [[social network|network]] studies: over {{nowrap|100 studies}} have observed homophily in some form or another, and they establish that similarity is associated with connection.<ref name=birds>{{cite journal | last1 = McPherson | first1 = M. | last2 = Smith-Lovin | first2 = L. | last3 = Cook | first3 = J. M. | year = 2001 | title = Birds of a Feather: Homophily in Social Networks | journal = Annual Review of Sociology | volume = 27 | issue = | pages = 415–444 | doi=10.1146/annurev.soc.27.1.415}}</ref> The categories on which homophily occurs include [[Age group|age]], [[gender]], [[social class|class]], and organizational role.<ref name=aaron>{{cite news|last=Retica|first=Aaron|title=Homophily|url=https://www.nytimes.com/2006/12/10/magazine/10Section2a.t-4.html|newspaper=New York Times|date=10 December 2006}}</ref>

Homophily (from Ancient Greek: homoû, 'together' + philíē, 'friendship, love') is the tendency of individuals to associate and bond with similar others, as in the proverb "birds of a feather flock together." The presence of homophily has been discovered in a vast array of network studies: over have observed homophily in some form or another, and they establish that similarity is associated with connection. The categories on which homophily occurs include age, gender, class, and organizational role.

'''同质性 Homophily '''（源自古希腊语：homoû，“ together” +philíē，“ friendship，love”）是个体趋向于与相似的其他人交往和联系的一种倾向，就像谚语“羽毛鸟聚在一起”。在各种各样的网络研究中都发现了同质性的存在：过去以某种形式或其他形式观察到同质性，并且他们确定相似性与连接有关。同形发生的类别包括年龄，性别，阶级和组织角色。

The opposite of homophily is [[heterophily]] or [[intermingling]]. Individuals in homophilic relationships share common characteristics (beliefs, [[values]], education, etc.) that make communication and relationship formation easier. Homophily between [[Breeding pair|mated pairs]] in animals has been extensively studied in the field of [[evolutionary biology]], where it is known as [[assortative mating|''assortative mating'']]. Homophily between mated pairs is common within natural animal mating populations.<ref name="Assortative Mating in Animals">{{cite journal |last1=Jiang |first1=Yuexin |last2=Bolnick |first2=Daniel I. |last3=Kirkpatrick |first3=Mark |title=Assortative Mating in Animals |journal=The American Naturalist |date=June 2013 |volume=181 |issue=6 |pages=E125–38 |doi=10.1086/670160 |pmid=23669548 |jstor=10.1086/670160 |ref=Assortative Mating in Animals|hdl=2152/31270 |hdl-access=free }}</ref>

The opposite of homophily is heterophily or intermingling. Individuals in homophilic relationships share common characteristics (beliefs, values, education, etc.) that make communication and relationship formation easier. Homophily between mated pairs in animals has been extensively studied in the field of evolutionary biology, where it is known as assortative mating. Homophily between mated pairs is common within natural animal mating populations.

同质性的相反是'''异构 heterophily '''或'''混合 Intermingling '''。处于亲密关系中的个人具有共同的特征（信念，价值观，教育程度等），这使沟通和关系的形成变得更加容易。动物的交配对之间的同质性已在进化生物学领域进行了广泛的研究，这被称为'''分类交配 Assortative Mating '''。在自然动物的交配种群中，交配对之间的同质性很普遍。

Homophily has a variety of consequences for social and economic outcomes, ranging from facilitating cooperation<ref name=":0" /> to slowing processes of [[Consensus decision-making|consensus-formation]],<ref name=":1">{{Cite journal|last=Golub and Jackson|first=|date=2012z|title=How homophily affects the speed of learning and best-response dynamics|url=http://www.bengolub.net/papers/homophily.pdf|journal=Quarterly Journal of Economics|volume=127(3)|pages=1287–1338|via=}}</ref> among others.

Homophily has a variety of consequences for social and economic outcomes, ranging from facilitating cooperation among others.

同质性对社会和经济成果有多种影响，包括促进彼此之间的合作。

==Types and dimensions==
类型和尺寸 

==Baseline vs. inbreeding==
基线与近交 

To test the relevance of homophily, researchers have distinguished between two types:<ref name=birds />

To test the relevance of homophily, researchers have distinguished between two types:

为了测试同的相关性，研究人员区分了两种类型：

* '''Baseline homophily''': simply the amount of homophily that would be expected by chance given an existing uneven distribution of people with varying characteristics; and

* '''Inbreeding homophily''': the amount of homophily over and above this expected value, typically due to personal preferences and choices.

===Status vs. value===

In their original formulation of homophily, [[Paul Lazarsfeld]] and [[Robert K. Merton]] (1954) distinguished between ''status homophily'' and ''value homophily'', find that individuals with similar [[social status]] characteristics are more likely to associate with each other than by chance:<ref name="lazar">[[Paul Lazarsfeld|Lazarsfeld, Paul F.]], and [[Robert K. Merton|Merton, Robert K.]] 1954. "Friendship as a Social Process: A Substantive and Methodological Analysis." Pp. 18–66 in ''Freedom and Control in Modern Society'', edited by M. Berger, T. Abel, and C. H. Page. New York: [[Van Nostrand Reinhold|Van Nostrand]]. </ref><ref name="birds" />

In their original formulation of homophily, Paul Lazarsfeld and Robert K. Merton (1954) distinguished between status homophily and value homophily, find that individuals with similar social status characteristics are more likely to associate with each other than by chance:

在他们关于同质性的原始构想中，Paul Lazarsfeld 和罗伯特·金·莫顿(1954)区分了身份同质性和价值同质性，发现具有相似社会地位特征的个体更有可能相互交往，而不是偶然交往:

* '''Status homophily''': includes both [[ascribed characteristics]] (e.g. race, ethnicity, sex, and age) and [[Acquired characteristic|acquired characteristics]] (e.g., religion and education).

* '''Value homophily''': involves association with others who think in similar ways, regardless of differences in status characteristics.

=== Dimensions ===

====Race and ethnicity====

[[Social network]]s in the United States today are strongly divided by [[Race (human categorization)|race]] and [[ethnicity]],<ref>{{Cite journal|last=Moody|first=James|date=November 2001|title=Race, School Integration, and Friendship Segregation in America|journal=American Journal of Sociology|language=en|volume=107|issue=3|pages=679–716|doi=10.1086/338954|issn=0002-9602}}</ref> which account for the greatest proportion of ''inbreeding'' homophily (though classification by these criteria can be problematic in [[sociology]] due to fuzzy boundaries and different definitions of race).

Social networks in the United States today are strongly divided by race and ethnicity, which account for the greatest proportion of inbreeding homophily (though classification by these criteria can be problematic in sociology due to fuzzy boundaries and different definitions of race).

今天，美国的社交网络按种族和族裔划分得很清楚，在近亲繁殖的同质性中，种族和族裔所占的比例最大(尽管由于种族界限模糊和定义不同，按这些标准进行分类在社会学中可能会有问题)。

Smaller groups have lower [[Racial diversity|diversity]] simply due to the number of members. This tends to give racial and ethnic minority groups a higher ''baseline'' homophily. Race and ethnicity also correlates with [[educational attainment]] and occupation, which further increase baseline homophily.<ref name="birds" />

Smaller groups have lower diversity simply due to the number of members. This tends to give racial and ethnic minority groups a higher baseline homophily. Race and ethnicity also correlates with educational attainment and occupation, which further increase baseline homophily.

较小的群体的多样性较低，仅仅是因为成员的数量。这往往使少数种族群体和少数民族群体具有较高的基线同质性。种族和民族也与学历和职业相关，这进一步增加了基线同质性。

====Sex and gender====

In regard to [[sex]] and [[gender]], ''baseline'' homophily of networks is relatively low compared to race and ethnicity. Men and women frequently live together, and are both large and equally-sized populations. Most sex homophily is of the inbreeding type.<ref name="birds" /> Especially in schools, students tend to have a high gender homophily.<ref>{{Cite journal|last=Shrum|first=Wesley|last2=Cheek|first2=Neil H.|last3=Hunter|first3=Saundra MacD.|date=October 1988|title=Friendship in School: Gender and Racial Homophily|journal=Sociology of Education|volume=61|issue=4|pages=227|doi=10.2307/2112441|issn=0038-0407|jstor=2112441}}</ref>

In regard to sex and gender, baseline homophily of networks is relatively low compared to race and ethnicity. Men and women frequently live together, and are both large and equally-sized populations. Most sex homophily is of the inbreeding type.

在性和性别方面，与种族和族裔相比，网络的基线同质性相对较低。男人和女人经常住在一起，人口众多，数量相等。大多数性别同质性为近交型。

====Age====

Most age homophily is of the ''baseline'' type. An interesting pattern of inbreeding age homophily for groups of different ages was found by Marsden (1988).<ref>Marsden, P. V. 1988. "Homogeneity in confiding relations." ''[[Social Networks (journal)|Social Networks]]'' 10:57–76.</ref> It indicated a strong relationship between someone's age and the social distance to other people with regard to confiding in someone. For example, the larger age gap someone had, the smaller chances that they were confided by others with lower ages to "discuss important matters."<ref name="birds" />

Most age homophily is of the baseline type. An interesting pattern of inbreeding age homophily for groups of different ages was found by Marsden (1988). It indicated a strong relationship between someone's age and the social distance to other people with regard to confiding in someone. For example, the larger age gap someone had, the smaller chances that they were confided by others with lower ages to "discuss important matters."

大多数年龄相同的是基线类型。马斯登(1988)发现了不同年龄组近交年龄相同的有趣模式。它表明了一个人的年龄和社会距离之间的强烈关系，以其他人在信任某人。例如，一个人的年龄差距越大，他们被年龄较小的人吐露“讨论重要事情”的机会就越小

====Religion====

Homophily based on religion is due to both ''baseline'' and ''inbreeding'' homophily.<ref name="birds" />

Homophily based on religion is due to both baseline and inbreeding homophily.

基于宗教的同质性既有基线性，也有近亲繁殖的同质性。

====Education, occupation and social class====

Family of birth accounts for considerable baseline homophily with respect to education, occupation, and [[social class]].<ref name="birds" />

Family of birth accounts for considerable baseline homophily with respect to education, occupation, and social class.

出生家庭在教育、职业和社会阶层方面占有相当大的基线同质性。

==== Social media ====

As [[social network]]s are largely divided by race, [[Social networking service|social-networking websites]] like [[Facebook]] also foster homophilic atmospheres.<ref>{{Cite journal|last=Aiello|first=Luca Maria|last2=Barrat|first2=Alain|last3=Schifanella|first3=Rossano|last4=Cattuto|first4=Ciro|last5=Markines|first5=Benjamin|last6=Menczer|first6=Filippo|date=2012-05-01|title=Friendship prediction and homophily in social media|journal=ACM Transactions on the Web|volume=6|issue=2|pages=1–33|doi=10.1145/2180861.2180866|issn=1559-1131}}</ref> When a Facebook user '[[Facebook like button|likes]]' or interacts with an article or post of a certain [[ideology]], Facebook continues to show that user posts of that similar ideology (which Facebook believes they will be drawn to). In a research article, McPherson, Smith-Lovin, and Cook (2003) write that [[Homogeneity and heterogeneity|homogeneous]] [[Personal network|personal networks]] result in limited "social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience."<ref>{{Cite journal |last=McPherson |first=Miller |last2=Smith-Lovin |first2=Lynn |last3=Cook |first3=James M. |date=2003-11-28 |title=Birds of a Feather: Homophily in Social Networks |journal=Annual Review of Sociology |language=en |volume=27 |issue=1 |pages=415–444 |doi=10.1146/annurev.soc.27.1.415}}</ref> This homophily can foster divides and [[Echo chamber (media)|echo chambers]] on social networking sites, where people of similar ideologies only interact with each other.

As social networks are largely divided by race, social-networking websites like Facebook also foster homophilic atmospheres. When a Facebook user 'likes' or interacts with an article or post of a certain ideology, Facebook continues to show that user posts of that similar ideology (which Facebook believes they will be drawn to). In a research article, McPherson, Smith-Lovin, and Cook (2003) write that homogeneous personal networks result in limited "social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience." This homophily can foster divides and echo chambers on social networking sites, where people of similar ideologies only interact with each other.

由于社交网络在很大程度上被种族划分开来，像 Facebook 这样的社交网站也培养了亲同性恋的氛围。当 Facebook 用户“喜欢”或与某种意识形态的文章或帖子互动时，Facebook 继续向用户展示类似意识形态的帖子(Facebook 相信他们会被这些帖子所吸引)。麦克弗森、史密斯-洛文和库克(2003年)在一篇研究文章中写道，同质化的个人网络造成了有限的“社会世界，这种社会世界对他们接收的信息、他们形成的态度以及他们所经历的互动有着强大的影响。”这种同质性可以在社交网站上形成分裂和回音室，在那里，具有相似意识形态的人只能相互交流。

==Causes and effects==

=== Causes ===

{{Unreferenced section|date=July 2020}}

'''Geography''': Baseline homophily often arises when the people who are located nearby also have similar characteristics. People are more likely to have contact with those who are geographically closer than those who are distant. Technology such as the telephone, e-mail, and social networks have reduced but not eliminated this effect.

Geography: Baseline homophily often arises when the people who are located nearby also have similar characteristics. People are more likely to have contact with those who are geographically closer than those who are distant. Technology such as the telephone, e-mail, and social networks have reduced but not eliminated this effect.

地理位置: 当周围的人也有相似的特征时，基线同质性常常出现。人们更有可能与那些地理上比较近的人接触，而不是那些距离较远的人。电话、电子邮件和社交网络等技术已经减少但并未消除这种影响。

'''Family ties''': [[Family relationships]] often produce relatively close, frequent contact among those who are at great geographic distance. These ties tend to decay slowly, but can be dramatically restructured when new marriages occur.

Family ties: Family relationships often produce relatively close, frequent contact among those who are at great geographic distance. These ties tend to decay slowly, but can be dramatically restructured when new marriages occur.

家庭关系: 家庭关系往往产生相对密切，频繁的接触之间的地理距离很远。这些关系往往会缓慢衰退，但当新的婚姻出现时，它们可以戏剧性地重组。

'''Organizations''': School, work, and volunteer activities provide the great majority of non-family ties. Many friendships, confiding relations, and social support ties are formed within voluntary groups. The social homogeneity of most organizations creates a strong baseline homophily in networks that are formed there.

Organizations: School, work, and volunteer activities provide the great majority of non-family ties. Many friendships, confiding relations, and social support ties are formed within voluntary groups. The social homogeneity of most organizations creates a strong baseline homophily in networks that are formed there.

组织: 学校、工作和志愿者活动提供了绝大多数的非家庭纽带。许多友谊、信任关系和社会支持关系都是在志愿者团体中形成的。大多数组织的社会同质性在形成的网络中创造了一个强大的基线同质性。

'''Isomorphic sources''': The connections between people who occupy equivalent roles will induce homophily in the system of network ties. This is common in three domains: [[workplace]] (e.g., all heads of [[HR department|HR departments]] will tend to associate with other HR heads), family (e.g., mothers tend to associate with other mothers), and informal networks.

Isomorphic sources: The connections between people who occupy equivalent roles will induce homophily in the system of network ties. This is common in three domains: workplace (e.g., all heads of HR departments will tend to associate with other HR heads), family (e.g., mothers tend to associate with other mothers), and informal networks.

同构性来源: 在网络关系系统中，身处同等角色的人之间的联系会导致同构性。这在三个领域很常见: 工作场所(例如，所有人力资源部门的负责人都倾向于与其他人力资源负责人联系)、家庭(例如，母亲倾向于与其他母亲联系)和非正式网络。

'''Cognitive processes''': People who have demographic similarity tend to own shared knowledge, and therefore they have a greater ease of communication and share cultural tastes, which can also generate homophily.

Cognitive processes: People who have demographic similarity tend to own shared knowledge, and therefore they have a greater ease of communication and share cultural tastes, which can also generate homophily.

认知过程: 人口统计学上相似的人倾向于拥有共享的知识，因此他们有更大的交流和共享的文化品味，这也可以产生同质性。

===Effects===

According to one study, [[perception]] of [[Interpersonal attraction|interpersonal]] similarity improves coordination and increase the expected payoff of interactions, above and beyond the effect of merely "liking others."<ref>{{cite journal|last1=Chierchia|first1=Gabriele|last2=Coricelli|first2=Giorgio|title=The impact of perceived similarity on tacit coordination: propensity for matching and aversion to decoupling choices|journal=Frontiers in Behavioral Neuroscience|doi=10.3389/fnbeh.2015.00202|pmid=26283940|pmc=4516978|volume=9|pages=202|year=2015}}</ref> Another study claims that homophily produces [[Social tolerance|tolerance]] and cooperation in [[Social space|social spaces]].<ref name=":0">{{cite journal|last1=Mark|first1=N. P.|title=Culture and competition: Homophily and distancing explanations for cultural niches|journal=American Sociological Review|date=2003|volume=68|issue=3|pages=319–345|jstor=1519727|doi=10.2307/1519727}}</ref> However, homophilic patterns can also restrict access to information or inclusion for minorities.<ref>{{Cite journal|last=Karimi|first=Fariba|last2=Génois|first2=Mathieu|last3=Wagner|first3=Claudia|last4=Singer|first4=Philipp|last5=Strohmaier|first5=Markus|date=2018-07-23|title=Homophily influences ranking of minorities in social networks|journal=Scientific Reports|volume=8|issue=1|pages=11077|doi=10.1038/s41598-018-29405-7|pmid=30038426|bibcode=2018NatSR...811077K|issn=2045-2322|doi-access=free}}</ref>

According to one study, perception of interpersonal similarity improves coordination and increase the expected payoff of interactions, above and beyond the effect of merely "liking others." Another study claims that homophily produces tolerance and cooperation in social spaces. However, homophilic patterns can also restrict access to information or inclusion for minorities.

根据一项研究，人际相似性感知可以改善协调性，提高互动的预期回报，远远超过仅仅“喜欢他人”的效果另一项研究声称，同质性在社会空间产生宽容与合作。然而，同源模式也会限制少数族裔获取信息或包容性。

The effects of homophily on the diffusion of information and behaviors are also complex. Some studies have claimed that homophily facilitates [[Access to information|access information]],<ref>{{cite arXiv|last1=Choudhury|first1=M. D|title="Birds of a feather": Does user homophily impact information diffusion in social media |date=2010 |eprint=1006.1702|class=cs.CY}}</ref> the diffusion of innovations and behaviors,<ref>{{cite journal|last1=Christakis|first1=N. A|last2=Fowler|first2=J. H.|title=The spread of obesity in a large social network over 32 years|journal=The New England Journal of Medicine|date=2007|volume=357|issue=4|pages=370–379|url=http://media.timesfreepress.com/docs/2010/01/Obesity_studies_0104.pdf|doi=10.1056/nejmsa066082|pmid=17652652|citeseerx=10.1.1.581.4893}}</ref> and the formation of [[Social norm|social norms]].<ref>{{cite journal|last1=Centola|first1=D|last2=R|first2=Willer|last3=M|first3=Macy|title=The emperor's dilemma: A computational model of self-enforcing norms. |journal=American Journal of Sociology|date=2005|volume=110|issue=4|pages=1009–1040|jstor=10.1086/427321|doi=10.1086/427321|url=https://repository.upenn.edu/cgi/viewcontent.cgi?article=1604&context=asc_papers}}</ref> Other studies, however, highlight mechanisms through which homophily can maintain disagreement, exacerbate [[Social polarization|polarization]] of opinions, and slow the formation of an overall [[Consensus decision-making|consensus]].<ref name=":1" /><ref>{{Cite journal|last=Centola D, Gonzalez-Avella JC, Eguiluz VM, San Miguel M.|first=|date=2007|title=Homophily, cultural drift, and the co-evolution of cultural groups|url=|journal=Journal of Conflict Resolution|volume=51(6)|pages=905–29|via=}}</ref>

The effects of homophily on the diffusion of information and behaviors are also complex. Some studies have claimed that homophily facilitates access information, the diffusion of innovations and behaviors, and the formation of social norms. Other studies, however, highlight mechanisms through which homophily can maintain disagreement, exacerbate polarization of opinions, and slow the formation of an overall consensus.

同质性对信息和行为扩散的影响也是复杂的。一些研究声称，同质性促进了信息的获取，创新产品渗透理论和行为，以及社会规范的形成。然而，其他的研究强调了同质性可以维持分歧，加剧意见的两极分化，并减缓整体共识形成的机制。

Homophily is a cause of [[Homogamy (sociology)|homogamy]]—marriage between people with similar characteristics.<ref name="fiore">Fiore, A. T. and Donath, J. S. (2005). [http://smg.media.mit.edu/papers/Fiore/fiore_donath_chi2005_short.pdf "Homophily in Online Dating: When Do You Like Someone Like Yourself?"]. ''MIT Media Lab''.</ref> Homophily is a [[fertility factor (demography)|fertility factor]]; an increased [[fertility]] is seen in people with a tendency to seek acquaintance among those with common characteristics.<ref name="fent2013" /> Governmental family policies have a decreased influence on [[Total fertility rate|fertility rates]] in such populations.<ref name="fent2013">{{cite journal|title=Family policies in the context of low fertility and social structure|url=http://www.demographic-research.org/Volumes/Vol29/37/|journal=Demographic Research|volume=29|issue=37|author1=Thomas Fent |author2=Belinda Aparicio Diaz |author3=Alexia Prskawetz |year=2013}}</ref>

Homophily is a cause of homogamy—marriage between people with similar characteristics. Homophily is a fertility factor; an increased fertility is seen in people with a tendency to seek acquaintance among those with common characteristics.

同质性是同质婚姻的一个原因ーー具有相似特征的人之间的婚姻。同质性是一种 F质粒，人们倾向于在具有共同特征的人群中寻求熟人，这种倾向会导致生育力的提高。

==See also==

*[[Groupthink]]

==References==

{{reflist|30em}}

{{World view}}

[[Category:Interpersonal relationships]]

Category:Interpersonal relationships

类别: 人际关系

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Homophily]]. Its edit history can be viewed at [[同质性/edithistory]]</noinclude>

[[Category:待整理页面]]

同质性

2020-09-25T11:50:50Z

923397935：

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

{{distinguish|Homophile}}

{{Short description|Process by which people befriend similar people}}{{expert needed|sociology|reason=statements about humans are unqualified|date=November 2011}}

{{Network science}}

'''Homophily''' (from [[Ancient Greek]]: ''[[:wiktionary:ὁμοῦ#Ancient Greek|homoû]]'', 'together' + ''philíē'', 'friendship, love') is the tendency of individuals to associate and [[human bonding|bond]] with similar others, as in the [[proverb]] "[[wikt:birds of a feather flock together|birds of a feather flock together]]."<ref>{{cite magazine |magazine=[[Foreign Affairs]] |first=Niall |last=Ferguson |quote=At the same time, birds of a feather flock together. Because of the phenomenon known as “homophily”, or attraction to similarity, social networks tend to form clusters of nodes with similar properties or attitudes. |url=https://www.foreignaffairs.com/articles/2017-08-15/false-prophecy-hyperconnection |title=The False Prophecy of Hyperconnection |date=August 15, 2017 |access-date=October 1, 2017}}</ref> The presence of homophily has been discovered in a vast array of [[social network|network]] studies: over {{nowrap|100 studies}} have observed homophily in some form or another, and they establish that similarity is associated with connection.<ref name=birds>{{cite journal | last1 = McPherson | first1 = M. | last2 = Smith-Lovin | first2 = L. | last3 = Cook | first3 = J. M. | year = 2001 | title = Birds of a Feather: Homophily in Social Networks | journal = Annual Review of Sociology | volume = 27 | issue = | pages = 415–444 | doi=10.1146/annurev.soc.27.1.415}}</ref> The categories on which homophily occurs include [[Age group|age]], [[gender]], [[social class|class]], and organizational role.<ref name=aaron>{{cite news|last=Retica|first=Aaron|title=Homophily|url=https://www.nytimes.com/2006/12/10/magazine/10Section2a.t-4.html|newspaper=New York Times|date=10 December 2006}}</ref>

Homophily (from Ancient Greek: homoû, 'together' + philíē, 'friendship, love') is the tendency of individuals to associate and bond with similar others, as in the proverb "birds of a feather flock together." The presence of homophily has been discovered in a vast array of network studies: over have observed homophily in some form or another, and they establish that similarity is associated with connection. The categories on which homophily occurs include age, gender, class, and organizational role.

'''同质性 Homophily '''（源自古希腊语：homoû，“ together” +philíē，“ friendship，love”）是个体趋向于与相似的其他人交往和联系的一种倾向，就像谚语“羽毛鸟聚在一起”。在各种各样的网络研究中都发现了同质性的存在：过去以某种形式或其他形式观察到同质性，并且他们确定相似性与连接有关。同形发生的类别包括年龄，性别，阶级和组织角色。

The opposite of homophily is [[heterophily]] or [[intermingling]]. Individuals in homophilic relationships share common characteristics (beliefs, [[values]], education, etc.) that make communication and relationship formation easier. Homophily between [[Breeding pair|mated pairs]] in animals has been extensively studied in the field of [[evolutionary biology]], where it is known as [[assortative mating|''assortative mating'']]. Homophily between mated pairs is common within natural animal mating populations.<ref name="Assortative Mating in Animals">{{cite journal |last1=Jiang |first1=Yuexin |last2=Bolnick |first2=Daniel I. |last3=Kirkpatrick |first3=Mark |title=Assortative Mating in Animals |journal=The American Naturalist |date=June 2013 |volume=181 |issue=6 |pages=E125–38 |doi=10.1086/670160 |pmid=23669548 |jstor=10.1086/670160 |ref=Assortative Mating in Animals|hdl=2152/31270 |hdl-access=free }}</ref>

The opposite of homophily is heterophily or intermingling. Individuals in homophilic relationships share common characteristics (beliefs, values, education, etc.) that make communication and relationship formation easier. Homophily between mated pairs in animals has been extensively studied in the field of evolutionary biology, where it is known as assortative mating. Homophily between mated pairs is common within natural animal mating populations.

同质性的相反是'''异构 heterophily '''或'''混合 Intermingling '''。处于亲密关系中的个人具有共同的特征（信念，价值观，教育程度等），这使沟通和关系的形成变得更加容易。动物的交配对之间的同质性已在进化生物学领域进行了广泛的研究，这被称为'''分类交配 Assortative Mating '''。在自然动物的交配种群中，交配对之间的同质性很普遍。

Homophily has a variety of consequences for social and economic outcomes, ranging from facilitating cooperation<ref name=":0" /> to slowing processes of [[Consensus decision-making|consensus-formation]],<ref name=":1">{{Cite journal|last=Golub and Jackson|first=|date=2012z|title=How homophily affects the speed of learning and best-response dynamics|url=http://www.bengolub.net/papers/homophily.pdf|journal=Quarterly Journal of Economics|volume=127(3)|pages=1287–1338|via=}}</ref> among others.

Homophily has a variety of consequences for social and economic outcomes, ranging from facilitating cooperation among others.

同质性对社会和经济成果有多种影响，包括促进彼此之间的合作。

==Types and dimensions==
类型和尺寸 

==Baseline vs. inbreeding==
基线与近交 

To test the relevance of homophily, researchers have distinguished between two types:<ref name=birds />

To test the relevance of homophily, researchers have distinguished between two types:

为了测试同的相关性，研究人员区分了两种类型：

* '''Baseline homophily''': simply the amount of homophily that would be expected by chance given an existing uneven distribution of people with varying characteristics; and

* '''Inbreeding homophily''': the amount of homophily over and above this expected value, typically due to personal preferences and choices.

===Status vs. value===

In their original formulation of homophily, [[Paul Lazarsfeld]] and [[Robert K. Merton]] (1954) distinguished between ''status homophily'' and ''value homophily'', find that individuals with similar [[social status]] characteristics are more likely to associate with each other than by chance:<ref name="lazar">[[Paul Lazarsfeld|Lazarsfeld, Paul F.]], and [[Robert K. Merton|Merton, Robert K.]] 1954. "Friendship as a Social Process: A Substantive and Methodological Analysis." Pp. 18–66 in ''Freedom and Control in Modern Society'', edited by M. Berger, T. Abel, and C. H. Page. New York: [[Van Nostrand Reinhold|Van Nostrand]]. </ref><ref name="birds" />

In their original formulation of homophily, Paul Lazarsfeld and Robert K. Merton (1954) distinguished between status homophily and value homophily, find that individuals with similar social status characteristics are more likely to associate with each other than by chance:

在他们关于同质性的原始构想中，Paul Lazarsfeld 和罗伯特·金·莫顿(1954)区分了身份同质性和价值同质性，发现具有相似社会地位特征的个体更有可能相互交往，而不是偶然交往:

* '''Status homophily''': includes both [[ascribed characteristics]] (e.g. race, ethnicity, sex, and age) and [[Acquired characteristic|acquired characteristics]] (e.g., religion and education).

* '''Value homophily''': involves association with others who think in similar ways, regardless of differences in status characteristics.

=== Dimensions ===

====Race and ethnicity====

[[Social network]]s in the United States today are strongly divided by [[Race (human categorization)|race]] and [[ethnicity]],<ref>{{Cite journal|last=Moody|first=James|date=November 2001|title=Race, School Integration, and Friendship Segregation in America|journal=American Journal of Sociology|language=en|volume=107|issue=3|pages=679–716|doi=10.1086/338954|issn=0002-9602}}</ref> which account for the greatest proportion of ''inbreeding'' homophily (though classification by these criteria can be problematic in [[sociology]] due to fuzzy boundaries and different definitions of race).

Social networks in the United States today are strongly divided by race and ethnicity, which account for the greatest proportion of inbreeding homophily (though classification by these criteria can be problematic in sociology due to fuzzy boundaries and different definitions of race).

今天，美国的社交网络按种族和族裔划分得很清楚，在近亲繁殖的同质性中，种族和族裔所占的比例最大(尽管由于种族界限模糊和定义不同，按这些标准进行分类在社会学中可能会有问题)。

Smaller groups have lower [[Racial diversity|diversity]] simply due to the number of members. This tends to give racial and ethnic minority groups a higher ''baseline'' homophily. Race and ethnicity also correlates with [[educational attainment]] and occupation, which further increase baseline homophily.<ref name="birds" />

Smaller groups have lower diversity simply due to the number of members. This tends to give racial and ethnic minority groups a higher baseline homophily. Race and ethnicity also correlates with educational attainment and occupation, which further increase baseline homophily.

较小的群体的多样性较低，仅仅是因为成员的数量。这往往使少数种族群体和少数民族群体具有较高的基线同质性。种族和民族也与学历和职业相关，这进一步增加了基线同质性。

====Sex and gender====

In regard to [[sex]] and [[gender]], ''baseline'' homophily of networks is relatively low compared to race and ethnicity. Men and women frequently live together, and are both large and equally-sized populations. Most sex homophily is of the inbreeding type.<ref name="birds" /> Especially in schools, students tend to have a high gender homophily.<ref>{{Cite journal|last=Shrum|first=Wesley|last2=Cheek|first2=Neil H.|last3=Hunter|first3=Saundra MacD.|date=October 1988|title=Friendship in School: Gender and Racial Homophily|journal=Sociology of Education|volume=61|issue=4|pages=227|doi=10.2307/2112441|issn=0038-0407|jstor=2112441}}</ref>

In regard to sex and gender, baseline homophily of networks is relatively low compared to race and ethnicity. Men and women frequently live together, and are both large and equally-sized populations. Most sex homophily is of the inbreeding type.

在性和性别方面，与种族和族裔相比，网络的基线同质性相对较低。男人和女人经常住在一起，人口众多，数量相等。大多数性别同质性为近交型。

====Age====

Most age homophily is of the ''baseline'' type. An interesting pattern of inbreeding age homophily for groups of different ages was found by Marsden (1988).<ref>Marsden, P. V. 1988. "Homogeneity in confiding relations." ''[[Social Networks (journal)|Social Networks]]'' 10:57–76.</ref> It indicated a strong relationship between someone's age and the social distance to other people with regard to confiding in someone. For example, the larger age gap someone had, the smaller chances that they were confided by others with lower ages to "discuss important matters."<ref name="birds" />

Most age homophily is of the baseline type. An interesting pattern of inbreeding age homophily for groups of different ages was found by Marsden (1988). It indicated a strong relationship between someone's age and the social distance to other people with regard to confiding in someone. For example, the larger age gap someone had, the smaller chances that they were confided by others with lower ages to "discuss important matters."

大多数年龄相同的是基线类型。马斯登(1988)发现了不同年龄组近交年龄相同的有趣模式。它表明了一个人的年龄和社会距离之间的强烈关系，以其他人在信任某人。例如，一个人的年龄差距越大，他们被年龄较小的人吐露“讨论重要事情”的机会就越小

====Religion====

Homophily based on religion is due to both ''baseline'' and ''inbreeding'' homophily.<ref name="birds" />

Homophily based on religion is due to both baseline and inbreeding homophily.

基于宗教的同质性既有基线性，也有近亲繁殖的同质性。

====Education, occupation and social class====

Family of birth accounts for considerable baseline homophily with respect to education, occupation, and [[social class]].<ref name="birds" />

Family of birth accounts for considerable baseline homophily with respect to education, occupation, and social class.

出生家庭在教育、职业和社会阶层方面占有相当大的基线同质性。

==== Social media ====

As [[social network]]s are largely divided by race, [[Social networking service|social-networking websites]] like [[Facebook]] also foster homophilic atmospheres.<ref>{{Cite journal|last=Aiello|first=Luca Maria|last2=Barrat|first2=Alain|last3=Schifanella|first3=Rossano|last4=Cattuto|first4=Ciro|last5=Markines|first5=Benjamin|last6=Menczer|first6=Filippo|date=2012-05-01|title=Friendship prediction and homophily in social media|journal=ACM Transactions on the Web|volume=6|issue=2|pages=1–33|doi=10.1145/2180861.2180866|issn=1559-1131}}</ref> When a Facebook user '[[Facebook like button|likes]]' or interacts with an article or post of a certain [[ideology]], Facebook continues to show that user posts of that similar ideology (which Facebook believes they will be drawn to). In a research article, McPherson, Smith-Lovin, and Cook (2003) write that [[Homogeneity and heterogeneity|homogeneous]] [[Personal network|personal networks]] result in limited "social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience."<ref>{{Cite journal |last=McPherson |first=Miller |last2=Smith-Lovin |first2=Lynn |last3=Cook |first3=James M. |date=2003-11-28 |title=Birds of a Feather: Homophily in Social Networks |journal=Annual Review of Sociology |language=en |volume=27 |issue=1 |pages=415–444 |doi=10.1146/annurev.soc.27.1.415}}</ref> This homophily can foster divides and [[Echo chamber (media)|echo chambers]] on social networking sites, where people of similar ideologies only interact with each other.

As social networks are largely divided by race, social-networking websites like Facebook also foster homophilic atmospheres. When a Facebook user 'likes' or interacts with an article or post of a certain ideology, Facebook continues to show that user posts of that similar ideology (which Facebook believes they will be drawn to). In a research article, McPherson, Smith-Lovin, and Cook (2003) write that homogeneous personal networks result in limited "social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience." This homophily can foster divides and echo chambers on social networking sites, where people of similar ideologies only interact with each other.

由于社交网络在很大程度上被种族划分开来，像 Facebook 这样的社交网站也培养了亲同性恋的氛围。当 Facebook 用户“喜欢”或与某种意识形态的文章或帖子互动时，Facebook 继续向用户展示类似意识形态的帖子(Facebook 相信他们会被这些帖子所吸引)。麦克弗森、史密斯-洛文和库克(2003年)在一篇研究文章中写道，同质化的个人网络造成了有限的“社会世界，这种社会世界对他们接收的信息、他们形成的态度以及他们所经历的互动有着强大的影响。”这种同质性可以在社交网站上形成分裂和回音室，在那里，具有相似意识形态的人只能相互交流。

==Causes and effects==

=== Causes ===

{{Unreferenced section|date=July 2020}}

'''Geography''': Baseline homophily often arises when the people who are located nearby also have similar characteristics. People are more likely to have contact with those who are geographically closer than those who are distant. Technology such as the telephone, e-mail, and social networks have reduced but not eliminated this effect.

Geography: Baseline homophily often arises when the people who are located nearby also have similar characteristics. People are more likely to have contact with those who are geographically closer than those who are distant. Technology such as the telephone, e-mail, and social networks have reduced but not eliminated this effect.

地理位置: 当周围的人也有相似的特征时，基线同质性常常出现。人们更有可能与那些地理上比较近的人接触，而不是那些距离较远的人。电话、电子邮件和社交网络等技术已经减少但并未消除这种影响。

'''Family ties''': [[Family relationships]] often produce relatively close, frequent contact among those who are at great geographic distance. These ties tend to decay slowly, but can be dramatically restructured when new marriages occur.

Family ties: Family relationships often produce relatively close, frequent contact among those who are at great geographic distance. These ties tend to decay slowly, but can be dramatically restructured when new marriages occur.

家庭关系: 家庭关系往往产生相对密切，频繁的接触之间的地理距离很远。这些关系往往会缓慢衰退，但当新的婚姻出现时，它们可以戏剧性地重组。

'''Organizations''': School, work, and volunteer activities provide the great majority of non-family ties. Many friendships, confiding relations, and social support ties are formed within voluntary groups. The social homogeneity of most organizations creates a strong baseline homophily in networks that are formed there.

Organizations: School, work, and volunteer activities provide the great majority of non-family ties. Many friendships, confiding relations, and social support ties are formed within voluntary groups. The social homogeneity of most organizations creates a strong baseline homophily in networks that are formed there.

组织: 学校、工作和志愿者活动提供了绝大多数的非家庭纽带。许多友谊、信任关系和社会支持关系都是在志愿者团体中形成的。大多数组织的社会同质性在形成的网络中创造了一个强大的基线同质性。

'''Isomorphic sources''': The connections between people who occupy equivalent roles will induce homophily in the system of network ties. This is common in three domains: [[workplace]] (e.g., all heads of [[HR department|HR departments]] will tend to associate with other HR heads), family (e.g., mothers tend to associate with other mothers), and informal networks.

Isomorphic sources: The connections between people who occupy equivalent roles will induce homophily in the system of network ties. This is common in three domains: workplace (e.g., all heads of HR departments will tend to associate with other HR heads), family (e.g., mothers tend to associate with other mothers), and informal networks.

同构性来源: 在网络关系系统中，身处同等角色的人之间的联系会导致同构性。这在三个领域很常见: 工作场所(例如，所有人力资源部门的负责人都倾向于与其他人力资源负责人联系)、家庭(例如，母亲倾向于与其他母亲联系)和非正式网络。

'''Cognitive processes''': People who have demographic similarity tend to own shared knowledge, and therefore they have a greater ease of communication and share cultural tastes, which can also generate homophily.

Cognitive processes: People who have demographic similarity tend to own shared knowledge, and therefore they have a greater ease of communication and share cultural tastes, which can also generate homophily.

认知过程: 人口统计学上相似的人倾向于拥有共享的知识，因此他们有更大的交流和共享的文化品味，这也可以产生同质性。

===Effects===

According to one study, [[perception]] of [[Interpersonal attraction|interpersonal]] similarity improves coordination and increase the expected payoff of interactions, above and beyond the effect of merely "liking others."<ref>{{cite journal|last1=Chierchia|first1=Gabriele|last2=Coricelli|first2=Giorgio|title=The impact of perceived similarity on tacit coordination: propensity for matching and aversion to decoupling choices|journal=Frontiers in Behavioral Neuroscience|doi=10.3389/fnbeh.2015.00202|pmid=26283940|pmc=4516978|volume=9|pages=202|year=2015}}</ref> Another study claims that homophily produces [[Social tolerance|tolerance]] and cooperation in [[Social space|social spaces]].<ref name=":0">{{cite journal|last1=Mark|first1=N. P.|title=Culture and competition: Homophily and distancing explanations for cultural niches|journal=American Sociological Review|date=2003|volume=68|issue=3|pages=319–345|jstor=1519727|doi=10.2307/1519727}}</ref> However, homophilic patterns can also restrict access to information or inclusion for minorities.<ref>{{Cite journal|last=Karimi|first=Fariba|last2=Génois|first2=Mathieu|last3=Wagner|first3=Claudia|last4=Singer|first4=Philipp|last5=Strohmaier|first5=Markus|date=2018-07-23|title=Homophily influences ranking of minorities in social networks|journal=Scientific Reports|volume=8|issue=1|pages=11077|doi=10.1038/s41598-018-29405-7|pmid=30038426|bibcode=2018NatSR...811077K|issn=2045-2322|doi-access=free}}</ref>

According to one study, perception of interpersonal similarity improves coordination and increase the expected payoff of interactions, above and beyond the effect of merely "liking others." Another study claims that homophily produces tolerance and cooperation in social spaces. However, homophilic patterns can also restrict access to information or inclusion for minorities.

根据一项研究，人际相似性感知可以改善协调性，提高互动的预期回报，远远超过仅仅“喜欢他人”的效果另一项研究声称，同质性在社会空间产生宽容与合作。然而，同源模式也会限制少数族裔获取信息或包容性。

The effects of homophily on the diffusion of information and behaviors are also complex. Some studies have claimed that homophily facilitates [[Access to information|access information]],<ref>{{cite arXiv|last1=Choudhury|first1=M. D|title="Birds of a feather": Does user homophily impact information diffusion in social media |date=2010 |eprint=1006.1702|class=cs.CY}}</ref> the diffusion of innovations and behaviors,<ref>{{cite journal|last1=Christakis|first1=N. A|last2=Fowler|first2=J. H.|title=The spread of obesity in a large social network over 32 years|journal=The New England Journal of Medicine|date=2007|volume=357|issue=4|pages=370–379|url=http://media.timesfreepress.com/docs/2010/01/Obesity_studies_0104.pdf|doi=10.1056/nejmsa066082|pmid=17652652|citeseerx=10.1.1.581.4893}}</ref> and the formation of [[Social norm|social norms]].<ref>{{cite journal|last1=Centola|first1=D|last2=R|first2=Willer|last3=M|first3=Macy|title=The emperor's dilemma: A computational model of self-enforcing norms. |journal=American Journal of Sociology|date=2005|volume=110|issue=4|pages=1009–1040|jstor=10.1086/427321|doi=10.1086/427321|url=https://repository.upenn.edu/cgi/viewcontent.cgi?article=1604&context=asc_papers}}</ref> Other studies, however, highlight mechanisms through which homophily can maintain disagreement, exacerbate [[Social polarization|polarization]] of opinions, and slow the formation of an overall [[Consensus decision-making|consensus]].<ref name=":1" /><ref>{{Cite journal|last=Centola D, Gonzalez-Avella JC, Eguiluz VM, San Miguel M.|first=|date=2007|title=Homophily, cultural drift, and the co-evolution of cultural groups|url=|journal=Journal of Conflict Resolution|volume=51(6)|pages=905–29|via=}}</ref>

The effects of homophily on the diffusion of information and behaviors are also complex. Some studies have claimed that homophily facilitates access information, the diffusion of innovations and behaviors, and the formation of social norms. Other studies, however, highlight mechanisms through which homophily can maintain disagreement, exacerbate polarization of opinions, and slow the formation of an overall consensus.

同质性对信息和行为扩散的影响也是复杂的。一些研究声称，同质性促进了信息的获取，创新产品渗透理论和行为，以及社会规范的形成。然而，其他的研究强调了同质性可以维持分歧，加剧意见的两极分化，并减缓整体共识形成的机制。

Homophily is a cause of [[Homogamy (sociology)|homogamy]]—marriage between people with similar characteristics.<ref name="fiore">Fiore, A. T. and Donath, J. S. (2005). [http://smg.media.mit.edu/papers/Fiore/fiore_donath_chi2005_short.pdf "Homophily in Online Dating: When Do You Like Someone Like Yourself?"]. ''MIT Media Lab''.</ref> Homophily is a [[fertility factor (demography)|fertility factor]]; an increased [[fertility]] is seen in people with a tendency to seek acquaintance among those with common characteristics.<ref name="fent2013" /> Governmental family policies have a decreased influence on [[Total fertility rate|fertility rates]] in such populations.<ref name="fent2013">{{cite journal|title=Family policies in the context of low fertility and social structure|url=http://www.demographic-research.org/Volumes/Vol29/37/|journal=Demographic Research|volume=29|issue=37|author1=Thomas Fent |author2=Belinda Aparicio Diaz |author3=Alexia Prskawetz |year=2013}}</ref>

Homophily is a cause of homogamy—marriage between people with similar characteristics. Homophily is a fertility factor; an increased fertility is seen in people with a tendency to seek acquaintance among those with common characteristics.

同质性是同质婚姻的一个原因ーー具有相似特征的人之间的婚姻。同质性是一种 F质粒，人们倾向于在具有共同特征的人群中寻求熟人，这种倾向会导致生育力的提高。

==See also==

*[[Groupthink]]

==References==

{{reflist|30em}}

{{World view}}

[[Category:Interpersonal relationships]]

Category:Interpersonal relationships

类别: 人际关系

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Homophily]]. Its edit history can be viewed at [[同质性/edithistory]]</noinclude>

[[Category:待整理页面]]

同质性

2020-09-25T11:45:03Z

923397935：

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

{{distinguish|Homophile}}

{{Short description|Process by which people befriend similar people}}{{expert needed|sociology|reason=statements about humans are unqualified|date=November 2011}}

{{Network science}}

'''Homophily''' (from [[Ancient Greek]]: ''[[:wiktionary:ὁμοῦ#Ancient Greek|homoû]]'', 'together' + ''philíē'', 'friendship, love') is the tendency of individuals to associate and [[human bonding|bond]] with similar others, as in the [[proverb]] "[[wikt:birds of a feather flock together|birds of a feather flock together]]."<ref>{{cite magazine |magazine=[[Foreign Affairs]] |first=Niall |last=Ferguson |quote=At the same time, birds of a feather flock together. Because of the phenomenon known as “homophily”, or attraction to similarity, social networks tend to form clusters of nodes with similar properties or attitudes. |url=https://www.foreignaffairs.com/articles/2017-08-15/false-prophecy-hyperconnection |title=The False Prophecy of Hyperconnection |date=August 15, 2017 |access-date=October 1, 2017}}</ref> The presence of homophily has been discovered in a vast array of [[social network|network]] studies: over {{nowrap|100 studies}} have observed homophily in some form or another, and they establish that similarity is associated with connection.<ref name=birds>{{cite journal | last1 = McPherson | first1 = M. | last2 = Smith-Lovin | first2 = L. | last3 = Cook | first3 = J. M. | year = 2001 | title = Birds of a Feather: Homophily in Social Networks | journal = Annual Review of Sociology | volume = 27 | issue = | pages = 415–444 | doi=10.1146/annurev.soc.27.1.415}}</ref> The categories on which homophily occurs include [[Age group|age]], [[gender]], [[social class|class]], and organizational role.<ref name=aaron>{{cite news|last=Retica|first=Aaron|title=Homophily|url=https://www.nytimes.com/2006/12/10/magazine/10Section2a.t-4.html|newspaper=New York Times|date=10 December 2006}}</ref>

Homophily (from Ancient Greek: homoû, 'together' + philíē, 'friendship, love') is the tendency of individuals to associate and bond with similar others, as in the proverb "birds of a feather flock together." The presence of homophily has been discovered in a vast array of network studies: over have observed homophily in some form or another, and they establish that similarity is associated with connection. The categories on which homophily occurs include age, gender, class, and organizational role.

同质性（源自古希腊语：homoû，“ together” +philíē，“ friendship，love”）是个体趋向于与相似的其他人交往和联系的一种倾向，就像谚语“羽毛鸟聚在一起”。在各种各样的网络研究中都发现了同质性的存在：过去以某种形式或其他形式观察到同质性，并且他们确定相似性与连接有关。同形发生的类别包括年龄，性别，阶级和组织角色。

The opposite of homophily is [[heterophily]] or [[intermingling]]. Individuals in homophilic relationships share common characteristics (beliefs, [[values]], education, etc.) that make communication and relationship formation easier. Homophily between [[Breeding pair|mated pairs]] in animals has been extensively studied in the field of [[evolutionary biology]], where it is known as [[assortative mating|''assortative mating'']]. Homophily between mated pairs is common within natural animal mating populations.<ref name="Assortative Mating in Animals">{{cite journal |last1=Jiang |first1=Yuexin |last2=Bolnick |first2=Daniel I. |last3=Kirkpatrick |first3=Mark |title=Assortative Mating in Animals |journal=The American Naturalist |date=June 2013 |volume=181 |issue=6 |pages=E125–38 |doi=10.1086/670160 |pmid=23669548 |jstor=10.1086/670160 |ref=Assortative Mating in Animals|hdl=2152/31270 |hdl-access=free }}</ref>

The opposite of homophily is heterophily or intermingling. Individuals in homophilic relationships share common characteristics (beliefs, values, education, etc.) that make communication and relationship formation easier. Homophily between mated pairs in animals has been extensively studied in the field of evolutionary biology, where it is known as assortative mating. Homophily between mated pairs is common within natural animal mating populations.

同质性的反义词是异质性或混合性。在同性恋关系中的个体具有共同的特征(信仰、价值观、教育等)让沟通和关系更容易形成。动物交配对之间的相似性已经在进化生物学领域得到了广泛的研究，在那里它被称为选型交配。在自然的动物交配种群中，交配对之间的相同点是很常见的。

Homophily has a variety of consequences for social and economic outcomes, ranging from facilitating cooperation<ref name=":0" /> to slowing processes of [[Consensus decision-making|consensus-formation]],<ref name=":1">{{Cite journal|last=Golub and Jackson|first=|date=2012z|title=How homophily affects the speed of learning and best-response dynamics|url=http://www.bengolub.net/papers/homophily.pdf|journal=Quarterly Journal of Economics|volume=127(3)|pages=1287–1338|via=}}</ref> among others.

Homophily has a variety of consequences for social and economic outcomes, ranging from facilitating cooperation among others.

同质性对社会和经济结果有各种各样的影响，从促进其他方面的合作。

==Types and dimensions==

===Baseline vs. inbreeding===

To test the relevance of homophily, researchers have distinguished between two types:<ref name=birds />

To test the relevance of homophily, researchers have distinguished between two types:

为了检验同质性的相关性，研究人员区分了两种类型:

* '''Baseline homophily''': simply the amount of homophily that would be expected by chance given an existing uneven distribution of people with varying characteristics; and

* '''Inbreeding homophily''': the amount of homophily over and above this expected value, typically due to personal preferences and choices.

===Status vs. value===

In their original formulation of homophily, [[Paul Lazarsfeld]] and [[Robert K. Merton]] (1954) distinguished between ''status homophily'' and ''value homophily'', find that individuals with similar [[social status]] characteristics are more likely to associate with each other than by chance:<ref name="lazar">[[Paul Lazarsfeld|Lazarsfeld, Paul F.]], and [[Robert K. Merton|Merton, Robert K.]] 1954. "Friendship as a Social Process: A Substantive and Methodological Analysis." Pp. 18–66 in ''Freedom and Control in Modern Society'', edited by M. Berger, T. Abel, and C. H. Page. New York: [[Van Nostrand Reinhold|Van Nostrand]]. </ref><ref name="birds" />

In their original formulation of homophily, Paul Lazarsfeld and Robert K. Merton (1954) distinguished between status homophily and value homophily, find that individuals with similar social status characteristics are more likely to associate with each other than by chance:

在他们关于同质性的原始构想中，Paul Lazarsfeld 和罗伯特·金·莫顿(1954)区分了身份同质性和价值同质性，发现具有相似社会地位特征的个体更有可能相互交往，而不是偶然交往:

* '''Status homophily''': includes both [[ascribed characteristics]] (e.g. race, ethnicity, sex, and age) and [[Acquired characteristic|acquired characteristics]] (e.g., religion and education).

* '''Value homophily''': involves association with others who think in similar ways, regardless of differences in status characteristics.

=== Dimensions ===

====Race and ethnicity====

[[Social network]]s in the United States today are strongly divided by [[Race (human categorization)|race]] and [[ethnicity]],<ref>{{Cite journal|last=Moody|first=James|date=November 2001|title=Race, School Integration, and Friendship Segregation in America|journal=American Journal of Sociology|language=en|volume=107|issue=3|pages=679–716|doi=10.1086/338954|issn=0002-9602}}</ref> which account for the greatest proportion of ''inbreeding'' homophily (though classification by these criteria can be problematic in [[sociology]] due to fuzzy boundaries and different definitions of race).

Social networks in the United States today are strongly divided by race and ethnicity, which account for the greatest proportion of inbreeding homophily (though classification by these criteria can be problematic in sociology due to fuzzy boundaries and different definitions of race).

今天，美国的社交网络按种族和族裔划分得很清楚，在近亲繁殖的同质性中，种族和族裔所占的比例最大(尽管由于种族界限模糊和定义不同，按这些标准进行分类在社会学中可能会有问题)。

Smaller groups have lower [[Racial diversity|diversity]] simply due to the number of members. This tends to give racial and ethnic minority groups a higher ''baseline'' homophily. Race and ethnicity also correlates with [[educational attainment]] and occupation, which further increase baseline homophily.<ref name="birds" />

Smaller groups have lower diversity simply due to the number of members. This tends to give racial and ethnic minority groups a higher baseline homophily. Race and ethnicity also correlates with educational attainment and occupation, which further increase baseline homophily.

较小的群体的多样性较低，仅仅是因为成员的数量。这往往使少数种族群体和少数民族群体具有较高的基线同质性。种族和民族也与学历和职业相关，这进一步增加了基线同质性。

====Sex and gender====

In regard to [[sex]] and [[gender]], ''baseline'' homophily of networks is relatively low compared to race and ethnicity. Men and women frequently live together, and are both large and equally-sized populations. Most sex homophily is of the inbreeding type.<ref name="birds" /> Especially in schools, students tend to have a high gender homophily.<ref>{{Cite journal|last=Shrum|first=Wesley|last2=Cheek|first2=Neil H.|last3=Hunter|first3=Saundra MacD.|date=October 1988|title=Friendship in School: Gender and Racial Homophily|journal=Sociology of Education|volume=61|issue=4|pages=227|doi=10.2307/2112441|issn=0038-0407|jstor=2112441}}</ref>

In regard to sex and gender, baseline homophily of networks is relatively low compared to race and ethnicity. Men and women frequently live together, and are both large and equally-sized populations. Most sex homophily is of the inbreeding type.

在性和性别方面，与种族和族裔相比，网络的基线同质性相对较低。男人和女人经常住在一起，人口众多，数量相等。大多数性别同质性为近交型。

====Age====

Most age homophily is of the ''baseline'' type. An interesting pattern of inbreeding age homophily for groups of different ages was found by Marsden (1988).<ref>Marsden, P. V. 1988. "Homogeneity in confiding relations." ''[[Social Networks (journal)|Social Networks]]'' 10:57–76.</ref> It indicated a strong relationship between someone's age and the social distance to other people with regard to confiding in someone. For example, the larger age gap someone had, the smaller chances that they were confided by others with lower ages to "discuss important matters."<ref name="birds" />

Most age homophily is of the baseline type. An interesting pattern of inbreeding age homophily for groups of different ages was found by Marsden (1988). It indicated a strong relationship between someone's age and the social distance to other people with regard to confiding in someone. For example, the larger age gap someone had, the smaller chances that they were confided by others with lower ages to "discuss important matters."

大多数年龄相同的是基线类型。马斯登(1988)发现了不同年龄组近交年龄相同的有趣模式。它表明了一个人的年龄和社会距离之间的强烈关系，以其他人在信任某人。例如，一个人的年龄差距越大，他们被年龄较小的人吐露“讨论重要事情”的机会就越小

====Religion====

Homophily based on religion is due to both ''baseline'' and ''inbreeding'' homophily.<ref name="birds" />

Homophily based on religion is due to both baseline and inbreeding homophily.

基于宗教的同质性既有基线性，也有近亲繁殖的同质性。

====Education, occupation and social class====

Family of birth accounts for considerable baseline homophily with respect to education, occupation, and [[social class]].<ref name="birds" />

Family of birth accounts for considerable baseline homophily with respect to education, occupation, and social class.

出生家庭在教育、职业和社会阶层方面占有相当大的基线同质性。

==== Social media ====

As [[social network]]s are largely divided by race, [[Social networking service|social-networking websites]] like [[Facebook]] also foster homophilic atmospheres.<ref>{{Cite journal|last=Aiello|first=Luca Maria|last2=Barrat|first2=Alain|last3=Schifanella|first3=Rossano|last4=Cattuto|first4=Ciro|last5=Markines|first5=Benjamin|last6=Menczer|first6=Filippo|date=2012-05-01|title=Friendship prediction and homophily in social media|journal=ACM Transactions on the Web|volume=6|issue=2|pages=1–33|doi=10.1145/2180861.2180866|issn=1559-1131}}</ref> When a Facebook user '[[Facebook like button|likes]]' or interacts with an article or post of a certain [[ideology]], Facebook continues to show that user posts of that similar ideology (which Facebook believes they will be drawn to). In a research article, McPherson, Smith-Lovin, and Cook (2003) write that [[Homogeneity and heterogeneity|homogeneous]] [[Personal network|personal networks]] result in limited "social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience."<ref>{{Cite journal |last=McPherson |first=Miller |last2=Smith-Lovin |first2=Lynn |last3=Cook |first3=James M. |date=2003-11-28 |title=Birds of a Feather: Homophily in Social Networks |journal=Annual Review of Sociology |language=en |volume=27 |issue=1 |pages=415–444 |doi=10.1146/annurev.soc.27.1.415}}</ref> This homophily can foster divides and [[Echo chamber (media)|echo chambers]] on social networking sites, where people of similar ideologies only interact with each other.

As social networks are largely divided by race, social-networking websites like Facebook also foster homophilic atmospheres. When a Facebook user 'likes' or interacts with an article or post of a certain ideology, Facebook continues to show that user posts of that similar ideology (which Facebook believes they will be drawn to). In a research article, McPherson, Smith-Lovin, and Cook (2003) write that homogeneous personal networks result in limited "social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience." This homophily can foster divides and echo chambers on social networking sites, where people of similar ideologies only interact with each other.

由于社交网络在很大程度上被种族划分开来，像 Facebook 这样的社交网站也培养了亲同性恋的氛围。当 Facebook 用户“喜欢”或与某种意识形态的文章或帖子互动时，Facebook 继续向用户展示类似意识形态的帖子(Facebook 相信他们会被这些帖子所吸引)。麦克弗森、史密斯-洛文和库克(2003年)在一篇研究文章中写道，同质化的个人网络造成了有限的“社会世界，这种社会世界对他们接收的信息、他们形成的态度以及他们所经历的互动有着强大的影响。”这种同质性可以在社交网站上形成分裂和回音室，在那里，具有相似意识形态的人只能相互交流。

==Causes and effects==

=== Causes ===

{{Unreferenced section|date=July 2020}}

'''Geography''': Baseline homophily often arises when the people who are located nearby also have similar characteristics. People are more likely to have contact with those who are geographically closer than those who are distant. Technology such as the telephone, e-mail, and social networks have reduced but not eliminated this effect.

Geography: Baseline homophily often arises when the people who are located nearby also have similar characteristics. People are more likely to have contact with those who are geographically closer than those who are distant. Technology such as the telephone, e-mail, and social networks have reduced but not eliminated this effect.

地理位置: 当周围的人也有相似的特征时，基线同质性常常出现。人们更有可能与那些地理上比较近的人接触，而不是那些距离较远的人。电话、电子邮件和社交网络等技术已经减少但并未消除这种影响。

'''Family ties''': [[Family relationships]] often produce relatively close, frequent contact among those who are at great geographic distance. These ties tend to decay slowly, but can be dramatically restructured when new marriages occur.

Family ties: Family relationships often produce relatively close, frequent contact among those who are at great geographic distance. These ties tend to decay slowly, but can be dramatically restructured when new marriages occur.

家庭关系: 家庭关系往往产生相对密切，频繁的接触之间的地理距离很远。这些关系往往会缓慢衰退，但当新的婚姻出现时，它们可以戏剧性地重组。

'''Organizations''': School, work, and volunteer activities provide the great majority of non-family ties. Many friendships, confiding relations, and social support ties are formed within voluntary groups. The social homogeneity of most organizations creates a strong baseline homophily in networks that are formed there.

Organizations: School, work, and volunteer activities provide the great majority of non-family ties. Many friendships, confiding relations, and social support ties are formed within voluntary groups. The social homogeneity of most organizations creates a strong baseline homophily in networks that are formed there.

组织: 学校、工作和志愿者活动提供了绝大多数的非家庭纽带。许多友谊、信任关系和社会支持关系都是在志愿者团体中形成的。大多数组织的社会同质性在形成的网络中创造了一个强大的基线同质性。

'''Isomorphic sources''': The connections between people who occupy equivalent roles will induce homophily in the system of network ties. This is common in three domains: [[workplace]] (e.g., all heads of [[HR department|HR departments]] will tend to associate with other HR heads), family (e.g., mothers tend to associate with other mothers), and informal networks.

Isomorphic sources: The connections between people who occupy equivalent roles will induce homophily in the system of network ties. This is common in three domains: workplace (e.g., all heads of HR departments will tend to associate with other HR heads), family (e.g., mothers tend to associate with other mothers), and informal networks.

同构性来源: 在网络关系系统中，身处同等角色的人之间的联系会导致同构性。这在三个领域很常见: 工作场所(例如，所有人力资源部门的负责人都倾向于与其他人力资源负责人联系)、家庭(例如，母亲倾向于与其他母亲联系)和非正式网络。

'''Cognitive processes''': People who have demographic similarity tend to own shared knowledge, and therefore they have a greater ease of communication and share cultural tastes, which can also generate homophily.

Cognitive processes: People who have demographic similarity tend to own shared knowledge, and therefore they have a greater ease of communication and share cultural tastes, which can also generate homophily.

认知过程: 人口统计学上相似的人倾向于拥有共享的知识，因此他们有更大的交流和共享的文化品味，这也可以产生同质性。

===Effects===

According to one study, [[perception]] of [[Interpersonal attraction|interpersonal]] similarity improves coordination and increase the expected payoff of interactions, above and beyond the effect of merely "liking others."<ref>{{cite journal|last1=Chierchia|first1=Gabriele|last2=Coricelli|first2=Giorgio|title=The impact of perceived similarity on tacit coordination: propensity for matching and aversion to decoupling choices|journal=Frontiers in Behavioral Neuroscience|doi=10.3389/fnbeh.2015.00202|pmid=26283940|pmc=4516978|volume=9|pages=202|year=2015}}</ref> Another study claims that homophily produces [[Social tolerance|tolerance]] and cooperation in [[Social space|social spaces]].<ref name=":0">{{cite journal|last1=Mark|first1=N. P.|title=Culture and competition: Homophily and distancing explanations for cultural niches|journal=American Sociological Review|date=2003|volume=68|issue=3|pages=319–345|jstor=1519727|doi=10.2307/1519727}}</ref> However, homophilic patterns can also restrict access to information or inclusion for minorities.<ref>{{Cite journal|last=Karimi|first=Fariba|last2=Génois|first2=Mathieu|last3=Wagner|first3=Claudia|last4=Singer|first4=Philipp|last5=Strohmaier|first5=Markus|date=2018-07-23|title=Homophily influences ranking of minorities in social networks|journal=Scientific Reports|volume=8|issue=1|pages=11077|doi=10.1038/s41598-018-29405-7|pmid=30038426|bibcode=2018NatSR...811077K|issn=2045-2322|doi-access=free}}</ref>

According to one study, perception of interpersonal similarity improves coordination and increase the expected payoff of interactions, above and beyond the effect of merely "liking others." Another study claims that homophily produces tolerance and cooperation in social spaces. However, homophilic patterns can also restrict access to information or inclusion for minorities.

根据一项研究，人际相似性感知可以改善协调性，提高互动的预期回报，远远超过仅仅“喜欢他人”的效果另一项研究声称，同质性在社会空间产生宽容与合作。然而，同源模式也会限制少数族裔获取信息或包容性。

The effects of homophily on the diffusion of information and behaviors are also complex. Some studies have claimed that homophily facilitates [[Access to information|access information]],<ref>{{cite arXiv|last1=Choudhury|first1=M. D|title="Birds of a feather": Does user homophily impact information diffusion in social media |date=2010 |eprint=1006.1702|class=cs.CY}}</ref> the diffusion of innovations and behaviors,<ref>{{cite journal|last1=Christakis|first1=N. A|last2=Fowler|first2=J. H.|title=The spread of obesity in a large social network over 32 years|journal=The New England Journal of Medicine|date=2007|volume=357|issue=4|pages=370–379|url=http://media.timesfreepress.com/docs/2010/01/Obesity_studies_0104.pdf|doi=10.1056/nejmsa066082|pmid=17652652|citeseerx=10.1.1.581.4893}}</ref> and the formation of [[Social norm|social norms]].<ref>{{cite journal|last1=Centola|first1=D|last2=R|first2=Willer|last3=M|first3=Macy|title=The emperor's dilemma: A computational model of self-enforcing norms. |journal=American Journal of Sociology|date=2005|volume=110|issue=4|pages=1009–1040|jstor=10.1086/427321|doi=10.1086/427321|url=https://repository.upenn.edu/cgi/viewcontent.cgi?article=1604&context=asc_papers}}</ref> Other studies, however, highlight mechanisms through which homophily can maintain disagreement, exacerbate [[Social polarization|polarization]] of opinions, and slow the formation of an overall [[Consensus decision-making|consensus]].<ref name=":1" /><ref>{{Cite journal|last=Centola D, Gonzalez-Avella JC, Eguiluz VM, San Miguel M.|first=|date=2007|title=Homophily, cultural drift, and the co-evolution of cultural groups|url=|journal=Journal of Conflict Resolution|volume=51(6)|pages=905–29|via=}}</ref>

The effects of homophily on the diffusion of information and behaviors are also complex. Some studies have claimed that homophily facilitates access information, the diffusion of innovations and behaviors, and the formation of social norms. Other studies, however, highlight mechanisms through which homophily can maintain disagreement, exacerbate polarization of opinions, and slow the formation of an overall consensus.

同质性对信息和行为扩散的影响也是复杂的。一些研究声称，同质性促进了信息的获取，创新产品渗透理论和行为，以及社会规范的形成。然而，其他的研究强调了同质性可以维持分歧，加剧意见的两极分化，并减缓整体共识形成的机制。

Homophily is a cause of [[Homogamy (sociology)|homogamy]]—marriage between people with similar characteristics.<ref name="fiore">Fiore, A. T. and Donath, J. S. (2005). [http://smg.media.mit.edu/papers/Fiore/fiore_donath_chi2005_short.pdf "Homophily in Online Dating: When Do You Like Someone Like Yourself?"]. ''MIT Media Lab''.</ref> Homophily is a [[fertility factor (demography)|fertility factor]]; an increased [[fertility]] is seen in people with a tendency to seek acquaintance among those with common characteristics.<ref name="fent2013" /> Governmental family policies have a decreased influence on [[Total fertility rate|fertility rates]] in such populations.<ref name="fent2013">{{cite journal|title=Family policies in the context of low fertility and social structure|url=http://www.demographic-research.org/Volumes/Vol29/37/|journal=Demographic Research|volume=29|issue=37|author1=Thomas Fent |author2=Belinda Aparicio Diaz |author3=Alexia Prskawetz |year=2013}}</ref>

Homophily is a cause of homogamy—marriage between people with similar characteristics. Homophily is a fertility factor; an increased fertility is seen in people with a tendency to seek acquaintance among those with common characteristics.

同质性是同质婚姻的一个原因ーー具有相似特征的人之间的婚姻。同质性是一种 F质粒，人们倾向于在具有共同特征的人群中寻求熟人，这种倾向会导致生育力的提高。

==See also==

*[[Groupthink]]

==References==

{{reflist|30em}}

{{World view}}

[[Category:Interpersonal relationships]]

Category:Interpersonal relationships

类别: 人际关系

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Homophily]]. Its edit history can be viewed at [[同质性/edithistory]]</noinclude>

[[Category:待整理页面]]

三元闭包

2020-09-25T11:36:55Z

923397935：/* Strong Triadic Closure Property and local bridges */

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本词条由信白初步翻译 

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.

'''三合闭包 Triadic Closure '''是'''社会网络理论 Social Network Theory '''中的一个概念，最早由德国社会学家格奥尔格·西梅尔（Georg Simmel）在其1908年的著作《社会学：社会形式的调查》中提出。三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。这个属性太极端了，无法在非常大的'''复杂网络 Complex Network '''中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

==History==
历史 

Triadic closure was made popular by [[Mark Granovetter]] in his 1973 article ''The Strength of Weak Ties''.<ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> There he synthesized the theory of [[cognitive balance]] first introduced by [[Fritz Heider]] in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱链接的力量》（The Strength of Weak Ties）中使三元闭包变得流行。在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的'''认知平衡理论 The Theory Of Cognitive Balance '''，该理论对Simmelian的社交网络有所了解。一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。如果没有关闭三个人的闭合，那么连接到两个人的人都将要关闭，以便在关系网络中实现闭合。

==Measurements==
测量 

{{Unreferenced section|date=September 2009}}

The two most common measures of triadic closure for a graph are (in no particular order) the [[clustering coefficient]] and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一张图的三元闭合的两个最常见的度量是（不按特定顺序）该图的'''聚类系数 Clustering Coefficient '''和'''可传递性 Transitivity '''。

==Clustering coefficient==
聚类系数 

One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

衡量三元闭包的一种方法是聚类系数，如下所示：

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

令<math>G =（V，E）</math>是无向的'''简单图 Simple Graph '''（即没有自环或多个边的图），其中V为顶点集，E为边集。另外，令<math>N = |V|</math>和<math>M = |E|</math>分别表示G中顶点和边的数量，并令<math>d_i</math> 是顶点的度i。

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

我们可以在三个顶点<math>i</math>，<math>j</math>和<math>k</math>中定义一个三角形，以使其具有以下三个边的集合：{（i ，j），（j，k），（i，k）}。

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

我们也可以将顶点<math>i</math>所涉及的三角形的数量定义为<math>\delta（i）</math>，并且，由于每个三角形都被计数了三次，因此我们可以表示 G中的三角形为<math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>。

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

假设三元闭包成立，则一个三元组的形成仅需要两个牢固的边缘。因此，在顶点<math>i</math>的三元组关闭假设下应该出现的理论三元组的数量为<math>\tau（i）= \binom{d_i}{2}</math>，假设<math>d_i \ge 2</math>。我们可以表示<math>\tau（G）= \frac{1}{3} \sum_{i\in V} \ \tau（i）</math>。

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于具有<math>d_i \ge 2</math>的顶点<math>i</math>，顶点<math>i</math>的聚类系数<math>c（i)</math> 是封闭的顶点<math>i</math>的三元组分数，可以测量为<math>\frac{\delta（i）}{\tau（i）}</math>。因此，图<math> G </math>的聚类系数<math> C（G）</math>由<math> C（G）=\frac {1}{N_2} \sum_{i \in V,d_i \ge 2}c（i）</math>，其中<math>N_2</math>是度数至少为2的节点数。

==Transitivity==
传递性 

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

关于三元闭包的另一种度量是可传递性，定义为<math>T（G）= \frac{3\delta（G）}{\tau（G）}</math>。

==Causes and effects==
'''因果 Causes And Effects ''' 

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

在信任网络中，由于传递属性，可能会发生三元闭包。如果节点A信任节点B，并且节点B信任节点C，则节点A将具有信任节点C的基础。在社交网络中，会发生强烈的三元闭包，因为具有共同邻居B的节点A和C拥有更多的机会来满足并因此建立起相对薄弱的纽带。节点B还具有将A和C聚在一起以减少两个单独关系中的潜在压力的动机。

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

遵循此原理的网络将高度互连，并且具有很高的聚类系数。但是，不遵循该原理的网络将连接不良，一旦包含负面关系，网络可能会变得不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.<ref name=Easley/>

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

三合闭包是网络如何随着时间演变的良好模型。尽管简单图论倾向于在某个时间点分析网络，但应用三元闭包原理可以预测网络内联系的发展并显示连通性的发展。

In [[social networks]], triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三合闭包有利于合作行为，但是当通过
referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

从现有联系中推荐人时，合作者的平均全球比例要小于个人从总体人口中随机选择新联系时的比例。对此的两个可能的影响是结构和信息结构。结构构造来自于倾向于高度可聚性的倾向。信息结构来自这样一个假设：与随机的陌生人不同，一个人对朋友的朋友有所了解。

==Strong Triadic Closure Property and local bridges==
'''强三元闭包性 Strong Triadic Closure Property '''和'''本地网桥 Local Bridges ''' 

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A [[local bridge]] occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性是，如果一个节点与两个邻居有牢固的联系，则这些邻居之间必须至少有一条弱联系。另一方面，当节点充当两个其他未连接的相邻节点之间的网守时，则发生本地网桥。在遵循“强三元闭合性”的网络中，本地网桥中涉及的节点之间的联系之一必须是弱联系。

==Proof by contradiction==
'''矛盾证明 Proof By Contradiction ''' 

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.<ref name=Easley/>

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

假设节点B是节点A和C之间的本地桥梁，以使所涉及的节点之间没有弱联系。因此，B与A和C都具有牢固的联系。根据“强三元闭包性”的定义，节点A和C之间将形成弱联系。但是，这与B是本地网守的事实相矛盾。因此，本地网桥中涉及的至少一个节点需要是弱连接，以防止三元闭包的发生。

==References==

{{Reflist}}

{{Social networking}}

{{DEFAULTSORT:Triadic Closure}}

[[Category:Social systems]]

Category:Social systems

类别: 社会系统

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Triadic closure]]. Its edit history can be viewed at [[三元闭包/edithistory]]</noinclude>

[[Category:待整理页面]]

三元闭包

2020-09-25T11:31:20Z

923397935：/* Causes and effects */

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'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.

'''三合闭包 Triadic Closure '''是'''社会网络理论 Social Network Theory '''中的一个概念，最早由德国社会学家格奥尔格·西梅尔（Georg Simmel）在其1908年的著作《社会学：社会形式的调查》中提出。三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。这个属性太极端了，无法在非常大的'''复杂网络 Complex Network '''中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

==History==
历史 

Triadic closure was made popular by [[Mark Granovetter]] in his 1973 article ''The Strength of Weak Ties''.<ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> There he synthesized the theory of [[cognitive balance]] first introduced by [[Fritz Heider]] in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱链接的力量》（The Strength of Weak Ties）中使三元闭包变得流行。在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的'''认知平衡理论 The Theory Of Cognitive Balance '''，该理论对Simmelian的社交网络有所了解。一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。如果没有关闭三个人的闭合，那么连接到两个人的人都将要关闭，以便在关系网络中实现闭合。

==Measurements==
测量 

{{Unreferenced section|date=September 2009}}

The two most common measures of triadic closure for a graph are (in no particular order) the [[clustering coefficient]] and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一张图的三元闭合的两个最常见的度量是（不按特定顺序）该图的'''聚类系数 Clustering Coefficient '''和'''可传递性 Transitivity '''。

==Clustering coefficient==
聚类系数 

One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

衡量三元闭包的一种方法是聚类系数，如下所示：

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

令<math>G =（V，E）</math>是无向的'''简单图 Simple Graph '''（即没有自环或多个边的图），其中V为顶点集，E为边集。另外，令<math>N = |V|</math>和<math>M = |E|</math>分别表示G中顶点和边的数量，并令<math>d_i</math> 是顶点的度i。

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

我们可以在三个顶点<math>i</math>，<math>j</math>和<math>k</math>中定义一个三角形，以使其具有以下三个边的集合：{（i ，j），（j，k），（i，k）}。

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

我们也可以将顶点<math>i</math>所涉及的三角形的数量定义为<math>\delta（i）</math>，并且，由于每个三角形都被计数了三次，因此我们可以表示 G中的三角形为<math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>。

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

假设三元闭包成立，则一个三元组的形成仅需要两个牢固的边缘。因此，在顶点<math>i</math>的三元组关闭假设下应该出现的理论三元组的数量为<math>\tau（i）= \binom{d_i}{2}</math>，假设<math>d_i \ge 2</math>。我们可以表示<math>\tau（G）= \frac{1}{3} \sum_{i\in V} \ \tau（i）</math>。

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于具有<math>d_i \ge 2</math>的顶点<math>i</math>，顶点<math>i</math>的聚类系数<math>c（i)</math> 是封闭的顶点<math>i</math>的三元组分数，可以测量为<math>\frac{\delta（i）}{\tau（i）}</math>。因此，图<math> G </math>的聚类系数<math> C（G）</math>由<math> C（G）=\frac {1}{N_2} \sum_{i \in V,d_i \ge 2}c（i）</math>，其中<math>N_2</math>是度数至少为2的节点数。

==Transitivity==
传递性 

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

关于三元闭包的另一种度量是可传递性，定义为<math>T（G）= \frac{3\delta（G）}{\tau（G）}</math>。

==Causes and effects==
'''因果 Causes And Effects ''' 

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

在信任网络中，由于传递属性，可能会发生三元闭包。如果节点A信任节点B，并且节点B信任节点C，则节点A将具有信任节点C的基础。在社交网络中，会发生强烈的三元闭包，因为具有共同邻居B的节点A和C拥有更多的机会来满足并因此建立起相对薄弱的纽带。节点B还具有将A和C聚在一起以减少两个单独关系中的潜在压力的动机。

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

遵循此原理的网络将高度互连，并且具有很高的聚类系数。但是，不遵循该原理的网络将连接不良，一旦包含负面关系，网络可能会变得不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.<ref name=Easley/>

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

三合闭包是网络如何随着时间演变的良好模型。尽管简单图论倾向于在某个时间点分析网络，但应用三元闭包原理可以预测网络内联系的发展并显示连通性的发展。

In [[social networks]], triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三合闭包有利于合作行为，但是当通过
referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

从现有联系中推荐人时，合作者的平均全球比例要小于个人从总体人口中随机选择新联系时的比例。对此的两个可能的影响是结构和信息结构。结构构造来自于倾向于高度可聚性的倾向。信息结构来自这样一个假设：与随机的陌生人不同，一个人对朋友的朋友有所了解。

==Strong Triadic Closure Property and local bridges==

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A [[local bridge]] occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性质是指，如果一个节点与两个邻居之间有强关联，那么这两个邻居之间至少有弱关联。另一方面，当一个节点在两个没有连接的相邻节点之间充当网守时，就会出现本地桥接。在遵循强三元闭包性质的网络中，局部桥梁中涉及的节点之间的联系之一需要是弱联系。

===Proof by contradiction===

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.<ref name=Easley/>

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

设 b 节点是 a 节点和 c 节点之间的本地桥梁，这样所涉及的节点之间就不存在弱联系。因此，b 与 a 和 c 都有很强的联系。根据强三元闭包的定义，在节点 a 和 c 之间会产生弱联系。因此，局部桥梁中至少有一个节点需要是弱连接，以防止三分闭合发生。

==References==

{{Reflist}}

{{Social networking}}

{{DEFAULTSORT:Triadic Closure}}

[[Category:Social systems]]

Category:Social systems

类别: 社会系统

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Triadic closure]]. Its edit history can be viewed at [[三元闭包/edithistory]]</noinclude>

[[Category:待整理页面]]

三元闭包

2020-09-25T11:29:37Z

923397935：/* Clustering coefficient */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.

'''三合闭包 Triadic Closure '''是'''社会网络理论 Social Network Theory '''中的一个概念，最早由德国社会学家格奥尔格·西梅尔（Georg Simmel）在其1908年的著作《社会学：社会形式的调查》中提出。三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。这个属性太极端了，无法在非常大的'''复杂网络 Complex Network '''中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

==History==
历史 

Triadic closure was made popular by [[Mark Granovetter]] in his 1973 article ''The Strength of Weak Ties''.<ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> There he synthesized the theory of [[cognitive balance]] first introduced by [[Fritz Heider]] in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱链接的力量》（The Strength of Weak Ties）中使三元闭包变得流行。在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的'''认知平衡理论 The Theory Of Cognitive Balance '''，该理论对Simmelian的社交网络有所了解。一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。如果没有关闭三个人的闭合，那么连接到两个人的人都将要关闭，以便在关系网络中实现闭合。

==Measurements==
测量 

{{Unreferenced section|date=September 2009}}

The two most common measures of triadic closure for a graph are (in no particular order) the [[clustering coefficient]] and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一张图的三元闭合的两个最常见的度量是（不按特定顺序）该图的'''聚类系数 Clustering Coefficient '''和'''可传递性 Transitivity '''。

==Clustering coefficient==
聚类系数 

One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

衡量三元闭包的一种方法是聚类系数，如下所示：

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

令<math>G =（V，E）</math>是无向的'''简单图 Simple Graph '''（即没有自环或多个边的图），其中V为顶点集，E为边集。另外，令<math>N = |V|</math>和<math>M = |E|</math>分别表示G中顶点和边的数量，并令<math>d_i</math> 是顶点的度i。

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

我们可以在三个顶点<math>i</math>，<math>j</math>和<math>k</math>中定义一个三角形，以使其具有以下三个边的集合：{（i ，j），（j，k），（i，k）}。

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

我们也可以将顶点<math>i</math>所涉及的三角形的数量定义为<math>\delta（i）</math>，并且，由于每个三角形都被计数了三次，因此我们可以表示 G中的三角形为<math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>。

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

假设三元闭包成立，则一个三元组的形成仅需要两个牢固的边缘。因此，在顶点<math>i</math>的三元组关闭假设下应该出现的理论三元组的数量为<math>\tau（i）= \binom{d_i}{2}</math>，假设<math>d_i \ge 2</math>。我们可以表示<math>\tau（G）= \frac{1}{3} \sum_{i\in V} \ \tau（i）</math>。

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于具有<math>d_i \ge 2</math>的顶点<math>i</math>，顶点<math>i</math>的聚类系数<math>c（i)</math> 是封闭的顶点<math>i</math>的三元组分数，可以测量为<math>\frac{\delta（i）}{\tau（i）}</math>。因此，图<math> G </math>的聚类系数<math> C（G）</math>由<math> C（G）=\frac {1}{N_2} \sum_{i \in V,d_i \ge 2}c（i）</math>，其中<math>N_2</math>是度数至少为2的节点数。

==Transitivity==
传递性 

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

关于三元闭包的另一种度量是可传递性，定义为<math>T（G）= \frac{3\delta（G）}{\tau（G）}</math>。

==Causes and effects==
'''因果 Causes And Effects ''' 

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

在信任网络中，由于传递属性，可能会发生三元闭包。如果节点A信任节点B，并且节点B信任节点C，则节点A将具有信任节点C的基础。在社交网络中，会发生强烈的三元闭包，因为具有共同邻居B的节点A和C拥有更多的机会来满足并因此建立起相对薄弱的纽带。节点B还具有将A和C聚在一起以减少两个单独关系中的潜在压力的动机。

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

遵循此原理的网络将高度互连，并且具有很高的群集系数。但是，不遵循该原理的网络连接不良，一旦包含负面关系，网络可能会变得不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.<ref name=Easley/>

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

三元闭包是一个很好的模型，用来说明网络将如何随着时间的推移而演化。简单的图论倾向于在一个时间点上分析网络，而应用三并元闭包原理可以预测网络中关系的发展，并显示连通性的进展。

In [[social networks]], triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三元闭包促进了合作行为，但是当新的连接是通过

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

根据现有关系的推荐，合作者的全球平均比例比个人从总体人口中随机选择新关系的比例要低。两个可能的影响是通过结构和信息化建设。结构构造起源于向高集聚性的倾向。这种信息结构来自于这样一种假设，即一个人知道一个朋友的朋友的某些事情，而不是一个随机的陌生人。

==Strong Triadic Closure Property and local bridges==

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A [[local bridge]] occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性质是指，如果一个节点与两个邻居之间有强关联，那么这两个邻居之间至少有弱关联。另一方面，当一个节点在两个没有连接的相邻节点之间充当网守时，就会出现本地桥接。在遵循强三元闭包性质的网络中，局部桥梁中涉及的节点之间的联系之一需要是弱联系。

===Proof by contradiction===

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.<ref name=Easley/>

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

设 b 节点是 a 节点和 c 节点之间的本地桥梁，这样所涉及的节点之间就不存在弱联系。因此，b 与 a 和 c 都有很强的联系。根据强三元闭包的定义，在节点 a 和 c 之间会产生弱联系。因此，局部桥梁中至少有一个节点需要是弱连接，以防止三分闭合发生。

==References==

{{Reflist}}

{{Social networking}}

{{DEFAULTSORT:Triadic Closure}}

[[Category:Social systems]]

Category:Social systems

类别: 社会系统

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Triadic closure]]. Its edit history can be viewed at [[三元闭包/edithistory]]</noinclude>

[[Category:待整理页面]]

三元闭包

2020-09-25T11:29:15Z

923397935：/* Causes and effects */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.

'''三合闭包 Triadic Closure '''是'''社会网络理论 Social Network Theory '''中的一个概念，最早由德国社会学家格奥尔格·西梅尔（Georg Simmel）在其1908年的著作《社会学：社会形式的调查》中提出。三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。这个属性太极端了，无法在非常大的'''复杂网络 Complex Network '''中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

==History==
历史 

Triadic closure was made popular by [[Mark Granovetter]] in his 1973 article ''The Strength of Weak Ties''.<ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> There he synthesized the theory of [[cognitive balance]] first introduced by [[Fritz Heider]] in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱链接的力量》（The Strength of Weak Ties）中使三元闭包变得流行。在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的'''认知平衡理论 The Theory Of Cognitive Balance '''，该理论对Simmelian的社交网络有所了解。一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。如果没有关闭三个人的闭合，那么连接到两个人的人都将要关闭，以便在关系网络中实现闭合。

==Measurements==
测量 

{{Unreferenced section|date=September 2009}}

The two most common measures of triadic closure for a graph are (in no particular order) the [[clustering coefficient]] and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一张图的三元闭合的两个最常见的度量是（不按特定顺序）该图的'''聚类系数 Clustering Coefficient '''和'''可传递性 Transitivity '''。

==Clustering coefficient==
聚类系数
One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

衡量三元闭包的一种方法是聚类系数，如下所示：

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

令<math>G =（V，E）</math>是无向的'''简单图 Simple Graph '''（即没有自环或多个边的图），其中V为顶点集，E为边集。另外，令<math>N = |V|</math>和<math>M = |E|</math>分别表示G中顶点和边的数量，并令<math>d_i</math> 是顶点的度i。

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

我们可以在三个顶点<math>i</math>，<math>j</math>和<math>k</math>中定义一个三角形，以使其具有以下三个边的集合：{（i ，j），（j，k），（i，k）}。

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

我们也可以将顶点<math>i</math>所涉及的三角形的数量定义为<math>\delta（i）</math>，并且，由于每个三角形都被计数了三次，因此我们可以表示 G中的三角形为<math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>。

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

假设三元闭包成立，则一个三元组的形成仅需要两个牢固的边缘。因此，在顶点<math>i</math>的三元组关闭假设下应该出现的理论三元组的数量为<math>\tau（i）= \binom{d_i}{2}</math>，假设<math>d_i \ge 2</math>。我们可以表示<math>\tau（G）= \frac{1}{3} \sum_{i\in V} \ \tau（i）</math>。

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于具有<math>d_i \ge 2</math>的顶点<math>i</math>，顶点<math>i</math>的聚类系数<math>c（i)</math> 是封闭的顶点<math>i</math>的三元组分数，可以测量为<math>\frac{\delta（i）}{\tau（i）}</math>。因此，图<math> G </math>的聚类系数<math> C（G）</math>由<math> C（G）=\frac {1}{N_2} \sum_{i \in V,d_i \ge 2}c（i）</math>，其中<math>N_2</math>是度数至少为2的节点数。

==Transitivity==
传递性 

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

关于三元闭包的另一种度量是可传递性，定义为<math>T（G）= \frac{3\delta（G）}{\tau（G）}</math>。

==Causes and effects==
'''因果 Causes And Effects ''' 

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

在信任网络中，由于传递属性，可能会发生三元闭包。如果节点A信任节点B，并且节点B信任节点C，则节点A将具有信任节点C的基础。在社交网络中，会发生强烈的三元闭包，因为具有共同邻居B的节点A和C拥有更多的机会来满足并因此建立起相对薄弱的纽带。节点B还具有将A和C聚在一起以减少两个单独关系中的潜在压力的动机。

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

遵循此原理的网络将高度互连，并且具有很高的群集系数。但是，不遵循该原理的网络连接不良，一旦包含负面关系，网络可能会变得不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.<ref name=Easley/>

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

三元闭包是一个很好的模型，用来说明网络将如何随着时间的推移而演化。简单的图论倾向于在一个时间点上分析网络，而应用三并元闭包原理可以预测网络中关系的发展，并显示连通性的进展。

In [[social networks]], triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三元闭包促进了合作行为，但是当新的连接是通过

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

根据现有关系的推荐，合作者的全球平均比例比个人从总体人口中随机选择新关系的比例要低。两个可能的影响是通过结构和信息化建设。结构构造起源于向高集聚性的倾向。这种信息结构来自于这样一种假设，即一个人知道一个朋友的朋友的某些事情，而不是一个随机的陌生人。

==Strong Triadic Closure Property and local bridges==

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A [[local bridge]] occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性质是指，如果一个节点与两个邻居之间有强关联，那么这两个邻居之间至少有弱关联。另一方面，当一个节点在两个没有连接的相邻节点之间充当网守时，就会出现本地桥接。在遵循强三元闭包性质的网络中，局部桥梁中涉及的节点之间的联系之一需要是弱联系。

===Proof by contradiction===

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.<ref name=Easley/>

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

设 b 节点是 a 节点和 c 节点之间的本地桥梁，这样所涉及的节点之间就不存在弱联系。因此，b 与 a 和 c 都有很强的联系。根据强三元闭包的定义，在节点 a 和 c 之间会产生弱联系。因此，局部桥梁中至少有一个节点需要是弱连接，以防止三分闭合发生。

==References==

{{Reflist}}

{{Social networking}}

{{DEFAULTSORT:Triadic Closure}}

[[Category:Social systems]]

Category:Social systems

类别: 社会系统

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Triadic closure]]. Its edit history can be viewed at [[三元闭包/edithistory]]</noinclude>

[[Category:待整理页面]]

三元闭包

2020-09-25T11:26:49Z

923397935：/* Clustering coefficient */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.

'''三合闭包 Triadic Closure '''是'''社会网络理论 Social Network Theory '''中的一个概念，最早由德国社会学家格奥尔格·西梅尔（Georg Simmel）在其1908年的著作《社会学：社会形式的调查》中提出。三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。这个属性太极端了，无法在非常大的'''复杂网络 Complex Network '''中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

==History==
历史 

Triadic closure was made popular by [[Mark Granovetter]] in his 1973 article ''The Strength of Weak Ties''.<ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> There he synthesized the theory of [[cognitive balance]] first introduced by [[Fritz Heider]] in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱链接的力量》（The Strength of Weak Ties）中使三元闭包变得流行。在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的'''认知平衡理论 The Theory Of Cognitive Balance '''，该理论对Simmelian的社交网络有所了解。一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。如果没有关闭三个人的闭合，那么连接到两个人的人都将要关闭，以便在关系网络中实现闭合。

==Measurements==
测量 

{{Unreferenced section|date=September 2009}}

The two most common measures of triadic closure for a graph are (in no particular order) the [[clustering coefficient]] and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一张图的三元闭合的两个最常见的度量是（不按特定顺序）该图的'''聚类系数 Clustering Coefficient '''和'''可传递性 Transitivity '''。

==Clustering coefficient==
聚类系数
One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

衡量三元闭包的一种方法是聚类系数，如下所示：

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

令<math>G =（V，E）</math>是无向的'''简单图 Simple Graph '''（即没有自环或多个边的图），其中V为顶点集，E为边集。另外，令<math>N = |V|</math>和<math>M = |E|</math>分别表示G中顶点和边的数量，并令<math>d_i</math> 是顶点的度i。

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

我们可以在三个顶点<math>i</math>，<math>j</math>和<math>k</math>中定义一个三角形，以使其具有以下三个边的集合：{（i ，j），（j，k），（i，k）}。

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

我们也可以将顶点<math>i</math>所涉及的三角形的数量定义为<math>\delta（i）</math>，并且，由于每个三角形都被计数了三次，因此我们可以表示 G中的三角形为<math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>。

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

假设三元闭包成立，则一个三元组的形成仅需要两个牢固的边缘。因此，在顶点<math>i</math>的三元组关闭假设下应该出现的理论三元组的数量为<math>\tau（i）= \binom{d_i}{2}</math>，假设<math>d_i \ge 2</math>。我们可以表示<math>\tau（G）= \frac{1}{3} \sum_{i\in V} \ \tau（i）</math>。

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于具有<math>d_i \ge 2</math>的顶点<math>i</math>，顶点<math>i</math>的聚类系数<math>c（i)</math> 是封闭的顶点<math>i</math>的三元组分数，可以测量为<math>\frac{\delta（i）}{\tau（i）}</math>。因此，图<math> G </math>的聚类系数<math> C（G）</math>由<math> C（G）=\frac {1}{N_2} \sum_{i \in V,d_i \ge 2}c（i）</math>，其中<math>N_2</math>是度数至少为2的节点数。

==Transitivity==
传递性 

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

关于三元闭包的另一种度量是可传递性，定义为<math>T（G）= \frac{3\delta（G）}{\tau（G）}</math>。

==Causes and effects==

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

在信任网络中，由于三元闭包具有传递性，容易产生三元闭包。如果节点 a 信任节点 b，而节点 b 信任节点 c，则节点 a 将有信任节点 c 的基础。在社交网络中，强三元闭包的出现是因为与共同邻居 b 相遇的节点 a 和 c 有更多的机会相遇，从而产生至少个弱关系。节点 b 也有动机将 a 和 c 结合在一起，以减少两个独立关系中的潜在压力。

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

遵循这一原则的网络变得高度互联，并具有非常高的聚类系数。然而，不遵循这一原则的网络结果是连接不良，一旦包括负面关系，就可能遭受不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.<ref name=Easley/>

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

三元闭包是一个很好的模型，用来说明网络将如何随着时间的推移而演化。简单的图论倾向于在一个时间点上分析网络，而应用三并元闭包原理可以预测网络中关系的发展，并显示连通性的进展。

In [[social networks]], triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三元闭包促进了合作行为，但是当新的连接是通过

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

根据现有关系的推荐，合作者的全球平均比例比个人从总体人口中随机选择新关系的比例要低。两个可能的影响是通过结构和信息化建设。结构构造起源于向高集聚性的倾向。这种信息结构来自于这样一种假设，即一个人知道一个朋友的朋友的某些事情，而不是一个随机的陌生人。

==Strong Triadic Closure Property and local bridges==

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A [[local bridge]] occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性质是指，如果一个节点与两个邻居之间有强关联，那么这两个邻居之间至少有弱关联。另一方面，当一个节点在两个没有连接的相邻节点之间充当网守时，就会出现本地桥接。在遵循强三元闭包性质的网络中，局部桥梁中涉及的节点之间的联系之一需要是弱联系。

===Proof by contradiction===

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.<ref name=Easley/>

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

设 b 节点是 a 节点和 c 节点之间的本地桥梁，这样所涉及的节点之间就不存在弱联系。因此，b 与 a 和 c 都有很强的联系。根据强三元闭包的定义，在节点 a 和 c 之间会产生弱联系。因此，局部桥梁中至少有一个节点需要是弱连接，以防止三分闭合发生。

==References==

{{Reflist}}

{{Social networking}}

{{DEFAULTSORT:Triadic Closure}}

[[Category:Social systems]]

Category:Social systems

类别: 社会系统

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Triadic closure]]. Its edit history can be viewed at [[三元闭包/edithistory]]</noinclude>

[[Category:待整理页面]]

三元闭包

2020-09-25T11:25:57Z

923397935：/* Clustering coefficient */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.

'''三合闭包 Triadic Closure '''是'''社会网络理论 Social Network Theory '''中的一个概念，最早由德国社会学家格奥尔格·西梅尔（Georg Simmel）在其1908年的著作《社会学：社会形式的调查》中提出。三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。这个属性太极端了，无法在非常大的'''复杂网络 Complex Network '''中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

==History==
历史 

Triadic closure was made popular by [[Mark Granovetter]] in his 1973 article ''The Strength of Weak Ties''.<ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> There he synthesized the theory of [[cognitive balance]] first introduced by [[Fritz Heider]] in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱链接的力量》（The Strength of Weak Ties）中使三元闭包变得流行。在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的'''认知平衡理论 The Theory Of Cognitive Balance '''，该理论对Simmelian的社交网络有所了解。一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。如果没有关闭三个人的闭合，那么连接到两个人的人都将要关闭，以便在关系网络中实现闭合。

==Measurements==
测量 

{{Unreferenced section|date=September 2009}}

The two most common measures of triadic closure for a graph are (in no particular order) the [[clustering coefficient]] and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一张图的三元闭合的两个最常见的度量是（不按特定顺序）该图的'''聚类系数 Clustering Coefficient '''和'''可传递性 Transitivity '''。

==Clustering coefficient==
聚类系数
One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

衡量三元闭包的一种方法是聚类系数，如下所示：

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

令<math>G =（V，E）</math>是无向的'''简单图 Simple Graph '''（即没有自环或多个边的图），其中V为顶点集，E为边集。另外，令<math>N = |V|</math>和<math>M = |E|</math>分别表示G中顶点和边的数量，并令<math>d_i</math> 是顶点的度i。

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

我们可以在三个顶点<math>i</math>，<math>j</math>和<math>k</math>中定义一个三角形，以使其具有以下三个边的集合：{（i ，j），（j，k），（i，k）}。

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

我们也可以将顶点<math>i</math>所涉及的三角形的数量定义为<math>\delta（i）</math>，并且，由于每个三角形都被计数了三次，因此我们可以表示 G中的三角形为<math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>。

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

假设三元闭包成立，则一个三元组的形成仅需要两个牢固的边缘。因此，在顶点<math>i</math>的三元组关闭假设下应该出现的理论三元组的数量为<math>\tau（i）= \binom{d_i}{2}</math>，假设<math>d_i \ge 2</math>。我们可以表示<math>\tau（G）= \frac{1}{3} \sum_{i\in V} \ \tau（i）</math>。

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于具有<math>d_i\ge 2</math>的顶点<math>i</ math>，顶点<math>i</math>的聚类系数<math>c（i)</math> 是封闭的顶点<math>i</math>的三元组分数，可以测量为<math>\frac{\delta（i）}{\tau（i）}</math>。因此，图<math> G </math>的聚类系数<math> C（G）</math>由<math> C（G）=\frac {1}{N_2} \sum_{i \in V,d_i \ge 2}c（i）</math>，其中<math>N_2</math>是度数至少为2的节点数。

==Transitivity==
传递性 

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

关于三元闭包的另一种度量是可传递性，定义为<math>T（G）= \frac{3\delta（G）}{\tau（G）}</math>。

==Causes and effects==

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

在信任网络中，由于三元闭包具有传递性，容易产生三元闭包。如果节点 a 信任节点 b，而节点 b 信任节点 c，则节点 a 将有信任节点 c 的基础。在社交网络中，强三元闭包的出现是因为与共同邻居 b 相遇的节点 a 和 c 有更多的机会相遇，从而产生至少个弱关系。节点 b 也有动机将 a 和 c 结合在一起，以减少两个独立关系中的潜在压力。

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

遵循这一原则的网络变得高度互联，并具有非常高的聚类系数。然而，不遵循这一原则的网络结果是连接不良，一旦包括负面关系，就可能遭受不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.<ref name=Easley/>

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

三元闭包是一个很好的模型，用来说明网络将如何随着时间的推移而演化。简单的图论倾向于在一个时间点上分析网络，而应用三并元闭包原理可以预测网络中关系的发展，并显示连通性的进展。

In [[social networks]], triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三元闭包促进了合作行为，但是当新的连接是通过

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

根据现有关系的推荐，合作者的全球平均比例比个人从总体人口中随机选择新关系的比例要低。两个可能的影响是通过结构和信息化建设。结构构造起源于向高集聚性的倾向。这种信息结构来自于这样一种假设，即一个人知道一个朋友的朋友的某些事情，而不是一个随机的陌生人。

==Strong Triadic Closure Property and local bridges==

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A [[local bridge]] occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性质是指，如果一个节点与两个邻居之间有强关联，那么这两个邻居之间至少有弱关联。另一方面，当一个节点在两个没有连接的相邻节点之间充当网守时，就会出现本地桥接。在遵循强三元闭包性质的网络中，局部桥梁中涉及的节点之间的联系之一需要是弱联系。

===Proof by contradiction===

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.<ref name=Easley/>

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

设 b 节点是 a 节点和 c 节点之间的本地桥梁，这样所涉及的节点之间就不存在弱联系。因此，b 与 a 和 c 都有很强的联系。根据强三元闭包的定义，在节点 a 和 c 之间会产生弱联系。因此，局部桥梁中至少有一个节点需要是弱连接，以防止三分闭合发生。

==References==

{{Reflist}}

{{Social networking}}

{{DEFAULTSORT:Triadic Closure}}

[[Category:Social systems]]

Category:Social systems

类别: 社会系统

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Triadic closure]]. Its edit history can be viewed at [[三元闭包/edithistory]]</noinclude>

[[Category:待整理页面]]

三元闭包

2020-09-25T11:24:34Z

923397935：/* Clustering coefficient */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.

'''三合闭包 Triadic Closure '''是'''社会网络理论 Social Network Theory '''中的一个概念，最早由德国社会学家格奥尔格·西梅尔（Georg Simmel）在其1908年的著作《社会学：社会形式的调查》中提出。三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。这个属性太极端了，无法在非常大的'''复杂网络 Complex Network '''中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

==History==
历史 

Triadic closure was made popular by [[Mark Granovetter]] in his 1973 article ''The Strength of Weak Ties''.<ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> There he synthesized the theory of [[cognitive balance]] first introduced by [[Fritz Heider]] in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱链接的力量》（The Strength of Weak Ties）中使三元闭包变得流行。在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的'''认知平衡理论 The Theory Of Cognitive Balance '''，该理论对Simmelian的社交网络有所了解。一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。如果没有关闭三个人的闭合，那么连接到两个人的人都将要关闭，以便在关系网络中实现闭合。

==Measurements==
测量 

{{Unreferenced section|date=September 2009}}

The two most common measures of triadic closure for a graph are (in no particular order) the [[clustering coefficient]] and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一张图的三元闭合的两个最常见的度量是（不按特定顺序）该图的'''聚类系数 Clustering Coefficient '''和'''可传递性 Transitivity '''。

==Clustering coefficient==
聚类系数
One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

衡量三元闭包的一种方法是聚类系数，如下所示：

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

令<math>G =（V，E）</math>是无向的'''简单图 Simple Graph '''（即没有自环或多个边的图），其中V为顶点集，E为边集。另外，令<math>N = |V|</math>和<math>M = |E|</math>分别表示G中顶点和边的数量，并令<math>d_i</math> 是顶点的度i。

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

我们可以在三个顶点<math>i</math>，<math>j</math>和<math>k</math>中定义一个三角形，以使其具有以下三个边的集合：{（i ，j），（j，k），（i，k）}。

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

我们也可以将顶点<math>i</math>所涉及的三角形的数量定义为<math>\delta（i）</math>，并且，由于每个三角形都被计数了三次，因此我们可以表示 G中的三角形为<math>\delta（G）= \frac{1}{3} \sum_{i\in V} \ \ delta（i）</math>。

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

假设三元闭包成立，则一个三元组的形成仅需要两个牢固的边缘。因此，在顶点<math>i</math>的三元组关闭假设下应该出现的理论三元组的数量为<math>\tau（i）= \binom{d_i}{2}</math>，假设<math>d_i \ge 2</math>。我们可以表示<math>\tau（G）= \frac{1}{3} \sum_{i\in V} \ \tau（i）</math>。

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于具有<math>d_i\ge 2</math>的顶点<math>i</ math>，顶点<math>i</math>的聚类系数<math>c（i)</math> 是封闭的顶点<math>i</math>的三元组分数，可以测量为<math>\frac{\delta（i）}{\tau（i）}</math>。因此，图<math> G </math>的聚类系数<math> C（G）</math>由<math> C（G）=\frac {1}{N_2} \sum_{i \in V,d_i \ge 2}c（i）</math>，其中<math>N_2</math>是度数至少为2的节点数。

==Transitivity==
传递性 

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

关于三元闭包的另一种度量是可传递性，定义为<math>T（G）= \frac{3\delta（G）}{\tau（G）}</math>。

==Causes and effects==

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

在信任网络中，由于三元闭包具有传递性，容易产生三元闭包。如果节点 a 信任节点 b，而节点 b 信任节点 c，则节点 a 将有信任节点 c 的基础。在社交网络中，强三元闭包的出现是因为与共同邻居 b 相遇的节点 a 和 c 有更多的机会相遇，从而产生至少个弱关系。节点 b 也有动机将 a 和 c 结合在一起，以减少两个独立关系中的潜在压力。

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

遵循这一原则的网络变得高度互联，并具有非常高的聚类系数。然而，不遵循这一原则的网络结果是连接不良，一旦包括负面关系，就可能遭受不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.<ref name=Easley/>

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

三元闭包是一个很好的模型，用来说明网络将如何随着时间的推移而演化。简单的图论倾向于在一个时间点上分析网络，而应用三并元闭包原理可以预测网络中关系的发展，并显示连通性的进展。

In [[social networks]], triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三元闭包促进了合作行为，但是当新的连接是通过

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

根据现有关系的推荐，合作者的全球平均比例比个人从总体人口中随机选择新关系的比例要低。两个可能的影响是通过结构和信息化建设。结构构造起源于向高集聚性的倾向。这种信息结构来自于这样一种假设，即一个人知道一个朋友的朋友的某些事情，而不是一个随机的陌生人。

==Strong Triadic Closure Property and local bridges==

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A [[local bridge]] occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性质是指，如果一个节点与两个邻居之间有强关联，那么这两个邻居之间至少有弱关联。另一方面，当一个节点在两个没有连接的相邻节点之间充当网守时，就会出现本地桥接。在遵循强三元闭包性质的网络中，局部桥梁中涉及的节点之间的联系之一需要是弱联系。

===Proof by contradiction===

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.<ref name=Easley/>

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

设 b 节点是 a 节点和 c 节点之间的本地桥梁，这样所涉及的节点之间就不存在弱联系。因此，b 与 a 和 c 都有很强的联系。根据强三元闭包的定义，在节点 a 和 c 之间会产生弱联系。因此，局部桥梁中至少有一个节点需要是弱连接，以防止三分闭合发生。

==References==

{{Reflist}}

{{Social networking}}

{{DEFAULTSORT:Triadic Closure}}

[[Category:Social systems]]

Category:Social systems

类别: 社会系统

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Triadic closure]]. Its edit history can be viewed at [[三元闭包/edithistory]]</noinclude>

[[Category:待整理页面]]

三元闭包

2020-09-25T11:23:40Z

923397935：/* Clustering coefficient */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.

'''三合闭包 Triadic Closure '''是'''社会网络理论 Social Network Theory '''中的一个概念，最早由德国社会学家格奥尔格·西梅尔（Georg Simmel）在其1908年的著作《社会学：社会形式的调查》中提出。三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。这个属性太极端了，无法在非常大的'''复杂网络 Complex Network '''中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

==History==
历史 

Triadic closure was made popular by [[Mark Granovetter]] in his 1973 article ''The Strength of Weak Ties''.<ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> There he synthesized the theory of [[cognitive balance]] first introduced by [[Fritz Heider]] in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱链接的力量》（The Strength of Weak Ties）中使三元闭包变得流行。在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的'''认知平衡理论 The Theory Of Cognitive Balance '''，该理论对Simmelian的社交网络有所了解。一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。如果没有关闭三个人的闭合，那么连接到两个人的人都将要关闭，以便在关系网络中实现闭合。

==Measurements==
测量 

{{Unreferenced section|date=September 2009}}

The two most common measures of triadic closure for a graph are (in no particular order) the [[clustering coefficient]] and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一张图的三元闭合的两个最常见的度量是（不按特定顺序）该图的'''聚类系数 Clustering Coefficient '''和'''可传递性 Transitivity '''。

==Clustering coefficient==
聚类系数
One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

衡量三元闭包的一种方法是聚类系数，如下所示：

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

令<math>G =（V，E）</math>是无向的'''简单图 Simple Graph '''（即没有自环或多个边的图），其中V为顶点集，E为边集。另外，令<math>N = |V|</math>和<math>M = |E|</math>分别表示G中顶点和边的数量，并令<math>d_i</math> 是顶点的度i。

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

我们可以在三个顶点<math>i</math>，<math>j</math>和<math>k</math>中定义一个三角形，以使其具有以下三个边的集合：{（i ，j），（j，k），（i，k）}。

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

我们也可以将顶点<math>i</math>所涉及的三角形的数量定义为<math>\delta（i）</math>，并且，由于每个三角形都被计数了三次，因此我们可以表示 G中的三角形为<math>\delta（G）= \frac{1}{3} \sum_{i\in V} \ \ delta（i</math>。

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

假设三元闭包成立，则一个三元组的形成仅需要两个牢固的边缘。因此，在顶点<math>i</math>的三元组关闭假设下应该出现的理论三元组的数量为<math>\tau（i）= \binom{d_i}{2}</math>，假设<math>d_i \ge 2</math>。我们可以表示<math>\tau（G）= \frac{1}{3} \sum_{i\in V} \ \tau（i）</math>。

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于具有<math>d_i\ge 2</math>的顶点<math>i</ math>，顶点<math>i</math>的聚类系数<math>c（i)</math> 是封闭的顶点<math>i</math>的三元组分数，可以测量为<math>\frac{\delta（i）}{\tau（i）}</math>。因此，图<math> G </math>的聚类系数<math> C（G）</math>由<math> C（G）=\frac {1}{N_2} \sum_{i \in V,d_i \ge 2}c（i）</math>，其中<math>N_2</math>是度数至少为2的节点数。

==Transitivity==
传递性 

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

关于三元闭包的另一种度量是可传递性，定义为<math>T（G）= \frac{3\delta（G）}{\tau（G）}</math>。

==Causes and effects==

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

在信任网络中，由于三元闭包具有传递性，容易产生三元闭包。如果节点 a 信任节点 b，而节点 b 信任节点 c，则节点 a 将有信任节点 c 的基础。在社交网络中，强三元闭包的出现是因为与共同邻居 b 相遇的节点 a 和 c 有更多的机会相遇，从而产生至少个弱关系。节点 b 也有动机将 a 和 c 结合在一起，以减少两个独立关系中的潜在压力。

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

遵循这一原则的网络变得高度互联，并具有非常高的聚类系数。然而，不遵循这一原则的网络结果是连接不良，一旦包括负面关系，就可能遭受不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.<ref name=Easley/>

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

三元闭包是一个很好的模型，用来说明网络将如何随着时间的推移而演化。简单的图论倾向于在一个时间点上分析网络，而应用三并元闭包原理可以预测网络中关系的发展，并显示连通性的进展。

In [[social networks]], triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三元闭包促进了合作行为，但是当新的连接是通过

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

根据现有关系的推荐，合作者的全球平均比例比个人从总体人口中随机选择新关系的比例要低。两个可能的影响是通过结构和信息化建设。结构构造起源于向高集聚性的倾向。这种信息结构来自于这样一种假设，即一个人知道一个朋友的朋友的某些事情，而不是一个随机的陌生人。

==Strong Triadic Closure Property and local bridges==

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A [[local bridge]] occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性质是指，如果一个节点与两个邻居之间有强关联，那么这两个邻居之间至少有弱关联。另一方面，当一个节点在两个没有连接的相邻节点之间充当网守时，就会出现本地桥接。在遵循强三元闭包性质的网络中，局部桥梁中涉及的节点之间的联系之一需要是弱联系。

===Proof by contradiction===

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.<ref name=Easley/>

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

设 b 节点是 a 节点和 c 节点之间的本地桥梁，这样所涉及的节点之间就不存在弱联系。因此，b 与 a 和 c 都有很强的联系。根据强三元闭包的定义，在节点 a 和 c 之间会产生弱联系。因此，局部桥梁中至少有一个节点需要是弱连接，以防止三分闭合发生。

==References==

{{Reflist}}

{{Social networking}}

{{DEFAULTSORT:Triadic Closure}}

[[Category:Social systems]]

Category:Social systems

类别: 社会系统

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Triadic closure]]. Its edit history can be viewed at [[三元闭包/edithistory]]</noinclude>

[[Category:待整理页面]]

三元闭包

2020-09-25T11:19:06Z

923397935：/* Clustering coefficient */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.

'''三合闭包 Triadic Closure '''是'''社会网络理论 Social Network Theory '''中的一个概念，最早由德国社会学家格奥尔格·西梅尔（Georg Simmel）在其1908年的著作《社会学：社会形式的调查》中提出。三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。这个属性太极端了，无法在非常大的'''复杂网络 Complex Network '''中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

==History==
历史 

Triadic closure was made popular by [[Mark Granovetter]] in his 1973 article ''The Strength of Weak Ties''.<ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> There he synthesized the theory of [[cognitive balance]] first introduced by [[Fritz Heider]] in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱链接的力量》（The Strength of Weak Ties）中使三元闭包变得流行。在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的'''认知平衡理论 The Theory Of Cognitive Balance '''，该理论对Simmelian的社交网络有所了解。一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。如果没有关闭三个人的闭合，那么连接到两个人的人都将要关闭，以便在关系网络中实现闭合。

==Measurements==
测量 

{{Unreferenced section|date=September 2009}}

The two most common measures of triadic closure for a graph are (in no particular order) the [[clustering coefficient]] and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一张图的三元闭合的两个最常见的度量是（不按特定顺序）该图的'''聚类系数 Clustering Coefficient '''和'''可传递性 Transitivity '''。

==Clustering coefficient==
聚类系数
One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

衡量三元闭包的一种方法是聚类系数，如下所示：

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

令<math>G =（V，E）</math>是无向的简单图（即没有自环或多个边的图），其中V为顶点集，E为边集。另外，令<math>N = |V|</math>和<math>M = |E|</math>分别表示G中顶点和边的数量，并令<math>d_i</math> 是顶点的度i。

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

我们可以在三个顶点<math>i</math>，<math>j</math>和<math>k</math>中定义一个三角形，以使其具有以下三个边的集合：{（i ，j），（j，k），（i，k）}。

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

我们也可以将顶点<math>i</math>所涉及的三角形的数量定义为<math>\delta（i）</math>，并且，由于每个三角形都被计数了三次，因此我们可以表示 G中的三角形为<math>\delta（G）= \frac{1}{3} \sum_{i\in V} \ \ delta（i</math>。

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

假设三元闭包成立，则一个三元组的形成仅需要两个牢固的边缘。因此，在顶点<math>i</math>的三元组关闭假设下应该出现的理论三元组的数量为<math>\tau（i）= \binom{d_i}{2}</math>，假设<math>d_i \ge 2</math>。我们可以表示<math>\tau（G）= \frac{1}{3} \sum_{i\in V} \ \tau（i）</math>。

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于具有<math> d_i \ ge 2 </ math>的顶点<math> i </ math>，顶点<math> i </ math>的聚类系数<math> c（i）</ math> 是封闭的顶点<math> i </ math>的三元组分数，可以测量为<math> \ frac {\ delta（i）} {\ tau（i）} </ math>。因此，图<math> G </ math>的聚类系数<math> C（G）</ math>由<math> C（G）= \ frac {1} {N_2} \ sum_ {i \ 在V中，d_i \ ge 2} c（i）</ math>，其中<math> N_2 </ math>是度数至少为2的节点数。

===Transitivity===

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

存在三元闭包的另一个度量是传递性，定义为 < math > t (g) = frac {3 delta (g)}{ tau (g)} </math > 。

==Causes and effects==

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

在信任网络中，由于三元闭包具有传递性，容易产生三元闭包。如果节点 a 信任节点 b，而节点 b 信任节点 c，则节点 a 将有信任节点 c 的基础。在社交网络中，强三元闭包的出现是因为与共同邻居 b 相遇的节点 a 和 c 有更多的机会相遇，从而产生至少个弱关系。节点 b 也有动机将 a 和 c 结合在一起，以减少两个独立关系中的潜在压力。

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

遵循这一原则的网络变得高度互联，并具有非常高的聚类系数。然而，不遵循这一原则的网络结果是连接不良，一旦包括负面关系，就可能遭受不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.<ref name=Easley/>

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

三元闭包是一个很好的模型，用来说明网络将如何随着时间的推移而演化。简单的图论倾向于在一个时间点上分析网络，而应用三并元闭包原理可以预测网络中关系的发展，并显示连通性的进展。

In [[social networks]], triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三元闭包促进了合作行为，但是当新的连接是通过

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

根据现有关系的推荐，合作者的全球平均比例比个人从总体人口中随机选择新关系的比例要低。两个可能的影响是通过结构和信息化建设。结构构造起源于向高集聚性的倾向。这种信息结构来自于这样一种假设，即一个人知道一个朋友的朋友的某些事情，而不是一个随机的陌生人。

==Strong Triadic Closure Property and local bridges==

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A [[local bridge]] occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性质是指，如果一个节点与两个邻居之间有强关联，那么这两个邻居之间至少有弱关联。另一方面，当一个节点在两个没有连接的相邻节点之间充当网守时，就会出现本地桥接。在遵循强三元闭包性质的网络中，局部桥梁中涉及的节点之间的联系之一需要是弱联系。

===Proof by contradiction===

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.<ref name=Easley/>

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

设 b 节点是 a 节点和 c 节点之间的本地桥梁，这样所涉及的节点之间就不存在弱联系。因此，b 与 a 和 c 都有很强的联系。根据强三元闭包的定义，在节点 a 和 c 之间会产生弱联系。因此，局部桥梁中至少有一个节点需要是弱连接，以防止三分闭合发生。

==References==

{{Reflist}}

{{Social networking}}

{{DEFAULTSORT:Triadic Closure}}

[[Category:Social systems]]

Category:Social systems

类别: 社会系统

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Triadic closure]]. Its edit history can be viewed at [[三元闭包/edithistory]]</noinclude>

[[Category:待整理页面]]

三元闭包

2020-09-25T11:14:28Z

923397935：/* Measurements */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.

'''三合闭包 Triadic Closure '''是'''社会网络理论 Social Network Theory '''中的一个概念，最早由德国社会学家格奥尔格·西梅尔（Georg Simmel）在其1908年的著作《社会学：社会形式的调查》中提出。三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。这个属性太极端了，无法在非常大的'''复杂网络 Complex Network '''中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

==History==
历史 

Triadic closure was made popular by [[Mark Granovetter]] in his 1973 article ''The Strength of Weak Ties''.<ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> There he synthesized the theory of [[cognitive balance]] first introduced by [[Fritz Heider]] in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱链接的力量》（The Strength of Weak Ties）中使三元闭包变得流行。在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的'''认知平衡理论 The Theory Of Cognitive Balance '''，该理论对Simmelian的社交网络有所了解。一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。如果没有关闭三个人的闭合，那么连接到两个人的人都将要关闭，以便在关系网络中实现闭合。

==Measurements==
测量 

{{Unreferenced section|date=September 2009}}

The two most common measures of triadic closure for a graph are (in no particular order) the [[clustering coefficient]] and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一张图的三元闭合的两个最常见的度量是（不按特定顺序）该图的'''聚类系数 Clustering Coefficient '''和'''可传递性 Transitivity '''。

===Clustering coefficient===
聚类系数
One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

衡量三元闭包的一种方法是聚类系数，如下所示：

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

令<math>G =（V，E）</math>是无向的简单图（即没有自环或多个边的图），其中V为顶点集，E为边集。另外，令<math>N = |V|</math>和<math>M = |E|</math>分别表示G中顶点和边的数量，并令<math>d_i</math> 是顶点的度i。

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

我们可以定义三个顶点之间的三角形 < math > ，< math > j </math > ，< math > k </math > 是一个具有以下三个边的集合: {(i，j) ，(j，k) ，(i，k)}。

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

我们还可以定义顶点 < math > i </math > 所涉及的三角形的个数为 < math > delta (i) </math > ，并且，当每个三角形被计算三次时，我们可以将 g 中三角形的个数表示为 < math > delta (g) = frac {1}{3} sum _ i in v } delta (i) </math > 。

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

假设三元闭包成立，一个三元组只需要两条强边即可形成。因此，在三元闭包假设下，顶点 i </math > i </math > 的理论三元数应该是 < math > tau (i) = binom { d _ i }{2} </math > ，假设 < math > d _ i ge 2 </math > 。我们可以表示 < math > tau (g) = frac {1}{3} sum { i in v } tau (i) </math > 。

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于一个顶点 < math > i </math > 与 < math > d _ i ge 2 </math > 相关的集聚系数 </math > c (i) </math > i </math > 是顶点 < math > i </math > 的三元组的分数，它是封闭的，可以被测量为 < math > frac { delta (i)}{ tau (i)} </math > 。因此，图的 c (g) </math > g </math > 的集聚系数由 < math > c (g) = frac {1}{ n _ 2} sum { i in v，d _ i ge2} c (i) </math > 给出，其中 < math > n _ 2 </math > 是度数至少为2的节点数。

===Transitivity===

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

存在三元闭包的另一个度量是传递性，定义为 < math > t (g) = frac {3 delta (g)}{ tau (g)} </math > 。

==Causes and effects==

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

在信任网络中，由于三元闭包具有传递性，容易产生三元闭包。如果节点 a 信任节点 b，而节点 b 信任节点 c，则节点 a 将有信任节点 c 的基础。在社交网络中，强三元闭包的出现是因为与共同邻居 b 相遇的节点 a 和 c 有更多的机会相遇，从而产生至少个弱关系。节点 b 也有动机将 a 和 c 结合在一起，以减少两个独立关系中的潜在压力。

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

遵循这一原则的网络变得高度互联，并具有非常高的聚类系数。然而，不遵循这一原则的网络结果是连接不良，一旦包括负面关系，就可能遭受不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.<ref name=Easley/>

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

三元闭包是一个很好的模型，用来说明网络将如何随着时间的推移而演化。简单的图论倾向于在一个时间点上分析网络，而应用三并元闭包原理可以预测网络中关系的发展，并显示连通性的进展。

In [[social networks]], triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三元闭包促进了合作行为，但是当新的连接是通过

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

根据现有关系的推荐，合作者的全球平均比例比个人从总体人口中随机选择新关系的比例要低。两个可能的影响是通过结构和信息化建设。结构构造起源于向高集聚性的倾向。这种信息结构来自于这样一种假设，即一个人知道一个朋友的朋友的某些事情，而不是一个随机的陌生人。

==Strong Triadic Closure Property and local bridges==

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A [[local bridge]] occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性质是指，如果一个节点与两个邻居之间有强关联，那么这两个邻居之间至少有弱关联。另一方面，当一个节点在两个没有连接的相邻节点之间充当网守时，就会出现本地桥接。在遵循强三元闭包性质的网络中，局部桥梁中涉及的节点之间的联系之一需要是弱联系。

===Proof by contradiction===

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.<ref name=Easley/>

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

设 b 节点是 a 节点和 c 节点之间的本地桥梁，这样所涉及的节点之间就不存在弱联系。因此，b 与 a 和 c 都有很强的联系。根据强三元闭包的定义，在节点 a 和 c 之间会产生弱联系。因此，局部桥梁中至少有一个节点需要是弱连接，以防止三分闭合发生。

==References==

{{Reflist}}

{{Social networking}}

{{DEFAULTSORT:Triadic Closure}}

[[Category:Social systems]]

Category:Social systems

类别: 社会系统

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Triadic closure]]. Its edit history can be viewed at [[三元闭包/edithistory]]</noinclude>

[[Category:待整理页面]]

三元闭包

2020-09-25T11:09:32Z

923397935：/* Measurements */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.

'''三合闭包 Triadic Closure '''是'''社会网络理论 Social Network Theory '''中的一个概念，最早由德国社会学家格奥尔格·西梅尔（Georg Simmel）在其1908年的著作《社会学：社会形式的调查》中提出。三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。这个属性太极端了，无法在非常大的'''复杂网络 Complex Network '''中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

==History==
历史 

Triadic closure was made popular by [[Mark Granovetter]] in his 1973 article ''The Strength of Weak Ties''.<ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> There he synthesized the theory of [[cognitive balance]] first introduced by [[Fritz Heider]] in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱链接的力量》（The Strength of Weak Ties）中使三元闭包变得流行。在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的'''认知平衡理论 The Theory Of Cognitive Balance '''，该理论对Simmelian的社交网络有所了解。一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。如果没有关闭三个人的闭合，那么连接到两个人的人都将要关闭，以便在关系网络中实现闭合。

==Measurements==
测量 

{{Unreferenced section|date=September 2009}}

The two most common measures of triadic closure for a graph are (in no particular order) the [[clustering coefficient]] and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一张图的三闭合的两个最常见的度量是（不按特定顺序）该图的聚类系数和可传递性。

===Clustering coefficient===

One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

三元闭合的一个衡量标准是集聚系数，具体如下:

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

设 g = (v，e) </math > 是一个无向简单图(即没有自回路或多条边的图) ，v 是顶点集，e 是边集。同时，让 < math > n = | v | </math > 和 < math > m = | e | </math > 分别表示 g 中的顶点数和边数，让 < math > di </math > 表示顶点的度数。

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

我们可以定义三个顶点之间的三角形 < math > ，< math > j </math > ，< math > k </math > 是一个具有以下三个边的集合: {(i，j) ，(j，k) ，(i，k)}。

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

我们还可以定义顶点 < math > i </math > 所涉及的三角形的个数为 < math > delta (i) </math > ，并且，当每个三角形被计算三次时，我们可以将 g 中三角形的个数表示为 < math > delta (g) = frac {1}{3} sum _ i in v } delta (i) </math > 。

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

假设三元闭包成立，一个三元组只需要两条强边即可形成。因此，在三元闭包假设下，顶点 i </math > i </math > 的理论三元数应该是 < math > tau (i) = binom { d _ i }{2} </math > ，假设 < math > d _ i ge 2 </math > 。我们可以表示 < math > tau (g) = frac {1}{3} sum { i in v } tau (i) </math > 。

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于一个顶点 < math > i </math > 与 < math > d _ i ge 2 </math > 相关的集聚系数 </math > c (i) </math > i </math > 是顶点 < math > i </math > 的三元组的分数，它是封闭的，可以被测量为 < math > frac { delta (i)}{ tau (i)} </math > 。因此，图的 c (g) </math > g </math > 的集聚系数由 < math > c (g) = frac {1}{ n _ 2} sum { i in v，d _ i ge2} c (i) </math > 给出，其中 < math > n _ 2 </math > 是度数至少为2的节点数。

===Transitivity===

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

存在三元闭包的另一个度量是传递性，定义为 < math > t (g) = frac {3 delta (g)}{ tau (g)} </math > 。

==Causes and effects==

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

在信任网络中，由于三元闭包具有传递性，容易产生三元闭包。如果节点 a 信任节点 b，而节点 b 信任节点 c，则节点 a 将有信任节点 c 的基础。在社交网络中，强三元闭包的出现是因为与共同邻居 b 相遇的节点 a 和 c 有更多的机会相遇，从而产生至少个弱关系。节点 b 也有动机将 a 和 c 结合在一起，以减少两个独立关系中的潜在压力。

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

遵循这一原则的网络变得高度互联，并具有非常高的聚类系数。然而，不遵循这一原则的网络结果是连接不良，一旦包括负面关系，就可能遭受不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.<ref name=Easley/>

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

三元闭包是一个很好的模型，用来说明网络将如何随着时间的推移而演化。简单的图论倾向于在一个时间点上分析网络，而应用三并元闭包原理可以预测网络中关系的发展，并显示连通性的进展。

In [[social networks]], triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三元闭包促进了合作行为，但是当新的连接是通过

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

根据现有关系的推荐，合作者的全球平均比例比个人从总体人口中随机选择新关系的比例要低。两个可能的影响是通过结构和信息化建设。结构构造起源于向高集聚性的倾向。这种信息结构来自于这样一种假设，即一个人知道一个朋友的朋友的某些事情，而不是一个随机的陌生人。

==Strong Triadic Closure Property and local bridges==

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A [[local bridge]] occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性质是指，如果一个节点与两个邻居之间有强关联，那么这两个邻居之间至少有弱关联。另一方面，当一个节点在两个没有连接的相邻节点之间充当网守时，就会出现本地桥接。在遵循强三元闭包性质的网络中，局部桥梁中涉及的节点之间的联系之一需要是弱联系。

===Proof by contradiction===

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.<ref name=Easley/>

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

设 b 节点是 a 节点和 c 节点之间的本地桥梁，这样所涉及的节点之间就不存在弱联系。因此，b 与 a 和 c 都有很强的联系。根据强三元闭包的定义，在节点 a 和 c 之间会产生弱联系。因此，局部桥梁中至少有一个节点需要是弱连接，以防止三分闭合发生。

==References==

{{Reflist}}

{{Social networking}}

{{DEFAULTSORT:Triadic Closure}}

[[Category:Social systems]]

Category:Social systems

类别: 社会系统

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Triadic closure]]. Its edit history can be viewed at [[三元闭包/edithistory]]</noinclude>

[[Category:待整理页面]]

三元闭包

2020-09-25T11:09:07Z

923397935：

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.

'''三合闭包 Triadic Closure '''是'''社会网络理论 Social Network Theory '''中的一个概念，最早由德国社会学家格奥尔格·西梅尔（Georg Simmel）在其1908年的著作《社会学：社会形式的调查》中提出。三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。这个属性太极端了，无法在非常大的'''复杂网络 Complex Network '''中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

==History==
历史 

Triadic closure was made popular by [[Mark Granovetter]] in his 1973 article ''The Strength of Weak Ties''.<ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> There he synthesized the theory of [[cognitive balance]] first introduced by [[Fritz Heider]] in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱链接的力量》（The Strength of Weak Ties）中使三元闭包变得流行。在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的'''认知平衡理论 The Theory Of Cognitive Balance '''，该理论对Simmelian的社交网络有所了解。一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。如果没有关闭三个人的闭合，那么连接到两个人的人都将要关闭，以便在关系网络中实现闭合。

==Measurements==
测量 

{{Unreferenced section|date=September 2009}}

The two most common measures of triadic closure for a graph are (in no particular order) the [[clustering coefficient]] and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一张图的三闭合的两个最常见的度量是（不按特定顺序）该图的聚类系数和可传递性。

===Clustering coefficient===

One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

三元闭合的一个衡量标准是集聚系数，具体如下:

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

设 g = (v，e) </math > 是一个无向简单图(即没有自回路或多条边的图) ，v 是顶点集，e 是边集。同时，让 < math > n = | v | </math > 和 < math > m = | e | </math > 分别表示 g 中的顶点数和边数，让 < math > di </math > 表示顶点的度数。

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

我们可以定义三个顶点之间的三角形 < math > ，< math > j </math > ，< math > k </math > 是一个具有以下三个边的集合: {(i，j) ，(j，k) ，(i，k)}。

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

我们还可以定义顶点 < math > i </math > 所涉及的三角形的个数为 < math > delta (i) </math > ，并且，当每个三角形被计算三次时，我们可以将 g 中三角形的个数表示为 < math > delta (g) = frac {1}{3} sum _ i in v } delta (i) </math > 。

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

假设三元闭包成立，一个三元组只需要两条强边即可形成。因此，在三元闭包假设下，顶点 i </math > i </math > 的理论三元数应该是 < math > tau (i) = binom { d _ i }{2} </math > ，假设 < math > d _ i ge 2 </math > 。我们可以表示 < math > tau (g) = frac {1}{3} sum { i in v } tau (i) </math > 。

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于一个顶点 < math > i </math > 与 < math > d _ i ge 2 </math > 相关的集聚系数 </math > c (i) </math > i </math > 是顶点 < math > i </math > 的三元组的分数，它是封闭的，可以被测量为 < math > frac { delta (i)}{ tau (i)} </math > 。因此，图的 c (g) </math > g </math > 的集聚系数由 < math > c (g) = frac {1}{ n _ 2} sum { i in v，d _ i ge2} c (i) </math > 给出，其中 < math > n _ 2 </math > 是度数至少为2的节点数。

===Transitivity===

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

存在三元闭包的另一个度量是传递性，定义为 < math > t (g) = frac {3 delta (g)}{ tau (g)} </math > 。

==Causes and effects==

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

在信任网络中，由于三元闭包具有传递性，容易产生三元闭包。如果节点 a 信任节点 b，而节点 b 信任节点 c，则节点 a 将有信任节点 c 的基础。在社交网络中，强三元闭包的出现是因为与共同邻居 b 相遇的节点 a 和 c 有更多的机会相遇，从而产生至少个弱关系。节点 b 也有动机将 a 和 c 结合在一起，以减少两个独立关系中的潜在压力。

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

遵循这一原则的网络变得高度互联，并具有非常高的聚类系数。然而，不遵循这一原则的网络结果是连接不良，一旦包括负面关系，就可能遭受不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.<ref name=Easley/>

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

三元闭包是一个很好的模型，用来说明网络将如何随着时间的推移而演化。简单的图论倾向于在一个时间点上分析网络，而应用三并元闭包原理可以预测网络中关系的发展，并显示连通性的进展。

In [[social networks]], triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三元闭包促进了合作行为，但是当新的连接是通过

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

根据现有关系的推荐，合作者的全球平均比例比个人从总体人口中随机选择新关系的比例要低。两个可能的影响是通过结构和信息化建设。结构构造起源于向高集聚性的倾向。这种信息结构来自于这样一种假设，即一个人知道一个朋友的朋友的某些事情，而不是一个随机的陌生人。

==Strong Triadic Closure Property and local bridges==

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A [[local bridge]] occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性质是指，如果一个节点与两个邻居之间有强关联，那么这两个邻居之间至少有弱关联。另一方面，当一个节点在两个没有连接的相邻节点之间充当网守时，就会出现本地桥接。在遵循强三元闭包性质的网络中，局部桥梁中涉及的节点之间的联系之一需要是弱联系。

===Proof by contradiction===

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.<ref name=Easley/>

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

设 b 节点是 a 节点和 c 节点之间的本地桥梁，这样所涉及的节点之间就不存在弱联系。因此，b 与 a 和 c 都有很强的联系。根据强三元闭包的定义，在节点 a 和 c 之间会产生弱联系。因此，局部桥梁中至少有一个节点需要是弱连接，以防止三分闭合发生。

==References==

{{Reflist}}

{{Social networking}}

{{DEFAULTSORT:Triadic Closure}}

[[Category:Social systems]]

Category:Social systems

类别: 社会系统

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Triadic closure]]. Its edit history can be viewed at [[三元闭包/edithistory]]</noinclude>

[[Category:待整理页面]]

三元闭包

2020-09-25T11:06:59Z

923397935：/* History */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
本词条由信白初步翻译 

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.

'''三合闭包 Triadic Closure '''是'''社会网络理论 Social Network Theory '''中的一个概念，最早由德国社会学家格奥尔格·西梅尔（Georg Simmel）在其1908年的著作《社会学：社会形式的调查》中提出。三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。这个属性太极端了，无法在非常大的复杂网络中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

==History==
历史 

Triadic closure was made popular by [[Mark Granovetter]] in his 1973 article ''The Strength of Weak Ties''.<ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> There he synthesized the theory of [[cognitive balance]] first introduced by [[Fritz Heider]] in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱结的力量》（The Strength of Weak Ties）中使三元闭包变得流行。在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的'''认知平衡理论 The Theory Of Cognitive Balance '''，该理论对Simmelian的社交网络有所了解。一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。如果没有关闭三个人的闭合，那么连接到两个人的人都将要关闭，以便在关系网络中实现闭合。

==Measurements==

{{Unreferenced section|date=September 2009}}

The two most common measures of triadic closure for a graph are (in no particular order) the [[clustering coefficient]] and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一个图的三元闭包的两个最常见的度量是该图的集聚系数和传递性(没有特定的顺序)。

===Clustering coefficient===

One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

三元闭合的一个衡量标准是集聚系数，具体如下:

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

设 g = (v，e) </math > 是一个无向简单图(即没有自回路或多条边的图) ，v 是顶点集，e 是边集。同时，让 < math > n = | v | </math > 和 < math > m = | e | </math > 分别表示 g 中的顶点数和边数，让 < math > di </math > 表示顶点的度数。

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

我们可以定义三个顶点之间的三角形 < math > ，< math > j </math > ，< math > k </math > 是一个具有以下三个边的集合: {(i，j) ，(j，k) ，(i，k)}。

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

我们还可以定义顶点 < math > i </math > 所涉及的三角形的个数为 < math > delta (i) </math > ，并且，当每个三角形被计算三次时，我们可以将 g 中三角形的个数表示为 < math > delta (g) = frac {1}{3} sum _ i in v } delta (i) </math > 。

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

假设三元闭包成立，一个三元组只需要两条强边即可形成。因此，在三元闭包假设下，顶点 i </math > i </math > 的理论三元数应该是 < math > tau (i) = binom { d _ i }{2} </math > ，假设 < math > d _ i ge 2 </math > 。我们可以表示 < math > tau (g) = frac {1}{3} sum { i in v } tau (i) </math > 。

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于一个顶点 < math > i </math > 与 < math > d _ i ge 2 </math > 相关的集聚系数 </math > c (i) </math > i </math > 是顶点 < math > i </math > 的三元组的分数，它是封闭的，可以被测量为 < math > frac { delta (i)}{ tau (i)} </math > 。因此，图的 c (g) </math > g </math > 的集聚系数由 < math > c (g) = frac {1}{ n _ 2} sum { i in v，d _ i ge2} c (i) </math > 给出，其中 < math > n _ 2 </math > 是度数至少为2的节点数。

===Transitivity===

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

存在三元闭包的另一个度量是传递性，定义为 < math > t (g) = frac {3 delta (g)}{ tau (g)} </math > 。

==Causes and effects==

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

在信任网络中，由于三元闭包具有传递性，容易产生三元闭包。如果节点 a 信任节点 b，而节点 b 信任节点 c，则节点 a 将有信任节点 c 的基础。在社交网络中，强三元闭包的出现是因为与共同邻居 b 相遇的节点 a 和 c 有更多的机会相遇，从而产生至少个弱关系。节点 b 也有动机将 a 和 c 结合在一起，以减少两个独立关系中的潜在压力。

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

遵循这一原则的网络变得高度互联，并具有非常高的聚类系数。然而，不遵循这一原则的网络结果是连接不良，一旦包括负面关系，就可能遭受不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.<ref name=Easley/>

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

三元闭包是一个很好的模型，用来说明网络将如何随着时间的推移而演化。简单的图论倾向于在一个时间点上分析网络，而应用三并元闭包原理可以预测网络中关系的发展，并显示连通性的进展。

In [[social networks]], triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三元闭包促进了合作行为，但是当新的连接是通过

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

根据现有关系的推荐，合作者的全球平均比例比个人从总体人口中随机选择新关系的比例要低。两个可能的影响是通过结构和信息化建设。结构构造起源于向高集聚性的倾向。这种信息结构来自于这样一种假设，即一个人知道一个朋友的朋友的某些事情，而不是一个随机的陌生人。

==Strong Triadic Closure Property and local bridges==

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A [[local bridge]] occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性质是指，如果一个节点与两个邻居之间有强关联，那么这两个邻居之间至少有弱关联。另一方面，当一个节点在两个没有连接的相邻节点之间充当网守时，就会出现本地桥接。在遵循强三元闭包性质的网络中，局部桥梁中涉及的节点之间的联系之一需要是弱联系。

===Proof by contradiction===

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.<ref name=Easley/>

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

设 b 节点是 a 节点和 c 节点之间的本地桥梁，这样所涉及的节点之间就不存在弱联系。因此，b 与 a 和 c 都有很强的联系。根据强三元闭包的定义，在节点 a 和 c 之间会产生弱联系。因此，局部桥梁中至少有一个节点需要是弱连接，以防止三分闭合发生。

==References==

{{Reflist}}

{{Social networking}}

{{DEFAULTSORT:Triadic Closure}}

[[Category:Social systems]]

Category:Social systems

类别: 社会系统

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Triadic closure]]. Its edit history can be viewed at [[三元闭包/edithistory]]</noinclude>

[[Category:待整理页面]]

三元闭包

2020-09-25T11:04:36Z

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本词条由信白初步翻译 

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.

'''三合闭包 Triadic Closure '''是'''社会网络理论 Social Network Theory '''中的一个概念，最早由德国社会学家格奥尔格·西梅尔（Georg Simmel）在其1908年的著作《社会学：社会形式的调查》中提出。三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。这个属性太极端了，无法在非常大的复杂网络中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

==History==
历史 

Triadic closure was made popular by [[Mark Granovetter]] in his 1973 article ''The Strength of Weak Ties''.<ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> There he synthesized the theory of [[cognitive balance]] first introduced by [[Fritz Heider]] in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱结的力量》（The Strength of Weak Ties）中使三合变得流行。在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的认知平衡理论，该理论对Simmelian的社交网络有所了解。一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。如果没有关闭三个人的三合会，那么连接到两个人的人都将要关闭此三合会，以便在关系网络中实现闭合。

==Measurements==

{{Unreferenced section|date=September 2009}}

The two most common measures of triadic closure for a graph are (in no particular order) the [[clustering coefficient]] and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一个图的三元闭包的两个最常见的度量是该图的集聚系数和传递性(没有特定的顺序)。

===Clustering coefficient===

One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

三元闭合的一个衡量标准是集聚系数，具体如下:

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

设 g = (v，e) </math > 是一个无向简单图(即没有自回路或多条边的图) ，v 是顶点集，e 是边集。同时，让 < math > n = | v | </math > 和 < math > m = | e | </math > 分别表示 g 中的顶点数和边数，让 < math > di </math > 表示顶点的度数。

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.

我们可以定义三个顶点之间的三角形 < math > ，< math > j </math > ，< math > k </math > 是一个具有以下三个边的集合: {(i，j) ，(j，k) ，(i，k)}。

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.

我们还可以定义顶点 < math > i </math > 所涉及的三角形的个数为 < math > delta (i) </math > ，并且，当每个三角形被计算三次时，我们可以将 g 中三角形的个数表示为 < math > delta (g) = frac {1}{3} sum _ i in v } delta (i) </math > 。

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

假设三元闭包成立，一个三元组只需要两条强边即可形成。因此，在三元闭包假设下，顶点 i </math > i </math > 的理论三元数应该是 < math > tau (i) = binom { d _ i }{2} </math > ，假设 < math > d _ i ge 2 </math > 。我们可以表示 < math > tau (g) = frac {1}{3} sum { i in v } tau (i) </math > 。

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于一个顶点 < math > i </math > 与 < math > d _ i ge 2 </math > 相关的集聚系数 </math > c (i) </math > i </math > 是顶点 < math > i </math > 的三元组的分数，它是封闭的，可以被测量为 < math > frac { delta (i)}{ tau (i)} </math > 。因此，图的 c (g) </math > g </math > 的集聚系数由 < math > c (g) = frac {1}{ n _ 2} sum { i in v，d _ i ge2} c (i) </math > 给出，其中 < math > n _ 2 </math > 是度数至少为2的节点数。

===Transitivity===

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

存在三元闭包的另一个度量是传递性，定义为 < math > t (g) = frac {3 delta (g)}{ tau (g)} </math > 。

==Causes and effects==

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

在信任网络中，由于三元闭包具有传递性，容易产生三元闭包。如果节点 a 信任节点 b，而节点 b 信任节点 c，则节点 a 将有信任节点 c 的基础。在社交网络中，强三元闭包的出现是因为与共同邻居 b 相遇的节点 a 和 c 有更多的机会相遇，从而产生至少个弱关系。节点 b 也有动机将 a 和 c 结合在一起，以减少两个独立关系中的潜在压力。

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

遵循这一原则的网络变得高度互联，并具有非常高的聚类系数。然而，不遵循这一原则的网络结果是连接不良，一旦包括负面关系，就可能遭受不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.<ref name=Easley/>

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

三元闭包是一个很好的模型，用来说明网络将如何随着时间的推移而演化。简单的图论倾向于在一个时间点上分析网络，而应用三并元闭包原理可以预测网络中关系的发展，并显示连通性的进展。

In [[social networks]], triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三元闭包促进了合作行为，但是当新的连接是通过

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

根据现有关系的推荐，合作者的全球平均比例比个人从总体人口中随机选择新关系的比例要低。两个可能的影响是通过结构和信息化建设。结构构造起源于向高集聚性的倾向。这种信息结构来自于这样一种假设，即一个人知道一个朋友的朋友的某些事情，而不是一个随机的陌生人。

==Strong Triadic Closure Property and local bridges==

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A [[local bridge]] occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性质是指，如果一个节点与两个邻居之间有强关联，那么这两个邻居之间至少有弱关联。另一方面，当一个节点在两个没有连接的相邻节点之间充当网守时，就会出现本地桥接。在遵循强三元闭包性质的网络中，局部桥梁中涉及的节点之间的联系之一需要是弱联系。

===Proof by contradiction===

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.<ref name=Easley/>

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

设 b 节点是 a 节点和 c 节点之间的本地桥梁，这样所涉及的节点之间就不存在弱联系。因此，b 与 a 和 c 都有很强的联系。根据强三元闭包的定义，在节点 a 和 c 之间会产生弱联系。因此，局部桥梁中至少有一个节点需要是弱连接，以防止三分闭合发生。

==References==

{{Reflist}}

{{Social networking}}

{{DEFAULTSORT:Triadic Closure}}

[[Category:Social systems]]

Category:Social systems

类别: 社会系统

[[Category:Sociological terminology]]

Category:Sociological terminology

类别: 社会学术语

<noinclude>

This page was moved from [[wikipedia:en:Triadic closure]]. Its edit history can be viewed at [[三元闭包/edithistory]]</noinclude>

[[Category:待整理页面]]

网络可控性

2020-09-25T10:51:49Z

923397935：/* Maximum Matching */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 

本词条由信白初步翻译 

[[Image:YYL1.png|right|thumb|Controlling a simple network.]]

Controlling a simple network.

图1：控制一个简单的网络。

'''Network Controllability''' is concerned about the structural [[controllability]] of a [[Graph (discrete mathematics)|network]]. Controllability describes our ability to guide a [[dynamical system]] from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al.<ref>{{Cite journal|last=Sharma|first=Ankush|last2=Cinti|first2=Caterina|last3=Capobianco|first3=Enrico|date=2017|title=Multitype Network-Guided Target Controllability in Phenotypically Characterized Osteosarcoma: Role of Tumor Microenvironment|journal=Frontiers in Immunology|language=English|volume=8|pages=918|doi=10.3389/fimmu.2017.00918|pmid=28824643|pmc=5536125|issn=1664-3224}}</ref> on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

Network Controllability is concerned about the structural controllability of a network. Controllability describes our ability to guide a dynamical system from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al. on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

'''网络可控性 Network Controllability '''研究的是网络的结构可控性。可控性描述了我们在有限时间内，通过合适的输入选择，引导动力系统从任何初始状态到任何期望的最终状态的能力。这个定义与我们直观的控制概念非常吻合。一般有向加权复杂网络的可控性是近年来世界范围内许多研究小组的重点研究课题。夏尔马等人最近在多种生物学网络(基因-基因、 miRNA-基因、蛋白质-蛋白质相互作用网络)上的研究，确定了骨肉瘤表型特征的控制目标，显示了基因和蛋白质在维持肿瘤微环境中的重要作用。

==Background==
背景 

Consider the canonical linear time-invariant dynamics on a complex network

Consider the canonical linear time-invariant dynamics on a complex network

考虑'''复杂网络 Complex Network '''上的正则线性动力学

<math>
\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)

\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)
</math>

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

其中，向量 <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> 表示用 <math>t</math> 时间捕获 <math>N</math> 个节点系统。

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the ''minimum'' number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al.<ref name="Liu-Nature-11">{{cite journal | last=Liu | first=Yang-Yu | last2=Slotine | first2=Jean-Jacques | last3=Barabási | first3=Albert-László | title=Controllability of complex networks | journal=Nature | publisher=Springer Science and Business Media LLC | volume=473 | issue=7346 | year=2011 | issn=0028-0836 | doi=10.1038/nature10011 | pages=167–173}}</ref> attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed<ref name="Liu-Nature-11"/> that the minimum number of inputs or driver nodes needed to maintain full control of the network is determined by the 'maximum matching’ in the network, that is, the maximum set of links that do not share start or end nodes. From this result, an analytical framework, based on the in-out degree distribution, was developed to predict <math>n_\mathrm{D} =N_\mathrm{D}/N </math> for scale-free and Erdős–Rényi Graphs.<ref name="Liu-Nature-11"/> However, more recently it has been demonstrated that network controllability (and other structure-only methods which use exclusively the connectivity of a graph, <math>\mathbf{A}</math>, to simplify the underlying dynamics), both undershoot and overshoot the number and which sets of driver nodes best control network dynamics, highlighting the importance of redundancy (e.g. canalization) and non-linear dynamics in determining control.<ref name="gates_rocha_scirep">{{cite journal | last=Gates | first=Alexander J. | last2=Rocha | first2=Luis M. | title=Control of complex networks requires both structure and dynamics | journal=Scientific Reports | publisher=Springer Science and Business Media LLC | volume=6 | issue=1 | date=2016-04-18 | issn=2045-2322 | doi=10.1038/srep24456 | page=24456|doi-access=free}}</ref>

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the minimum number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al. attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed

<math>N \times N</math> 矩阵 <math>\mathbf{A}</math> 描述了系统的接线图和元件之间的交互强度。<math>N \times M</math> 矩阵 <math>\mathbf{B}</math> 识别由外部控制器控制的节点。系统通过控制器强加给系统的时间相关向量 <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> 来控制。为了确定驱动节点的最小数目，用<math>N_\mathrm{D}</math>来表示，其控制足以完全控制系统的动力学进程，在这方面，Liu等人成功做到了'''将结构控制理论 Structural Control Theory '''、'''图论 Graph Theory '''和'''统计物理 Statistical Physics '''的工具的结合。

It is also notable, that Liu's et al. formulation<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math> for a chain graph and for a weak densely connected graph. Obviously, both these graphs have very different in and out degree distributions. A recent unpublished work,<ref name = "Arxiv_Close_Betw">{{cite arXiv|first=SJ |last=Banerjee |first2=S |last2=Roy|title=Key to Network Controllability|eprint=1209.3737}}</ref> questions whether [[Degree (graph theory)|degree]], which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued <ref name="Liu-Nature-11"/> (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown<ref name = "Arxiv_Close_Betw"/> that network controllability can be altered only by using [[betweenness centrality]] and [[closeness centrality]], without using [[degree (graph theory)]] or degree correlations at all.

It is also notable, that Liu's et al. formulation questions whether degree, which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown that network controllability can be altered only by using betweenness centrality and closeness centrality, without using degree (graph theory) or degree correlations at all.

同样值得关注的是，刘等人的发现。他们提出'''度 Degree '''是网络中一种纯粹的局部度量，能够完全描述网络的可控性，即便是稍微远一点的节点也能确定它对网络的可控性是否有影响。事实上，对于许多'''实词网络 Real-Word Networks '''，像'''食物网络 Food Webs '''、'''神经元网络 Neuronal Network ''' 和'''代谢网络 Metabolic Network '''，Liu等人计算的<math>{n_\mathrm{D}}^{real}</math><math> 和 {n_\mathrm{D}}^\mathrm{rand\_degree}</math> 的值并不匹配。值得注意的是。如果可控性主要是由度决定，那么为什么对于许多现实世界的网络来说，<math>{n_\mathrm{D}}^{real}</math> 和 <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> 如此不同？他们认为，这可能是由于度相关性的影响。然而，已有的研究表明，网络的可控性只能通过中间性和封闭性来改变，而完全不需要使用度（图论）或'''度关联 Degree Correlations '''。(arXiv:1203.5161v1)

[[File:YYL2.pdf|right|thumb|A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem,<ref name="Lin-74"/> since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.]]

A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem, since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.

示意图显示了定向网络的控制。对于给定的定向网络（图a），可以计算其最大匹配：没有共同头部或尾部的最大边缘集。最大匹配将由一组顶点不相交的有向路径和有向循环组成（请参见图b中的红色边缘）。如果一个节点是匹配边的头部，则该节点是匹配的（图b中的绿色节点）。否则，它是不匹配的（图b中的白色节点）。那些不匹配的节点是需要控制的节点，即驱动节点。通过向这些驱动节点注入信号，可以得到一组以起点为输入的有向路径（见图c）。这些路径被称为“茎”。得到的有向图称为U根因子连接。通过将定向周期“嫁接”到那些“茎”，人们就会得到“芽”。得到的有向图称为cacti（见图d）。根据结构可控性定理，由于存在跨越受控网络的cacti结构（参见图e），因此该系统是可控的。受控网络（图e）下的cacti结构（图d）是维持可控性的“骨架”。

==Structural Controllability==
'''结构可控性 Structural Controllability 

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, IEEE Trans. Auto. Contr. 19

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, ''IEEE Trans. Auto. Contr.'' '''21'''

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, IEEE Trans. Auto. Contr. 21

(1974) .</ref>and then extended by Shields and Pearson (1976) ref name"Shields-76"r.希尔兹和 j。培生，IEEE Trans。自动。女名女子名。21

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, IEEE Trans. Auto. Contr. 21

结构性质的概念最早是由 Lin (1974) <ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''(1974).</ref>
和另外由 Glover 和 Silverman (1976)派生。 <ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''(1976).</ref>

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976年)。 / ref '''主要的问题是，对于可变系统参数来说，缺乏可控性或可观测性是否是一般性的。在结构控制的框架下，系统参数可以是独立自由变量，也可以是固定零点。这对于物理系统的模型是一致的，因为参数值永远不会准确地知道，零值除外，零值表示交互或连接的缺失。

==Maximum Matching==
'''最大匹配 Maximum Matching ''' 

In graph theory, a [[matching (graph theory)|matching]] is a set of edges without common vertices. Liu et al.<ref name="Liu-Nature-11"/> extended this definition to directed graph, where a matching is a set of directed edges that do not share start or end vertices. It is easy to check that a matching of a directed graph composes of a set of vertex-disjoint simple paths and cycles. The maximum matching of a directed network can be efficiently calculated by working in the bipartite representation using the classical [[Hopcroft–Karp algorithm]], which runs in O(''E''{{radic|''N''}}) time in the worst case. For undirected graph, analytical solutions of the size and number of maximum matchings have been studied using the [[cavity method]] developed in statistical physics.<ref name="Zdeborova-06">[[Lenka Zdeborová|L. Zdeborová]] and M. Mezard, ''J. Stat. Mech.'' '''05''' (2006).</ref> Liu et al.<ref name="Liu-Nature-11"/> extended the calculations for directed graph. 

在图论中，匹配（图论）是一组没有共同顶点的边。Liu等人将此定义扩展到有向图，其中匹配是一组不共享起始或终止顶点的有向边。很容易检查有向图的匹配是否由一组顶点不相交的简单路径和周期组成。定向网络的最大匹配可以通过使用经典的'''Hopcroft-Karp算法 '''在'''二分表示 Bipartite Representation '''中工作来有效地计算，该算法在最坏情况下的时间复杂度是O(''E''{{radic|''N''}})。对于无向图，使用统计物理中开发的'''腔法 Cavity Method '''研究了最大匹配的大小和数量的分析解。<ref name=“Zdeborova-06”>[[Lenkazdeborová| L Zdeborová]]和M.Mezard，''J.Stat.Mech'' '''05'''（2006）。</ref>Liu等人<ref name=“Liu-Nature-11”/>扩展了有向图的计算范围。
In graph theory, a matching is a set of edges without common vertices. Liu et al. Liu et al. extended the calculations for directed graph.

By calculating the maximum matchings of a wide range of real networks, Liu et al.<ref name="Liu-Nature-11"/> asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the [[cavity method]]. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al.<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether ''control robustness" in networks is influenced more by using [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> over [[degree (graph theory)]] based metrics.

By calculating the maximum matchings of a wide range of real networks, Liu et al. asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the cavity method. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al. would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether control robustness" in networks is influenced more by using betweenness centrality and closeness centrality over degree (graph theory) based metrics.

通过计算各种真实网络的最大匹配，Liu等人断言驱动程序节点的数量主要由网络度分布<math>P(k_\mathrm{in}，k_\mathrm{out})</math>决定。他们还使用空腔方法计算了具有任意程度分布的网络集合的驱动节点的平均数量。有趣的是，对于'''链图 Chain Graph '''和'''弱密集连通图 Weak Densely Connected Graph '''，两者都具有非常不同的进出度分布；Liu等人的公式将预测<math>{n_\mathrm{D}}</math>的相同值。此外，对于许多真实词网络，即食物网、神经元和代谢网络，Liu等人计算的<math>{n_\mathrm{D}}^{real}</math> 和 <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> 值的不匹配。值得注意的是。如果可控性纯粹由度决定，那为什么<math>{n_\mathrm{D}}^{real}</math>和<math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math>对于许多现实世界的网络来说是如此不同？对于网络中的控制鲁棒性是否更多地受到使用基于度（图论）的度量的'''中介中心性 Betweenness Centrality '''和'''紧密度中心性 Closeness Centrality '''的影响，仍然是开放的。

While sparser graphs are more difficult to control,<ref name="Liu-Nature-11"/><ref name = "Arxiv_Close_Betw"/> it would obviously be interesting to find whether [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> or degree heterogeneity<ref name="Liu-Nature-11"/> plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

While sparser graphs are more difficult to control, it would obviously be interesting to find whether betweenness centrality and closeness centrality or degree heterogeneity plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

虽然'''稀疏图 Sparser Graph '''更难以控制，但发现中介中心性和紧密度中心性或'''度异质性 Degree Heterogeneity '''在决定具有几乎相似度分布的稀疏图的可控性中是否起更重要的作用显然是有趣的。

==Control of composite quantum systems and algebraic graph theory==
复合量子系统的控制与代数图论 

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively.<ref name="BG-07">{{cite journal | last=Burgarth | first=Daniel | last2=Giovannetti | first2=Vittorio | title=Full Control by Locally Induced Relaxation | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=99 | issue=10 | date=2007-09-05 | issn=0031-9007 | doi=10.1103/physrevlett.99.100501 | page=100501| arxiv=0704.3027 }}</ref> This framework permits to formulate Kalman's criterion with tools from [[algebraic graph theory]] via the [[minimum rank of a graph]] and related notions.<ref name="BDHSY-13">{{cite journal|last1=Burgarth|first1=Daniel|last2=D'Alessandro|first2=Domenico|last3=Hogben|first3=Leslie|author3-link=Leslie Hogben|last4=Severini|first4=Simone|last5=Young|first5=Michael|doi=10.1109/TAC.2013.2250075|issue=9|journal=IEEE Transactions on Automatic Control|mr=3101617|pages=2349–2354|title=Zero forcing, linear and quantum controllability for systems evolving on networks|volume=58|year=2013}}</ref><ref name="OT-15">S. O'Rourke, B. Touri, https://arxiv.org/abs/1511.05080.</ref>

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively. This framework permits to formulate Kalman's criterion with tools from algebraic graph theory via the minimum rank of a graph and related notions.

在复合量子系统的通用控制的背景下，还开发了网络控制理论，其中子系统及其相互作用分别与节点和链路相关联。该框架允许使用代数图论中的工具通过图的最小等级和相关概念来制定'''卡尔曼准则 Kalman's Criterion '''。

== See also ==

* [[Controllability Gramian]]

==References==

<references/>

== External links ==

* [http://barabasilab.neu.edu/projects/controllability/ The network controllability project website]

* [https://www.youtube.com/watch?v=9-q2qpOJfkg The video showing network controllability]





!-类别-

[[Category:Network theory]]

Category:Network theory

范畴: 网络理论

<noinclude>

This page was moved from [[wikipedia:en:Network controllability]]. Its edit history can be viewed at [[网络可控性/edithistory]]</noinclude>

[[Category:待整理页面]]

网络可控性

2020-09-25T10:51:04Z

923397935：/* Control of composite quantum systems and algebraic graph theory */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 

本词条由信白初步翻译 

[[Image:YYL1.png|right|thumb|Controlling a simple network.]]

Controlling a simple network.

图1：控制一个简单的网络。

'''Network Controllability''' is concerned about the structural [[controllability]] of a [[Graph (discrete mathematics)|network]]. Controllability describes our ability to guide a [[dynamical system]] from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al.<ref>{{Cite journal|last=Sharma|first=Ankush|last2=Cinti|first2=Caterina|last3=Capobianco|first3=Enrico|date=2017|title=Multitype Network-Guided Target Controllability in Phenotypically Characterized Osteosarcoma: Role of Tumor Microenvironment|journal=Frontiers in Immunology|language=English|volume=8|pages=918|doi=10.3389/fimmu.2017.00918|pmid=28824643|pmc=5536125|issn=1664-3224}}</ref> on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

Network Controllability is concerned about the structural controllability of a network. Controllability describes our ability to guide a dynamical system from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al. on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

'''网络可控性 Network Controllability '''研究的是网络的结构可控性。可控性描述了我们在有限时间内，通过合适的输入选择，引导动力系统从任何初始状态到任何期望的最终状态的能力。这个定义与我们直观的控制概念非常吻合。一般有向加权复杂网络的可控性是近年来世界范围内许多研究小组的重点研究课题。夏尔马等人最近在多种生物学网络(基因-基因、 miRNA-基因、蛋白质-蛋白质相互作用网络)上的研究，确定了骨肉瘤表型特征的控制目标，显示了基因和蛋白质在维持肿瘤微环境中的重要作用。

==Background==
背景 

Consider the canonical linear time-invariant dynamics on a complex network

Consider the canonical linear time-invariant dynamics on a complex network

考虑'''复杂网络 Complex Network '''上的正则线性动力学

<math>
\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)

\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)
</math>

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

其中，向量 <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> 表示用 <math>t</math> 时间捕获 <math>N</math> 个节点系统。

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the ''minimum'' number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al.<ref name="Liu-Nature-11">{{cite journal | last=Liu | first=Yang-Yu | last2=Slotine | first2=Jean-Jacques | last3=Barabási | first3=Albert-László | title=Controllability of complex networks | journal=Nature | publisher=Springer Science and Business Media LLC | volume=473 | issue=7346 | year=2011 | issn=0028-0836 | doi=10.1038/nature10011 | pages=167–173}}</ref> attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed<ref name="Liu-Nature-11"/> that the minimum number of inputs or driver nodes needed to maintain full control of the network is determined by the 'maximum matching’ in the network, that is, the maximum set of links that do not share start or end nodes. From this result, an analytical framework, based on the in-out degree distribution, was developed to predict <math>n_\mathrm{D} =N_\mathrm{D}/N </math> for scale-free and Erdős–Rényi Graphs.<ref name="Liu-Nature-11"/> However, more recently it has been demonstrated that network controllability (and other structure-only methods which use exclusively the connectivity of a graph, <math>\mathbf{A}</math>, to simplify the underlying dynamics), both undershoot and overshoot the number and which sets of driver nodes best control network dynamics, highlighting the importance of redundancy (e.g. canalization) and non-linear dynamics in determining control.<ref name="gates_rocha_scirep">{{cite journal | last=Gates | first=Alexander J. | last2=Rocha | first2=Luis M. | title=Control of complex networks requires both structure and dynamics | journal=Scientific Reports | publisher=Springer Science and Business Media LLC | volume=6 | issue=1 | date=2016-04-18 | issn=2045-2322 | doi=10.1038/srep24456 | page=24456|doi-access=free}}</ref>

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the minimum number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al. attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed

<math>N \times N</math> 矩阵 <math>\mathbf{A}</math> 描述了系统的接线图和元件之间的交互强度。<math>N \times M</math> 矩阵 <math>\mathbf{B}</math> 识别由外部控制器控制的节点。系统通过控制器强加给系统的时间相关向量 <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> 来控制。为了确定驱动节点的最小数目，用<math>N_\mathrm{D}</math>来表示，其控制足以完全控制系统的动力学进程，在这方面，Liu等人成功做到了'''将结构控制理论 Structural Control Theory '''、'''图论 Graph Theory '''和'''统计物理 Statistical Physics '''的工具的结合。

It is also notable, that Liu's et al. formulation<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math> for a chain graph and for a weak densely connected graph. Obviously, both these graphs have very different in and out degree distributions. A recent unpublished work,<ref name = "Arxiv_Close_Betw">{{cite arXiv|first=SJ |last=Banerjee |first2=S |last2=Roy|title=Key to Network Controllability|eprint=1209.3737}}</ref> questions whether [[Degree (graph theory)|degree]], which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued <ref name="Liu-Nature-11"/> (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown<ref name = "Arxiv_Close_Betw"/> that network controllability can be altered only by using [[betweenness centrality]] and [[closeness centrality]], without using [[degree (graph theory)]] or degree correlations at all.

It is also notable, that Liu's et al. formulation questions whether degree, which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown that network controllability can be altered only by using betweenness centrality and closeness centrality, without using degree (graph theory) or degree correlations at all.

同样值得关注的是，刘等人的发现。他们提出'''度 Degree '''是网络中一种纯粹的局部度量，能够完全描述网络的可控性，即便是稍微远一点的节点也能确定它对网络的可控性是否有影响。事实上，对于许多'''实词网络 Real-Word Networks '''，像'''食物网络 Food Webs '''、'''神经元网络 Neuronal Network ''' 和'''代谢网络 Metabolic Network '''，Liu等人计算的<math>{n_\mathrm{D}}^{real}</math><math> 和 {n_\mathrm{D}}^\mathrm{rand\_degree}</math> 的值并不匹配。值得注意的是。如果可控性主要是由度决定，那么为什么对于许多现实世界的网络来说，<math>{n_\mathrm{D}}^{real}</math> 和 <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> 如此不同？他们认为，这可能是由于度相关性的影响。然而，已有的研究表明，网络的可控性只能通过中间性和封闭性来改变，而完全不需要使用度（图论）或'''度关联 Degree Correlations '''。(arXiv:1203.5161v1)

[[File:YYL2.pdf|right|thumb|A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem,<ref name="Lin-74"/> since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.]]

A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem, since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.

示意图显示了定向网络的控制。对于给定的定向网络（图a），可以计算其最大匹配：没有共同头部或尾部的最大边缘集。最大匹配将由一组顶点不相交的有向路径和有向循环组成（请参见图b中的红色边缘）。如果一个节点是匹配边的头部，则该节点是匹配的（图b中的绿色节点）。否则，它是不匹配的（图b中的白色节点）。那些不匹配的节点是需要控制的节点，即驱动节点。通过向这些驱动节点注入信号，可以得到一组以起点为输入的有向路径（见图c）。这些路径被称为“茎”。得到的有向图称为U根因子连接。通过将定向周期“嫁接”到那些“茎”，人们就会得到“芽”。得到的有向图称为cacti（见图d）。根据结构可控性定理，由于存在跨越受控网络的cacti结构（参见图e），因此该系统是可控的。受控网络（图e）下的cacti结构（图d）是维持可控性的“骨架”。

==Structural Controllability==
'''结构可控性 Structural Controllability 

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, IEEE Trans. Auto. Contr. 19

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, ''IEEE Trans. Auto. Contr.'' '''21'''

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, IEEE Trans. Auto. Contr. 21

(1974) .</ref>and then extended by Shields and Pearson (1976) ref name"Shields-76"r.希尔兹和 j。培生，IEEE Trans。自动。女名女子名。21

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, IEEE Trans. Auto. Contr. 21

结构性质的概念最早是由 Lin (1974) <ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''(1974).</ref>
和另外由 Glover 和 Silverman (1976)派生。 <ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''(1976).</ref>

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976年)。 / ref '''主要的问题是，对于可变系统参数来说，缺乏可控性或可观测性是否是一般性的。在结构控制的框架下，系统参数可以是独立自由变量，也可以是固定零点。这对于物理系统的模型是一致的，因为参数值永远不会准确地知道，零值除外，零值表示交互或连接的缺失。

==Maximum Matching==
'''最大匹配 Maximum Matching ''' 

In graph theory, a [[matching (graph theory)|matching]] is a set of edges without common vertices. Liu et al.<ref name="Liu-Nature-11"/> extended this definition to directed graph, where a matching is a set of directed edges that do not share start or end vertices. It is easy to check that a matching of a directed graph composes of a set of vertex-disjoint simple paths and cycles. The maximum matching of a directed network can be efficiently calculated by working in the bipartite representation using the classical [[Hopcroft–Karp algorithm]], which runs in O(''E''{{radic|''N''}}) time in the worst case. For undirected graph, analytical solutions of the size and number of maximum matchings have been studied using the [[cavity method]] developed in statistical physics.<ref name="Zdeborova-06">[[Lenka Zdeborová|L. Zdeborová]] and M. Mezard, ''J. Stat. Mech.'' '''05''' (2006).</ref> Liu et al.<ref name="Liu-Nature-11"/> extended the calculations for directed graph.
在图论中，匹配（图论）是一组没有共同顶点的边。Liu等人将此定义扩展到有向图，其中匹配是一组不共享起始或终止顶点的有向边。很容易检查有向图的匹配是否由一组顶点不相交的简单路径和周期组成。定向网络的最大匹配可以通过使用经典的'''Hopcroft-Karp算法 '''在'''二分表示 Bipartite Representation '''中工作来有效地计算，该算法在最坏情况下的时间复杂度是O(''E''{{radic|''N''}})。对于无向图，使用统计物理中开发的'''腔法 Cavity Method '''研究了最大匹配的大小和数量的分析解。<ref name=“Zdeborova-06”>[[Lenkazdeborová| L Zdeborová]]和M.Mezard，''J.Stat.Mech'' '''05'''（2006）。</ref>Liu等人<ref name=“Liu-Nature-11”/>扩展了有向图的计算范围。
In graph theory, a matching is a set of edges without common vertices. Liu et al. Liu et al. extended the calculations for directed graph.

By calculating the maximum matchings of a wide range of real networks, Liu et al.<ref name="Liu-Nature-11"/> asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the [[cavity method]]. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al.<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether ''control robustness" in networks is influenced more by using [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> over [[degree (graph theory)]] based metrics.

By calculating the maximum matchings of a wide range of real networks, Liu et al. asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the cavity method. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al. would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether control robustness" in networks is influenced more by using betweenness centrality and closeness centrality over degree (graph theory) based metrics.

通过计算各种真实网络的最大匹配，Liu等人断言驱动程序节点的数量主要由网络度分布<math>P(k_\mathrm{in}，k_\mathrm{out})</math>决定。他们还使用空腔方法计算了具有任意程度分布的网络集合的驱动节点的平均数量。有趣的是，对于'''链图 Chain Graph '''和'''弱密集连通图 Weak Densely Connected Graph '''，两者都具有非常不同的进出度分布；Liu等人的公式将预测<math>{n_\mathrm{D}}</math>的相同值。此外，对于许多真实词网络，即食物网、神经元和代谢网络，Liu等人计算的<math>{n_\mathrm{D}}^{real}</math> 和 <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> 值的不匹配。值得注意的是。如果可控性纯粹由度决定，那为什么<math>{n_\mathrm{D}}^{real}</math>和<math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math>对于许多现实世界的网络来说是如此不同？对于网络中的控制鲁棒性是否更多地受到使用基于度（图论）的度量的'''中介中心性 Betweenness Centrality '''和'''紧密度中心性 Closeness Centrality '''的影响，仍然是开放的。

While sparser graphs are more difficult to control,<ref name="Liu-Nature-11"/><ref name = "Arxiv_Close_Betw"/> it would obviously be interesting to find whether [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> or degree heterogeneity<ref name="Liu-Nature-11"/> plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

While sparser graphs are more difficult to control, it would obviously be interesting to find whether betweenness centrality and closeness centrality or degree heterogeneity plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

虽然'''稀疏图 Sparser Graph '''更难以控制，但发现中介中心性和紧密度中心性或'''度异质性 Degree Heterogeneity '''在决定具有几乎相似度分布的稀疏图的可控性中是否起更重要的作用显然是有趣的。

==Control of composite quantum systems and algebraic graph theory==
复合量子系统的控制与代数图论 

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively.<ref name="BG-07">{{cite journal | last=Burgarth | first=Daniel | last2=Giovannetti | first2=Vittorio | title=Full Control by Locally Induced Relaxation | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=99 | issue=10 | date=2007-09-05 | issn=0031-9007 | doi=10.1103/physrevlett.99.100501 | page=100501| arxiv=0704.3027 }}</ref> This framework permits to formulate Kalman's criterion with tools from [[algebraic graph theory]] via the [[minimum rank of a graph]] and related notions.<ref name="BDHSY-13">{{cite journal|last1=Burgarth|first1=Daniel|last2=D'Alessandro|first2=Domenico|last3=Hogben|first3=Leslie|author3-link=Leslie Hogben|last4=Severini|first4=Simone|last5=Young|first5=Michael|doi=10.1109/TAC.2013.2250075|issue=9|journal=IEEE Transactions on Automatic Control|mr=3101617|pages=2349–2354|title=Zero forcing, linear and quantum controllability for systems evolving on networks|volume=58|year=2013}}</ref><ref name="OT-15">S. O'Rourke, B. Touri, https://arxiv.org/abs/1511.05080.</ref>

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively. This framework permits to formulate Kalman's criterion with tools from algebraic graph theory via the minimum rank of a graph and related notions.

在复合量子系统的通用控制的背景下，还开发了网络控制理论，其中子系统及其相互作用分别与节点和链路相关联。该框架允许使用代数图论中的工具通过图的最小等级和相关概念来制定'''卡尔曼准则 Kalman's Criterion '''。

== See also ==

* [[Controllability Gramian]]

==References==

<references/>

== External links ==

* [http://barabasilab.neu.edu/projects/controllability/ The network controllability project website]

* [https://www.youtube.com/watch?v=9-q2qpOJfkg The video showing network controllability]





!-类别-

[[Category:Network theory]]

Category:Network theory

范畴: 网络理论

<noinclude>

This page was moved from [[wikipedia:en:Network controllability]]. Its edit history can be viewed at [[网络可控性/edithistory]]</noinclude>

[[Category:待整理页面]]

网络可控性

2020-09-25T10:49:03Z

923397935：/* Maximum Matching */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 

本词条由信白初步翻译 

[[Image:YYL1.png|right|thumb|Controlling a simple network.]]

Controlling a simple network.

图1：控制一个简单的网络。

'''Network Controllability''' is concerned about the structural [[controllability]] of a [[Graph (discrete mathematics)|network]]. Controllability describes our ability to guide a [[dynamical system]] from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al.<ref>{{Cite journal|last=Sharma|first=Ankush|last2=Cinti|first2=Caterina|last3=Capobianco|first3=Enrico|date=2017|title=Multitype Network-Guided Target Controllability in Phenotypically Characterized Osteosarcoma: Role of Tumor Microenvironment|journal=Frontiers in Immunology|language=English|volume=8|pages=918|doi=10.3389/fimmu.2017.00918|pmid=28824643|pmc=5536125|issn=1664-3224}}</ref> on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

Network Controllability is concerned about the structural controllability of a network. Controllability describes our ability to guide a dynamical system from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al. on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

'''网络可控性 Network Controllability '''研究的是网络的结构可控性。可控性描述了我们在有限时间内，通过合适的输入选择，引导动力系统从任何初始状态到任何期望的最终状态的能力。这个定义与我们直观的控制概念非常吻合。一般有向加权复杂网络的可控性是近年来世界范围内许多研究小组的重点研究课题。夏尔马等人最近在多种生物学网络(基因-基因、 miRNA-基因、蛋白质-蛋白质相互作用网络)上的研究，确定了骨肉瘤表型特征的控制目标，显示了基因和蛋白质在维持肿瘤微环境中的重要作用。

==Background==
背景 

Consider the canonical linear time-invariant dynamics on a complex network

Consider the canonical linear time-invariant dynamics on a complex network

考虑'''复杂网络 Complex Network '''上的正则线性动力学

<math>
\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)

\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)
</math>

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

其中，向量 <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> 表示用 <math>t</math> 时间捕获 <math>N</math> 个节点系统。

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the ''minimum'' number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al.<ref name="Liu-Nature-11">{{cite journal | last=Liu | first=Yang-Yu | last2=Slotine | first2=Jean-Jacques | last3=Barabási | first3=Albert-László | title=Controllability of complex networks | journal=Nature | publisher=Springer Science and Business Media LLC | volume=473 | issue=7346 | year=2011 | issn=0028-0836 | doi=10.1038/nature10011 | pages=167–173}}</ref> attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed<ref name="Liu-Nature-11"/> that the minimum number of inputs or driver nodes needed to maintain full control of the network is determined by the 'maximum matching’ in the network, that is, the maximum set of links that do not share start or end nodes. From this result, an analytical framework, based on the in-out degree distribution, was developed to predict <math>n_\mathrm{D} =N_\mathrm{D}/N </math> for scale-free and Erdős–Rényi Graphs.<ref name="Liu-Nature-11"/> However, more recently it has been demonstrated that network controllability (and other structure-only methods which use exclusively the connectivity of a graph, <math>\mathbf{A}</math>, to simplify the underlying dynamics), both undershoot and overshoot the number and which sets of driver nodes best control network dynamics, highlighting the importance of redundancy (e.g. canalization) and non-linear dynamics in determining control.<ref name="gates_rocha_scirep">{{cite journal | last=Gates | first=Alexander J. | last2=Rocha | first2=Luis M. | title=Control of complex networks requires both structure and dynamics | journal=Scientific Reports | publisher=Springer Science and Business Media LLC | volume=6 | issue=1 | date=2016-04-18 | issn=2045-2322 | doi=10.1038/srep24456 | page=24456|doi-access=free}}</ref>

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the minimum number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al. attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed

<math>N \times N</math> 矩阵 <math>\mathbf{A}</math> 描述了系统的接线图和元件之间的交互强度。<math>N \times M</math> 矩阵 <math>\mathbf{B}</math> 识别由外部控制器控制的节点。系统通过控制器强加给系统的时间相关向量 <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> 来控制。为了确定驱动节点的最小数目，用<math>N_\mathrm{D}</math>来表示，其控制足以完全控制系统的动力学进程，在这方面，Liu等人成功做到了'''将结构控制理论 Structural Control Theory '''、'''图论 Graph Theory '''和'''统计物理 Statistical Physics '''的工具的结合。

It is also notable, that Liu's et al. formulation<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math> for a chain graph and for a weak densely connected graph. Obviously, both these graphs have very different in and out degree distributions. A recent unpublished work,<ref name = "Arxiv_Close_Betw">{{cite arXiv|first=SJ |last=Banerjee |first2=S |last2=Roy|title=Key to Network Controllability|eprint=1209.3737}}</ref> questions whether [[Degree (graph theory)|degree]], which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued <ref name="Liu-Nature-11"/> (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown<ref name = "Arxiv_Close_Betw"/> that network controllability can be altered only by using [[betweenness centrality]] and [[closeness centrality]], without using [[degree (graph theory)]] or degree correlations at all.

It is also notable, that Liu's et al. formulation questions whether degree, which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown that network controllability can be altered only by using betweenness centrality and closeness centrality, without using degree (graph theory) or degree correlations at all.

同样值得关注的是，刘等人的发现。他们提出'''度 Degree '''是网络中一种纯粹的局部度量，能够完全描述网络的可控性，即便是稍微远一点的节点也能确定它对网络的可控性是否有影响。事实上，对于许多'''实词网络 Real-Word Networks '''，像'''食物网络 Food Webs '''、'''神经元网络 Neuronal Network ''' 和'''代谢网络 Metabolic Network '''，Liu等人计算的<math>{n_\mathrm{D}}^{real}</math><math> 和 {n_\mathrm{D}}^\mathrm{rand\_degree}</math> 的值并不匹配。值得注意的是。如果可控性主要是由度决定，那么为什么对于许多现实世界的网络来说，<math>{n_\mathrm{D}}^{real}</math> 和 <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> 如此不同？他们认为，这可能是由于度相关性的影响。然而，已有的研究表明，网络的可控性只能通过中间性和封闭性来改变，而完全不需要使用度（图论）或'''度关联 Degree Correlations '''。(arXiv:1203.5161v1)

[[File:YYL2.pdf|right|thumb|A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem,<ref name="Lin-74"/> since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.]]

A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem, since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.

示意图显示了定向网络的控制。对于给定的定向网络（图a），可以计算其最大匹配：没有共同头部或尾部的最大边缘集。最大匹配将由一组顶点不相交的有向路径和有向循环组成（请参见图b中的红色边缘）。如果一个节点是匹配边的头部，则该节点是匹配的（图b中的绿色节点）。否则，它是不匹配的（图b中的白色节点）。那些不匹配的节点是需要控制的节点，即驱动节点。通过向这些驱动节点注入信号，可以得到一组以起点为输入的有向路径（见图c）。这些路径被称为“茎”。得到的有向图称为U根因子连接。通过将定向周期“嫁接”到那些“茎”，人们就会得到“芽”。得到的有向图称为cacti（见图d）。根据结构可控性定理，由于存在跨越受控网络的cacti结构（参见图e），因此该系统是可控的。受控网络（图e）下的cacti结构（图d）是维持可控性的“骨架”。

==Structural Controllability==
'''结构可控性 Structural Controllability 

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, IEEE Trans. Auto. Contr. 19

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, ''IEEE Trans. Auto. Contr.'' '''21'''

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, IEEE Trans. Auto. Contr. 21

(1974) .</ref>and then extended by Shields and Pearson (1976) ref name"Shields-76"r.希尔兹和 j。培生，IEEE Trans。自动。女名女子名。21

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, IEEE Trans. Auto. Contr. 21

结构性质的概念最早是由 Lin (1974) <ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''(1974).</ref>
和另外由 Glover 和 Silverman (1976)派生。 <ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''(1976).</ref>

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976年)。 / ref '''主要的问题是，对于可变系统参数来说，缺乏可控性或可观测性是否是一般性的。在结构控制的框架下，系统参数可以是独立自由变量，也可以是固定零点。这对于物理系统的模型是一致的，因为参数值永远不会准确地知道，零值除外，零值表示交互或连接的缺失。

==Maximum Matching==
'''最大匹配 Maximum Matching ''' 

In graph theory, a [[matching (graph theory)|matching]] is a set of edges without common vertices. Liu et al.<ref name="Liu-Nature-11"/> extended this definition to directed graph, where a matching is a set of directed edges that do not share start or end vertices. It is easy to check that a matching of a directed graph composes of a set of vertex-disjoint simple paths and cycles. The maximum matching of a directed network can be efficiently calculated by working in the bipartite representation using the classical [[Hopcroft–Karp algorithm]], which runs in O(''E''{{radic|''N''}}) time in the worst case. For undirected graph, analytical solutions of the size and number of maximum matchings have been studied using the [[cavity method]] developed in statistical physics.<ref name="Zdeborova-06">[[Lenka Zdeborová|L. Zdeborová]] and M. Mezard, ''J. Stat. Mech.'' '''05''' (2006).</ref> Liu et al.<ref name="Liu-Nature-11"/> extended the calculations for directed graph.
在图论中，匹配（图论）是一组没有共同顶点的边。Liu等人将此定义扩展到有向图，其中匹配是一组不共享起始或终止顶点的有向边。很容易检查有向图的匹配是否由一组顶点不相交的简单路径和周期组成。定向网络的最大匹配可以通过使用经典的'''Hopcroft-Karp算法 '''在'''二分表示 Bipartite Representation '''中工作来有效地计算，该算法在最坏情况下的时间复杂度是O(''E''{{radic|''N''}})。对于无向图，使用统计物理中开发的'''腔法 Cavity Method '''研究了最大匹配的大小和数量的分析解。<ref name=“Zdeborova-06”>[[Lenkazdeborová| L Zdeborová]]和M.Mezard，''J.Stat.Mech'' '''05'''（2006）。</ref>Liu等人<ref name=“Liu-Nature-11”/>扩展了有向图的计算范围。
In graph theory, a matching is a set of edges without common vertices. Liu et al. Liu et al. extended the calculations for directed graph.

By calculating the maximum matchings of a wide range of real networks, Liu et al.<ref name="Liu-Nature-11"/> asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the [[cavity method]]. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al.<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether ''control robustness" in networks is influenced more by using [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> over [[degree (graph theory)]] based metrics.

By calculating the maximum matchings of a wide range of real networks, Liu et al. asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the cavity method. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al. would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether control robustness" in networks is influenced more by using betweenness centrality and closeness centrality over degree (graph theory) based metrics.

通过计算各种真实网络的最大匹配，Liu等人断言驱动程序节点的数量主要由网络度分布<math>P(k_\mathrm{in}，k_\mathrm{out})</math>决定。他们还使用空腔方法计算了具有任意程度分布的网络集合的驱动节点的平均数量。有趣的是，对于'''链图 Chain Graph '''和'''弱密集连通图 Weak Densely Connected Graph '''，两者都具有非常不同的进出度分布；Liu等人的公式将预测<math>{n_\mathrm{D}}</math>的相同值。此外，对于许多真实词网络，即食物网、神经元和代谢网络，Liu等人计算的<math>{n_\mathrm{D}}^{real}</math> 和 <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> 值的不匹配。值得注意的是。如果可控性纯粹由度决定，那为什么<math>{n_\mathrm{D}}^{real}</math>和<math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math>对于许多现实世界的网络来说是如此不同？对于网络中的控制鲁棒性是否更多地受到使用基于度（图论）的度量的'''中介中心性 Betweenness Centrality '''和'''紧密度中心性 Closeness Centrality '''的影响，仍然是开放的。

While sparser graphs are more difficult to control,<ref name="Liu-Nature-11"/><ref name = "Arxiv_Close_Betw"/> it would obviously be interesting to find whether [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> or degree heterogeneity<ref name="Liu-Nature-11"/> plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

While sparser graphs are more difficult to control, it would obviously be interesting to find whether betweenness centrality and closeness centrality or degree heterogeneity plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

虽然'''稀疏图 Sparser Graph '''更难以控制，但发现中介中心性和紧密度中心性或'''度异质性 Degree Heterogeneity '''在决定具有几乎相似度分布的稀疏图的可控性中是否起更重要的作用显然是有趣的。

==Control of composite quantum systems and algebraic graph theory==

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively.<ref name="BG-07">{{cite journal | last=Burgarth | first=Daniel | last2=Giovannetti | first2=Vittorio | title=Full Control by Locally Induced Relaxation | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=99 | issue=10 | date=2007-09-05 | issn=0031-9007 | doi=10.1103/physrevlett.99.100501 | page=100501| arxiv=0704.3027 }}</ref> This framework permits to formulate Kalman's criterion with tools from [[algebraic graph theory]] via the [[minimum rank of a graph]] and related notions.<ref name="BDHSY-13">{{cite journal|last1=Burgarth|first1=Daniel|last2=D'Alessandro|first2=Domenico|last3=Hogben|first3=Leslie|author3-link=Leslie Hogben|last4=Severini|first4=Simone|last5=Young|first5=Michael|doi=10.1109/TAC.2013.2250075|issue=9|journal=IEEE Transactions on Automatic Control|mr=3101617|pages=2349–2354|title=Zero forcing, linear and quantum controllability for systems evolving on networks|volume=58|year=2013}}</ref><ref name="OT-15">S. O'Rourke, B. Touri, https://arxiv.org/abs/1511.05080.</ref>

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively. This framework permits to formulate Kalman's criterion with tools from algebraic graph theory via the minimum rank of a graph and related notions.

在复合量子系统的通用控制的背景下，也发展了网络控制理论，其中子系统及其相互作用分别与节点和链路相关联。这个框架允许用代数图论的工具通过图的最小秩和相关概念来建立卡尔曼准则。

== See also ==

* [[Controllability Gramian]]

==References==

<references/>

== External links ==

* [http://barabasilab.neu.edu/projects/controllability/ The network controllability project website]

* [https://www.youtube.com/watch?v=9-q2qpOJfkg The video showing network controllability]





!-类别-

[[Category:Network theory]]

Category:Network theory

范畴: 网络理论

<noinclude>

This page was moved from [[wikipedia:en:Network controllability]]. Its edit history can be viewed at [[网络可控性/edithistory]]</noinclude>

[[Category:待整理页面]]

网络可控性

2020-09-25T10:46:44Z

923397935：/* Maximum Matching */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 

本词条由信白初步翻译 

[[Image:YYL1.png|right|thumb|Controlling a simple network.]]

Controlling a simple network.

图1：控制一个简单的网络。

'''Network Controllability''' is concerned about the structural [[controllability]] of a [[Graph (discrete mathematics)|network]]. Controllability describes our ability to guide a [[dynamical system]] from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al.<ref>{{Cite journal|last=Sharma|first=Ankush|last2=Cinti|first2=Caterina|last3=Capobianco|first3=Enrico|date=2017|title=Multitype Network-Guided Target Controllability in Phenotypically Characterized Osteosarcoma: Role of Tumor Microenvironment|journal=Frontiers in Immunology|language=English|volume=8|pages=918|doi=10.3389/fimmu.2017.00918|pmid=28824643|pmc=5536125|issn=1664-3224}}</ref> on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

Network Controllability is concerned about the structural controllability of a network. Controllability describes our ability to guide a dynamical system from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al. on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

'''网络可控性 Network Controllability '''研究的是网络的结构可控性。可控性描述了我们在有限时间内，通过合适的输入选择，引导动力系统从任何初始状态到任何期望的最终状态的能力。这个定义与我们直观的控制概念非常吻合。一般有向加权复杂网络的可控性是近年来世界范围内许多研究小组的重点研究课题。夏尔马等人最近在多种生物学网络(基因-基因、 miRNA-基因、蛋白质-蛋白质相互作用网络)上的研究，确定了骨肉瘤表型特征的控制目标，显示了基因和蛋白质在维持肿瘤微环境中的重要作用。

==Background==
背景 

Consider the canonical linear time-invariant dynamics on a complex network

Consider the canonical linear time-invariant dynamics on a complex network

考虑'''复杂网络 Complex Network '''上的正则线性动力学

<math>
\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)

\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)
</math>

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

其中，向量 <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> 表示用 <math>t</math> 时间捕获 <math>N</math> 个节点系统。

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the ''minimum'' number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al.<ref name="Liu-Nature-11">{{cite journal | last=Liu | first=Yang-Yu | last2=Slotine | first2=Jean-Jacques | last3=Barabási | first3=Albert-László | title=Controllability of complex networks | journal=Nature | publisher=Springer Science and Business Media LLC | volume=473 | issue=7346 | year=2011 | issn=0028-0836 | doi=10.1038/nature10011 | pages=167–173}}</ref> attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed<ref name="Liu-Nature-11"/> that the minimum number of inputs or driver nodes needed to maintain full control of the network is determined by the 'maximum matching’ in the network, that is, the maximum set of links that do not share start or end nodes. From this result, an analytical framework, based on the in-out degree distribution, was developed to predict <math>n_\mathrm{D} =N_\mathrm{D}/N </math> for scale-free and Erdős–Rényi Graphs.<ref name="Liu-Nature-11"/> However, more recently it has been demonstrated that network controllability (and other structure-only methods which use exclusively the connectivity of a graph, <math>\mathbf{A}</math>, to simplify the underlying dynamics), both undershoot and overshoot the number and which sets of driver nodes best control network dynamics, highlighting the importance of redundancy (e.g. canalization) and non-linear dynamics in determining control.<ref name="gates_rocha_scirep">{{cite journal | last=Gates | first=Alexander J. | last2=Rocha | first2=Luis M. | title=Control of complex networks requires both structure and dynamics | journal=Scientific Reports | publisher=Springer Science and Business Media LLC | volume=6 | issue=1 | date=2016-04-18 | issn=2045-2322 | doi=10.1038/srep24456 | page=24456|doi-access=free}}</ref>

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the minimum number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al. attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed

<math>N \times N</math> 矩阵 <math>\mathbf{A}</math> 描述了系统的接线图和元件之间的交互强度。<math>N \times M</math> 矩阵 <math>\mathbf{B}</math> 识别由外部控制器控制的节点。系统通过控制器强加给系统的时间相关向量 <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> 来控制。为了确定驱动节点的最小数目，用<math>N_\mathrm{D}</math>来表示，其控制足以完全控制系统的动力学进程，在这方面，Liu等人成功做到了'''将结构控制理论 Structural Control Theory '''、'''图论 Graph Theory '''和'''统计物理 Statistical Physics '''的工具的结合。

It is also notable, that Liu's et al. formulation<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math> for a chain graph and for a weak densely connected graph. Obviously, both these graphs have very different in and out degree distributions. A recent unpublished work,<ref name = "Arxiv_Close_Betw">{{cite arXiv|first=SJ |last=Banerjee |first2=S |last2=Roy|title=Key to Network Controllability|eprint=1209.3737}}</ref> questions whether [[Degree (graph theory)|degree]], which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued <ref name="Liu-Nature-11"/> (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown<ref name = "Arxiv_Close_Betw"/> that network controllability can be altered only by using [[betweenness centrality]] and [[closeness centrality]], without using [[degree (graph theory)]] or degree correlations at all.

It is also notable, that Liu's et al. formulation questions whether degree, which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown that network controllability can be altered only by using betweenness centrality and closeness centrality, without using degree (graph theory) or degree correlations at all.

同样值得关注的是，刘等人的发现。他们提出'''度 Degree '''是网络中一种纯粹的局部度量，能够完全描述网络的可控性，即便是稍微远一点的节点也能确定它对网络的可控性是否有影响。事实上，对于许多'''实词网络 Real-Word Networks '''，像'''食物网络 Food Webs '''、'''神经元网络 Neuronal Network ''' 和'''代谢网络 Metabolic Network '''，Liu等人计算的<math>{n_\mathrm{D}}^{real}</math><math> 和 {n_\mathrm{D}}^\mathrm{rand\_degree}</math> 的值并不匹配。值得注意的是。如果可控性主要是由度决定，那么为什么对于许多现实世界的网络来说，<math>{n_\mathrm{D}}^{real}</math> 和 <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> 如此不同？他们认为，这可能是由于度相关性的影响。然而，已有的研究表明，网络的可控性只能通过中间性和封闭性来改变，而完全不需要使用度（图论）或'''度关联 Degree Correlations '''。(arXiv:1203.5161v1)

[[File:YYL2.pdf|right|thumb|A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem,<ref name="Lin-74"/> since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.]]

A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem, since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.

示意图显示了定向网络的控制。对于给定的定向网络（图a），可以计算其最大匹配：没有共同头部或尾部的最大边缘集。最大匹配将由一组顶点不相交的有向路径和有向循环组成（请参见图b中的红色边缘）。如果一个节点是匹配边的头部，则该节点是匹配的（图b中的绿色节点）。否则，它是不匹配的（图b中的白色节点）。那些不匹配的节点是需要控制的节点，即驱动节点。通过向这些驱动节点注入信号，可以得到一组以起点为输入的有向路径（见图c）。这些路径被称为“茎”。得到的有向图称为U根因子连接。通过将定向周期“嫁接”到那些“茎”，人们就会得到“芽”。得到的有向图称为cacti（见图d）。根据结构可控性定理，由于存在跨越受控网络的cacti结构（参见图e），因此该系统是可控的。受控网络（图e）下的cacti结构（图d）是维持可控性的“骨架”。

==Structural Controllability==
'''结构可控性 Structural Controllability 

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, IEEE Trans. Auto. Contr. 19

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, ''IEEE Trans. Auto. Contr.'' '''21'''

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, IEEE Trans. Auto. Contr. 21

(1974) .</ref>and then extended by Shields and Pearson (1976) ref name"Shields-76"r.希尔兹和 j。培生，IEEE Trans。自动。女名女子名。21

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, IEEE Trans. Auto. Contr. 21

结构性质的概念最早是由 Lin (1974) <ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''(1974).</ref>
和另外由 Glover 和 Silverman (1976)派生。 <ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''(1976).</ref>

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976年)。 / ref '''主要的问题是，对于可变系统参数来说，缺乏可控性或可观测性是否是一般性的。在结构控制的框架下，系统参数可以是独立自由变量，也可以是固定零点。这对于物理系统的模型是一致的，因为参数值永远不会准确地知道，零值除外，零值表示交互或连接的缺失。

==Maximum Matching==
'''最大匹配 Maximum Matching ''' 

In graph theory, a [[matching (graph theory)|matching]] is a set of edges without common vertices. Liu et al.<ref name="Liu-Nature-11"/> extended this definition to directed graph, where a matching is a set of directed edges that do not share start or end vertices. It is easy to check that a matching of a directed graph composes of a set of vertex-disjoint simple paths and cycles. The maximum matching of a directed network can be efficiently calculated by working in the bipartite representation using the classical [[Hopcroft–Karp algorithm]], which runs in O(''E''{{radic|''N''}}) time in the worst case. For undirected graph, analytical solutions of the size and number of maximum matchings have been studied using the [[cavity method]] developed in statistical physics.<ref name="Zdeborova-06">[[Lenka Zdeborová|L. Zdeborová]] and M. Mezard, ''J. Stat. Mech.'' '''05''' (2006).</ref> Liu et al.<ref name="Liu-Nature-11"/> extended the calculations for directed graph.
在图论中，匹配（图论）是一组没有共同顶点的边。Liu等人将此定义扩展到有向图，其中匹配是一组不共享起始或终止顶点的有向边。很容易检查有向图的匹配是否由一组顶点不相交的简单路径和周期组成。定向网络的最大匹配可以通过使用经典的'''Hopcroft-Karp算法 '''在'''二分表示 Bipartite Representation '''中工作来有效地计算，该算法在最坏情况下的时间复杂度是O(''E''{{radic|''N''}})。对于无向图，使用统计物理中开发的'''腔法 Cavity Method '''研究了最大匹配的大小和数量的分析解。<ref name=“Zdeborova-06”>[[Lenkazdeborová| L Zdeborová]]和M.Mezard，''J.Stat.Mech'' '''05'''（2006）。</ref>Liu等人<ref name=“Liu-Nature-11”/>扩展了有向图的计算范围。
In graph theory, a matching is a set of edges without common vertices. Liu et al. Liu et al. extended the calculations for directed graph.

By calculating the maximum matchings of a wide range of real networks, Liu et al.<ref name="Liu-Nature-11"/> asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the [[cavity method]]. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al.<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether ''control robustness" in networks is influenced more by using [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> over [[degree (graph theory)]] based metrics.

By calculating the maximum matchings of a wide range of real networks, Liu et al. asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the cavity method. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al. would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether control robustness" in networks is influenced more by using betweenness centrality and closeness centrality over degree (graph theory) based metrics.

通过计算各种真实网络的最大匹配，Liu等人断言驱动程序节点的数量主要由网络度分布<math>P(k_\mathrm{in}，k_\mathrm{out})</math>决定。他们还使用空腔方法计算了具有任意程度分布的网络集合的驱动节点的平均数量。有趣的是，对于'''链图 Chain Graph '''和'''弱密集连通图 Weak Densely Connected Graph '''，两者都具有非常不同的进出度分布；Liu等人的公式将预测<math>{n_\mathrm{D}}</math>的相同值。此外，对于许多真实词网络，即食物网、神经元和代谢网络，Liu等人计算的<math>{n_\mathrm{D}}^{real}</math> 和 <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> 值的不匹配。值得注意的是。如果可控性纯粹由度决定，那为什么<math>{n_\mathrm{D}}^{real}</math>和<math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math>对于许多现实世界的网络来说是如此不同？对于网络中的控制鲁棒性是否更多地受到使用基于度（图论）的度量的'''中介中心性 Betweenness Centrality '''和'''紧密度中心性 Closeness Centrality '''的影响，仍然是开放的。

While sparser graphs are more difficult to control,<ref name="Liu-Nature-11"/><ref name = "Arxiv_Close_Betw"/> it would obviously be interesting to find whether [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> or degree heterogeneity<ref name="Liu-Nature-11"/> plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

While sparser graphs are more difficult to control, it would obviously be interesting to find whether betweenness centrality and closeness centrality or degree heterogeneity plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

尽管稀疏图的可控性难以控制，但是研究具有几乎相似度分布的稀疏图的可控性中，是否存在介于介于中间中心性和封闭中心性或度异质性起着更重要的作用显然是很有意义的。

==Control of composite quantum systems and algebraic graph theory==

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively.<ref name="BG-07">{{cite journal | last=Burgarth | first=Daniel | last2=Giovannetti | first2=Vittorio | title=Full Control by Locally Induced Relaxation | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=99 | issue=10 | date=2007-09-05 | issn=0031-9007 | doi=10.1103/physrevlett.99.100501 | page=100501| arxiv=0704.3027 }}</ref> This framework permits to formulate Kalman's criterion with tools from [[algebraic graph theory]] via the [[minimum rank of a graph]] and related notions.<ref name="BDHSY-13">{{cite journal|last1=Burgarth|first1=Daniel|last2=D'Alessandro|first2=Domenico|last3=Hogben|first3=Leslie|author3-link=Leslie Hogben|last4=Severini|first4=Simone|last5=Young|first5=Michael|doi=10.1109/TAC.2013.2250075|issue=9|journal=IEEE Transactions on Automatic Control|mr=3101617|pages=2349–2354|title=Zero forcing, linear and quantum controllability for systems evolving on networks|volume=58|year=2013}}</ref><ref name="OT-15">S. O'Rourke, B. Touri, https://arxiv.org/abs/1511.05080.</ref>

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively. This framework permits to formulate Kalman's criterion with tools from algebraic graph theory via the minimum rank of a graph and related notions.

在复合量子系统的通用控制的背景下，也发展了网络控制理论，其中子系统及其相互作用分别与节点和链路相关联。这个框架允许用代数图论的工具通过图的最小秩和相关概念来建立卡尔曼准则。

== See also ==

* [[Controllability Gramian]]

==References==

<references/>

== External links ==

* [http://barabasilab.neu.edu/projects/controllability/ The network controllability project website]

* [https://www.youtube.com/watch?v=9-q2qpOJfkg The video showing network controllability]





!-类别-

[[Category:Network theory]]

Category:Network theory

范畴: 网络理论

<noinclude>

This page was moved from [[wikipedia:en:Network controllability]]. Its edit history can be viewed at [[网络可控性/edithistory]]</noinclude>

[[Category:待整理页面]]

网络可控性

2020-09-25T10:45:51Z

923397935：/* Maximum Matching */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 

本词条由信白初步翻译 

[[Image:YYL1.png|right|thumb|Controlling a simple network.]]

Controlling a simple network.

图1：控制一个简单的网络。

'''Network Controllability''' is concerned about the structural [[controllability]] of a [[Graph (discrete mathematics)|network]]. Controllability describes our ability to guide a [[dynamical system]] from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al.<ref>{{Cite journal|last=Sharma|first=Ankush|last2=Cinti|first2=Caterina|last3=Capobianco|first3=Enrico|date=2017|title=Multitype Network-Guided Target Controllability in Phenotypically Characterized Osteosarcoma: Role of Tumor Microenvironment|journal=Frontiers in Immunology|language=English|volume=8|pages=918|doi=10.3389/fimmu.2017.00918|pmid=28824643|pmc=5536125|issn=1664-3224}}</ref> on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

Network Controllability is concerned about the structural controllability of a network. Controllability describes our ability to guide a dynamical system from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al. on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

'''网络可控性 Network Controllability '''研究的是网络的结构可控性。可控性描述了我们在有限时间内，通过合适的输入选择，引导动力系统从任何初始状态到任何期望的最终状态的能力。这个定义与我们直观的控制概念非常吻合。一般有向加权复杂网络的可控性是近年来世界范围内许多研究小组的重点研究课题。夏尔马等人最近在多种生物学网络(基因-基因、 miRNA-基因、蛋白质-蛋白质相互作用网络)上的研究，确定了骨肉瘤表型特征的控制目标，显示了基因和蛋白质在维持肿瘤微环境中的重要作用。

==Background==
背景 

Consider the canonical linear time-invariant dynamics on a complex network

Consider the canonical linear time-invariant dynamics on a complex network

考虑'''复杂网络 Complex Network '''上的正则线性动力学

<math>
\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)

\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)
</math>

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

其中，向量 <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> 表示用 <math>t</math> 时间捕获 <math>N</math> 个节点系统。

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the ''minimum'' number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al.<ref name="Liu-Nature-11">{{cite journal | last=Liu | first=Yang-Yu | last2=Slotine | first2=Jean-Jacques | last3=Barabási | first3=Albert-László | title=Controllability of complex networks | journal=Nature | publisher=Springer Science and Business Media LLC | volume=473 | issue=7346 | year=2011 | issn=0028-0836 | doi=10.1038/nature10011 | pages=167–173}}</ref> attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed<ref name="Liu-Nature-11"/> that the minimum number of inputs or driver nodes needed to maintain full control of the network is determined by the 'maximum matching’ in the network, that is, the maximum set of links that do not share start or end nodes. From this result, an analytical framework, based on the in-out degree distribution, was developed to predict <math>n_\mathrm{D} =N_\mathrm{D}/N </math> for scale-free and Erdős–Rényi Graphs.<ref name="Liu-Nature-11"/> However, more recently it has been demonstrated that network controllability (and other structure-only methods which use exclusively the connectivity of a graph, <math>\mathbf{A}</math>, to simplify the underlying dynamics), both undershoot and overshoot the number and which sets of driver nodes best control network dynamics, highlighting the importance of redundancy (e.g. canalization) and non-linear dynamics in determining control.<ref name="gates_rocha_scirep">{{cite journal | last=Gates | first=Alexander J. | last2=Rocha | first2=Luis M. | title=Control of complex networks requires both structure and dynamics | journal=Scientific Reports | publisher=Springer Science and Business Media LLC | volume=6 | issue=1 | date=2016-04-18 | issn=2045-2322 | doi=10.1038/srep24456 | page=24456|doi-access=free}}</ref>

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the minimum number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al. attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed

<math>N \times N</math> 矩阵 <math>\mathbf{A}</math> 描述了系统的接线图和元件之间的交互强度。<math>N \times M</math> 矩阵 <math>\mathbf{B}</math> 识别由外部控制器控制的节点。系统通过控制器强加给系统的时间相关向量 <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> 来控制。为了确定驱动节点的最小数目，用<math>N_\mathrm{D}</math>来表示，其控制足以完全控制系统的动力学进程，在这方面，Liu等人成功做到了'''将结构控制理论 Structural Control Theory '''、'''图论 Graph Theory '''和'''统计物理 Statistical Physics '''的工具的结合。

It is also notable, that Liu's et al. formulation<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math> for a chain graph and for a weak densely connected graph. Obviously, both these graphs have very different in and out degree distributions. A recent unpublished work,<ref name = "Arxiv_Close_Betw">{{cite arXiv|first=SJ |last=Banerjee |first2=S |last2=Roy|title=Key to Network Controllability|eprint=1209.3737}}</ref> questions whether [[Degree (graph theory)|degree]], which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued <ref name="Liu-Nature-11"/> (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown<ref name = "Arxiv_Close_Betw"/> that network controllability can be altered only by using [[betweenness centrality]] and [[closeness centrality]], without using [[degree (graph theory)]] or degree correlations at all.

It is also notable, that Liu's et al. formulation questions whether degree, which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown that network controllability can be altered only by using betweenness centrality and closeness centrality, without using degree (graph theory) or degree correlations at all.

同样值得关注的是，刘等人的发现。他们提出'''度 Degree '''是网络中一种纯粹的局部度量，能够完全描述网络的可控性，即便是稍微远一点的节点也能确定它对网络的可控性是否有影响。事实上，对于许多'''实词网络 Real-Word Networks '''，像'''食物网络 Food Webs '''、'''神经元网络 Neuronal Network ''' 和'''代谢网络 Metabolic Network '''，Liu等人计算的<math>{n_\mathrm{D}}^{real}</math><math> 和 {n_\mathrm{D}}^\mathrm{rand\_degree}</math> 的值并不匹配。值得注意的是。如果可控性主要是由度决定，那么为什么对于许多现实世界的网络来说，<math>{n_\mathrm{D}}^{real}</math> 和 <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> 如此不同？他们认为，这可能是由于度相关性的影响。然而，已有的研究表明，网络的可控性只能通过中间性和封闭性来改变，而完全不需要使用度（图论）或'''度关联 Degree Correlations '''。(arXiv:1203.5161v1)

[[File:YYL2.pdf|right|thumb|A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem,<ref name="Lin-74"/> since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.]]

A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem, since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.

示意图显示了定向网络的控制。对于给定的定向网络（图a），可以计算其最大匹配：没有共同头部或尾部的最大边缘集。最大匹配将由一组顶点不相交的有向路径和有向循环组成（请参见图b中的红色边缘）。如果一个节点是匹配边的头部，则该节点是匹配的（图b中的绿色节点）。否则，它是不匹配的（图b中的白色节点）。那些不匹配的节点是需要控制的节点，即驱动节点。通过向这些驱动节点注入信号，可以得到一组以起点为输入的有向路径（见图c）。这些路径被称为“茎”。得到的有向图称为U根因子连接。通过将定向周期“嫁接”到那些“茎”，人们就会得到“芽”。得到的有向图称为cacti（见图d）。根据结构可控性定理，由于存在跨越受控网络的cacti结构（参见图e），因此该系统是可控的。受控网络（图e）下的cacti结构（图d）是维持可控性的“骨架”。

==Structural Controllability==
'''结构可控性 Structural Controllability 

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, IEEE Trans. Auto. Contr. 19

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, ''IEEE Trans. Auto. Contr.'' '''21'''

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, IEEE Trans. Auto. Contr. 21

(1974) .</ref>and then extended by Shields and Pearson (1976) ref name"Shields-76"r.希尔兹和 j。培生，IEEE Trans。自动。女名女子名。21

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, IEEE Trans. Auto. Contr. 21

结构性质的概念最早是由 Lin (1974) <ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''(1974).</ref>
和另外由 Glover 和 Silverman (1976)派生。 <ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''(1976).</ref>

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976年)。 / ref '''主要的问题是，对于可变系统参数来说，缺乏可控性或可观测性是否是一般性的。在结构控制的框架下，系统参数可以是独立自由变量，也可以是固定零点。这对于物理系统的模型是一致的，因为参数值永远不会准确地知道，零值除外，零值表示交互或连接的缺失。

==Maximum Matching==
'''最大匹配 Maximum Matching ''' 

In graph theory, a [[matching (graph theory)|matching]] is a set of edges without common vertices. Liu et al.<ref name="Liu-Nature-11"/> extended this definition to directed graph, where a matching is a set of directed edges that do not share start or end vertices. It is easy to check that a matching of a directed graph composes of a set of vertex-disjoint simple paths and cycles. The maximum matching of a directed network can be efficiently calculated by working in the bipartite representation using the classical [[Hopcroft–Karp algorithm]], which runs in O(''E''{{radic|''N''}}) time in the worst case. For undirected graph, analytical solutions of the size and number of maximum matchings have been studied using the [[cavity method]] developed in statistical physics.<ref name="Zdeborova-06">[[Lenka Zdeborová|L. Zdeborová]] and M. Mezard, ''J. Stat. Mech.'' '''05''' (2006).</ref> Liu et al.<ref name="Liu-Nature-11"/> extended the calculations for directed graph.
在图论中，匹配（图论）是一组没有共同顶点的边。Liu等人将此定义扩展到有向图，其中匹配是一组不共享起始或终止顶点的有向边。很容易检查有向图的匹配是否由一组顶点不相交的简单路径和周期组成。定向网络的最大匹配可以通过使用经典的'''Hopcroft-Karp算法 '''在'''二分表示 Bipartite Representation '''中工作来有效地计算，该算法在最坏情况下的时间复杂度是O(''E''{{radic|''N''}})。对于无向图，使用统计物理中开发的'''腔法 Cavity Method '''研究了最大匹配的大小和数量的分析解。<ref name=“Zdeborova-06”>[[Lenkazdeborová| L Zdeborová]]和M.Mezard，''J.Stat.Mech'' '''05'''（2006）。</ref>Liu等人<ref name=“Liu-Nature-11”/>扩展了有向图的计算范围。
In graph theory, a matching is a set of edges without common vertices. Liu et al. Liu et al. extended the calculations for directed graph.

By calculating the maximum matchings of a wide range of real networks, Liu et al.<ref name="Liu-Nature-11"/> asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the [[cavity method]]. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al.<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether ''control robustness" in networks is influenced more by using [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> over [[degree (graph theory)]] based metrics.

By calculating the maximum matchings of a wide range of real networks, Liu et al. asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the cavity method. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al. would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether control robustness" in networks is influenced more by using betweenness centrality and closeness centrality over degree (graph theory) based metrics.

通过计算各种真实网络的最大匹配，Liu等人断言驱动程序节点的数量主要由网络度分布<math>P(k_\mathrm{in}，k_\mathrm{out})</math>决定。他们还使用空腔方法计算了具有任意程度分布的网络集合的驱动节点的平均数量。有趣的是，对于'''链图 Chain Graph '''和'''弱密集连通图 Weak Densely Connected Graph '''，两者都具有非常不同的进出度分布；Liu等人的公式将预测<math>{n_\mathrm{D}}</math>的相同值。此外，对于许多真实词网络，即食物网、神经元和代谢网络，Liu等人计算的<math>{n_\mathrm{D}^{real}</math>和<math>{n_\mathrm{D}^\mathrm{rand\_degree}</math>值的不匹配。值得注意的是。如果可控性纯粹由度决定，那为什么<math>{n_\mathrm{D}}^{real}</math>和<math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math>对于许多现实世界的网络来说是如此不同？对于网络中的控制鲁棒性是否更多地受到使用基于度（图论）的度量的'''中介中心性 Betweenness Centrality '''和'''紧密度中心性 Closeness Centrality '''的影响，仍然是开放的。

While sparser graphs are more difficult to control,<ref name="Liu-Nature-11"/><ref name = "Arxiv_Close_Betw"/> it would obviously be interesting to find whether [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> or degree heterogeneity<ref name="Liu-Nature-11"/> plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

While sparser graphs are more difficult to control, it would obviously be interesting to find whether betweenness centrality and closeness centrality or degree heterogeneity plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

尽管稀疏图的可控性难以控制，但是研究具有几乎相似度分布的稀疏图的可控性中，是否存在介于介于中间中心性和封闭中心性或度异质性起着更重要的作用显然是很有意义的。

==Control of composite quantum systems and algebraic graph theory==

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively.<ref name="BG-07">{{cite journal | last=Burgarth | first=Daniel | last2=Giovannetti | first2=Vittorio | title=Full Control by Locally Induced Relaxation | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=99 | issue=10 | date=2007-09-05 | issn=0031-9007 | doi=10.1103/physrevlett.99.100501 | page=100501| arxiv=0704.3027 }}</ref> This framework permits to formulate Kalman's criterion with tools from [[algebraic graph theory]] via the [[minimum rank of a graph]] and related notions.<ref name="BDHSY-13">{{cite journal|last1=Burgarth|first1=Daniel|last2=D'Alessandro|first2=Domenico|last3=Hogben|first3=Leslie|author3-link=Leslie Hogben|last4=Severini|first4=Simone|last5=Young|first5=Michael|doi=10.1109/TAC.2013.2250075|issue=9|journal=IEEE Transactions on Automatic Control|mr=3101617|pages=2349–2354|title=Zero forcing, linear and quantum controllability for systems evolving on networks|volume=58|year=2013}}</ref><ref name="OT-15">S. O'Rourke, B. Touri, https://arxiv.org/abs/1511.05080.</ref>

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively. This framework permits to formulate Kalman's criterion with tools from algebraic graph theory via the minimum rank of a graph and related notions.

在复合量子系统的通用控制的背景下，也发展了网络控制理论，其中子系统及其相互作用分别与节点和链路相关联。这个框架允许用代数图论的工具通过图的最小秩和相关概念来建立卡尔曼准则。

== See also ==

* [[Controllability Gramian]]

==References==

<references/>

== External links ==

* [http://barabasilab.neu.edu/projects/controllability/ The network controllability project website]

* [https://www.youtube.com/watch?v=9-q2qpOJfkg The video showing network controllability]





!-类别-

[[Category:Network theory]]

Category:Network theory

范畴: 网络理论

<noinclude>

This page was moved from [[wikipedia:en:Network controllability]]. Its edit history can be viewed at [[网络可控性/edithistory]]</noinclude>

[[Category:待整理页面]]

网络可控性

2020-09-25T10:44:10Z

923397935：/* Maximum Matching */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 

本词条由信白初步翻译 

[[Image:YYL1.png|right|thumb|Controlling a simple network.]]

Controlling a simple network.

图1：控制一个简单的网络。

'''Network Controllability''' is concerned about the structural [[controllability]] of a [[Graph (discrete mathematics)|network]]. Controllability describes our ability to guide a [[dynamical system]] from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al.<ref>{{Cite journal|last=Sharma|first=Ankush|last2=Cinti|first2=Caterina|last3=Capobianco|first3=Enrico|date=2017|title=Multitype Network-Guided Target Controllability in Phenotypically Characterized Osteosarcoma: Role of Tumor Microenvironment|journal=Frontiers in Immunology|language=English|volume=8|pages=918|doi=10.3389/fimmu.2017.00918|pmid=28824643|pmc=5536125|issn=1664-3224}}</ref> on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

Network Controllability is concerned about the structural controllability of a network. Controllability describes our ability to guide a dynamical system from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al. on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

'''网络可控性 Network Controllability '''研究的是网络的结构可控性。可控性描述了我们在有限时间内，通过合适的输入选择，引导动力系统从任何初始状态到任何期望的最终状态的能力。这个定义与我们直观的控制概念非常吻合。一般有向加权复杂网络的可控性是近年来世界范围内许多研究小组的重点研究课题。夏尔马等人最近在多种生物学网络(基因-基因、 miRNA-基因、蛋白质-蛋白质相互作用网络)上的研究，确定了骨肉瘤表型特征的控制目标，显示了基因和蛋白质在维持肿瘤微环境中的重要作用。

==Background==
背景 

Consider the canonical linear time-invariant dynamics on a complex network

Consider the canonical linear time-invariant dynamics on a complex network

考虑'''复杂网络 Complex Network '''上的正则线性动力学

<math>
\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)

\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)
</math>

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

其中，向量 <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> 表示用 <math>t</math> 时间捕获 <math>N</math> 个节点系统。

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the ''minimum'' number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al.<ref name="Liu-Nature-11">{{cite journal | last=Liu | first=Yang-Yu | last2=Slotine | first2=Jean-Jacques | last3=Barabási | first3=Albert-László | title=Controllability of complex networks | journal=Nature | publisher=Springer Science and Business Media LLC | volume=473 | issue=7346 | year=2011 | issn=0028-0836 | doi=10.1038/nature10011 | pages=167–173}}</ref> attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed<ref name="Liu-Nature-11"/> that the minimum number of inputs or driver nodes needed to maintain full control of the network is determined by the 'maximum matching’ in the network, that is, the maximum set of links that do not share start or end nodes. From this result, an analytical framework, based on the in-out degree distribution, was developed to predict <math>n_\mathrm{D} =N_\mathrm{D}/N </math> for scale-free and Erdős–Rényi Graphs.<ref name="Liu-Nature-11"/> However, more recently it has been demonstrated that network controllability (and other structure-only methods which use exclusively the connectivity of a graph, <math>\mathbf{A}</math>, to simplify the underlying dynamics), both undershoot and overshoot the number and which sets of driver nodes best control network dynamics, highlighting the importance of redundancy (e.g. canalization) and non-linear dynamics in determining control.<ref name="gates_rocha_scirep">{{cite journal | last=Gates | first=Alexander J. | last2=Rocha | first2=Luis M. | title=Control of complex networks requires both structure and dynamics | journal=Scientific Reports | publisher=Springer Science and Business Media LLC | volume=6 | issue=1 | date=2016-04-18 | issn=2045-2322 | doi=10.1038/srep24456 | page=24456|doi-access=free}}</ref>

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the minimum number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al. attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed

<math>N \times N</math> 矩阵 <math>\mathbf{A}</math> 描述了系统的接线图和元件之间的交互强度。<math>N \times M</math> 矩阵 <math>\mathbf{B}</math> 识别由外部控制器控制的节点。系统通过控制器强加给系统的时间相关向量 <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> 来控制。为了确定驱动节点的最小数目，用<math>N_\mathrm{D}</math>来表示，其控制足以完全控制系统的动力学进程，在这方面，Liu等人成功做到了'''将结构控制理论 Structural Control Theory '''、'''图论 Graph Theory '''和'''统计物理 Statistical Physics '''的工具的结合。

It is also notable, that Liu's et al. formulation<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math> for a chain graph and for a weak densely connected graph. Obviously, both these graphs have very different in and out degree distributions. A recent unpublished work,<ref name = "Arxiv_Close_Betw">{{cite arXiv|first=SJ |last=Banerjee |first2=S |last2=Roy|title=Key to Network Controllability|eprint=1209.3737}}</ref> questions whether [[Degree (graph theory)|degree]], which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued <ref name="Liu-Nature-11"/> (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown<ref name = "Arxiv_Close_Betw"/> that network controllability can be altered only by using [[betweenness centrality]] and [[closeness centrality]], without using [[degree (graph theory)]] or degree correlations at all.

It is also notable, that Liu's et al. formulation questions whether degree, which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown that network controllability can be altered only by using betweenness centrality and closeness centrality, without using degree (graph theory) or degree correlations at all.

同样值得关注的是，刘等人的发现。他们提出'''度 Degree '''是网络中一种纯粹的局部度量，能够完全描述网络的可控性，即便是稍微远一点的节点也能确定它对网络的可控性是否有影响。事实上，对于许多'''实词网络 Real-Word Networks '''，像'''食物网络 Food Webs '''、'''神经元网络 Neuronal Network ''' 和'''代谢网络 Metabolic Network '''，Liu等人计算的<math>{n_\mathrm{D}}^{real}</math><math> 和 {n_\mathrm{D}}^\mathrm{rand\_degree}</math> 的值并不匹配。值得注意的是。如果可控性主要是由度决定，那么为什么对于许多现实世界的网络来说，<math>{n_\mathrm{D}}^{real}</math> 和 <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> 如此不同？他们认为，这可能是由于度相关性的影响。然而，已有的研究表明，网络的可控性只能通过中间性和封闭性来改变，而完全不需要使用度（图论）或'''度关联 Degree Correlations '''。(arXiv:1203.5161v1)

[[File:YYL2.pdf|right|thumb|A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem,<ref name="Lin-74"/> since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.]]

A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem, since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.

示意图显示了定向网络的控制。对于给定的定向网络（图a），可以计算其最大匹配：没有共同头部或尾部的最大边缘集。最大匹配将由一组顶点不相交的有向路径和有向循环组成（请参见图b中的红色边缘）。如果一个节点是匹配边的头部，则该节点是匹配的（图b中的绿色节点）。否则，它是不匹配的（图b中的白色节点）。那些不匹配的节点是需要控制的节点，即驱动节点。通过向这些驱动节点注入信号，可以得到一组以起点为输入的有向路径（见图c）。这些路径被称为“茎”。得到的有向图称为U根因子连接。通过将定向周期“嫁接”到那些“茎”，人们就会得到“芽”。得到的有向图称为cacti（见图d）。根据结构可控性定理，由于存在跨越受控网络的cacti结构（参见图e），因此该系统是可控的。受控网络（图e）下的cacti结构（图d）是维持可控性的“骨架”。

==Structural Controllability==
'''结构可控性 Structural Controllability 

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, IEEE Trans. Auto. Contr. 19

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, ''IEEE Trans. Auto. Contr.'' '''21'''

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, IEEE Trans. Auto. Contr. 21

(1974) .</ref>and then extended by Shields and Pearson (1976) ref name"Shields-76"r.希尔兹和 j。培生，IEEE Trans。自动。女名女子名。21

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, IEEE Trans. Auto. Contr. 21

结构性质的概念最早是由 Lin (1974) <ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''(1974).</ref>
和另外由 Glover 和 Silverman (1976)派生。 <ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''(1976).</ref>

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976年)。 / ref '''主要的问题是，对于可变系统参数来说，缺乏可控性或可观测性是否是一般性的。在结构控制的框架下，系统参数可以是独立自由变量，也可以是固定零点。这对于物理系统的模型是一致的，因为参数值永远不会准确地知道，零值除外，零值表示交互或连接的缺失。

==Maximum Matching==
'''最大匹配 Maximum Matching ''' 

In graph theory, a [[matching (graph theory)|matching]] is a set of edges without common vertices. Liu et al.<ref name="Liu-Nature-11"/> extended this definition to directed graph, where a matching is a set of directed edges that do not share start or end vertices. It is easy to check that a matching of a directed graph composes of a set of vertex-disjoint simple paths and cycles. The maximum matching of a directed network can be efficiently calculated by working in the bipartite representation using the classical [[Hopcroft–Karp algorithm]], which runs in O(''E''{{radic|''N''}}) time in the worst case. For undirected graph, analytical solutions of the size and number of maximum matchings have been studied using the [[cavity method]] developed in statistical physics.<ref name="Zdeborova-06">[[Lenka Zdeborová|L. Zdeborová]] and M. Mezard, ''J. Stat. Mech.'' '''05''' (2006).</ref> Liu et al.<ref name="Liu-Nature-11"/> extended the calculations for directed graph.
在图论中，匹配（图论）是一组没有共同顶点的边。Liu等人将此定义扩展到有向图，其中匹配是一组不共享起始或终止顶点的有向边。很容易检查有向图的匹配是否由一组顶点不相交的简单路径和周期组成。定向网络的最大匹配可以通过使用经典的'''Hopcroft-Karp算法 '''在'''二分表示 Bipartite Representation '''中工作来有效地计算，该算法在最坏情况下的时间复杂度是O(''E''{{radic|''N''}})。对于无向图，使用统计物理中开发的'''腔法 Cavity Method '''研究了最大匹配的大小和数量的分析解。<ref name=“Zdeborova-06”>[[Lenkazdeborová| L Zdeborová]]和M.Mezard，''J.Stat.Mech'' '''05'''（2006）。</ref>Liu等人<ref name=“Liu-Nature-11”/>扩展了有向图的计算范围。
In graph theory, a matching is a set of edges without common vertices. Liu et al. Liu et al. extended the calculations for directed graph.

By calculating the maximum matchings of a wide range of real networks, Liu et al.<ref name="Liu-Nature-11"/> asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the [[cavity method]]. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al.<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether ''control robustness" in networks is influenced more by using [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> over [[degree (graph theory)]] based metrics.

By calculating the maximum matchings of a wide range of real networks, Liu et al. asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the cavity method. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al. would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether control robustness" in networks is influenced more by using betweenness centrality and closeness centrality over degree (graph theory) based metrics.

通过计算各种真实网络的最大匹配，Liu等人断言驱动程序节点的数量主要由网络度分布<math>P(k_\mathrm{in}，k_\mathrm{out})</math>决定。他们还使用空腔方法计算了具有任意程度分布的网络集合的驱动节点的平均数量。有趣的是，对于'''链图 Chain Graph '''和'''弱密集连通图 Weak Densely Connected Graph '''，两者都具有非常不同的进出度分布；Liu等人的公式将预测<math>{n_\mathrm{D}}</math>的相同值。此外，对于许多真实词网络，即食物网、神经元和代谢网络，Liu等人计算的<math>{n/mathrm{D}^{real}</math>和<math>{n/mathrm{D}^\mathrm{rand\_degree}</math>值的不匹配。值得注意的是。如果可控性纯粹由度决定，那为什么<math>{n_\mathrm{D}}^{real}</math>和<math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math>对于许多现实世界的网络来说是如此不同？对于网络中的控制鲁棒性是否更多地受到使用基于度（图论）的度量的'''中介中心性 Betweenness Centrality '''和'''紧密度中心性 Closeness Centrality '''的影响，仍然是开放的。

While sparser graphs are more difficult to control,<ref name="Liu-Nature-11"/><ref name = "Arxiv_Close_Betw"/> it would obviously be interesting to find whether [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> or degree heterogeneity<ref name="Liu-Nature-11"/> plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

While sparser graphs are more difficult to control, it would obviously be interesting to find whether betweenness centrality and closeness centrality or degree heterogeneity plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

尽管稀疏图的可控性难以控制，但是研究具有几乎相似度分布的稀疏图的可控性中，是否存在介于介于中间中心性和封闭中心性或度异质性起着更重要的作用显然是很有意义的。

==Control of composite quantum systems and algebraic graph theory==

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively.<ref name="BG-07">{{cite journal | last=Burgarth | first=Daniel | last2=Giovannetti | first2=Vittorio | title=Full Control by Locally Induced Relaxation | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=99 | issue=10 | date=2007-09-05 | issn=0031-9007 | doi=10.1103/physrevlett.99.100501 | page=100501| arxiv=0704.3027 }}</ref> This framework permits to formulate Kalman's criterion with tools from [[algebraic graph theory]] via the [[minimum rank of a graph]] and related notions.<ref name="BDHSY-13">{{cite journal|last1=Burgarth|first1=Daniel|last2=D'Alessandro|first2=Domenico|last3=Hogben|first3=Leslie|author3-link=Leslie Hogben|last4=Severini|first4=Simone|last5=Young|first5=Michael|doi=10.1109/TAC.2013.2250075|issue=9|journal=IEEE Transactions on Automatic Control|mr=3101617|pages=2349–2354|title=Zero forcing, linear and quantum controllability for systems evolving on networks|volume=58|year=2013}}</ref><ref name="OT-15">S. O'Rourke, B. Touri, https://arxiv.org/abs/1511.05080.</ref>

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively. This framework permits to formulate Kalman's criterion with tools from algebraic graph theory via the minimum rank of a graph and related notions.

在复合量子系统的通用控制的背景下，也发展了网络控制理论，其中子系统及其相互作用分别与节点和链路相关联。这个框架允许用代数图论的工具通过图的最小秩和相关概念来建立卡尔曼准则。

== See also ==

* [[Controllability Gramian]]

==References==

<references/>

== External links ==

* [http://barabasilab.neu.edu/projects/controllability/ The network controllability project website]

* [https://www.youtube.com/watch?v=9-q2qpOJfkg The video showing network controllability]





!-类别-

[[Category:Network theory]]

Category:Network theory

范畴: 网络理论

<noinclude>

This page was moved from [[wikipedia:en:Network controllability]]. Its edit history can be viewed at [[网络可控性/edithistory]]</noinclude>

[[Category:待整理页面]]

网络可控性

2020-09-25T10:34:04Z

923397935：/* Maximum Matching */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 

本词条由信白初步翻译 

[[Image:YYL1.png|right|thumb|Controlling a simple network.]]

Controlling a simple network.

图1：控制一个简单的网络。

'''Network Controllability''' is concerned about the structural [[controllability]] of a [[Graph (discrete mathematics)|network]]. Controllability describes our ability to guide a [[dynamical system]] from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al.<ref>{{Cite journal|last=Sharma|first=Ankush|last2=Cinti|first2=Caterina|last3=Capobianco|first3=Enrico|date=2017|title=Multitype Network-Guided Target Controllability in Phenotypically Characterized Osteosarcoma: Role of Tumor Microenvironment|journal=Frontiers in Immunology|language=English|volume=8|pages=918|doi=10.3389/fimmu.2017.00918|pmid=28824643|pmc=5536125|issn=1664-3224}}</ref> on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

Network Controllability is concerned about the structural controllability of a network. Controllability describes our ability to guide a dynamical system from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al. on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

'''网络可控性 Network Controllability '''研究的是网络的结构可控性。可控性描述了我们在有限时间内，通过合适的输入选择，引导动力系统从任何初始状态到任何期望的最终状态的能力。这个定义与我们直观的控制概念非常吻合。一般有向加权复杂网络的可控性是近年来世界范围内许多研究小组的重点研究课题。夏尔马等人最近在多种生物学网络(基因-基因、 miRNA-基因、蛋白质-蛋白质相互作用网络)上的研究，确定了骨肉瘤表型特征的控制目标，显示了基因和蛋白质在维持肿瘤微环境中的重要作用。

==Background==
背景 

Consider the canonical linear time-invariant dynamics on a complex network

Consider the canonical linear time-invariant dynamics on a complex network

考虑'''复杂网络 Complex Network '''上的正则线性动力学

<math>
\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)

\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)
</math>

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

其中，向量 <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> 表示用 <math>t</math> 时间捕获 <math>N</math> 个节点系统。

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the ''minimum'' number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al.<ref name="Liu-Nature-11">{{cite journal | last=Liu | first=Yang-Yu | last2=Slotine | first2=Jean-Jacques | last3=Barabási | first3=Albert-László | title=Controllability of complex networks | journal=Nature | publisher=Springer Science and Business Media LLC | volume=473 | issue=7346 | year=2011 | issn=0028-0836 | doi=10.1038/nature10011 | pages=167–173}}</ref> attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed<ref name="Liu-Nature-11"/> that the minimum number of inputs or driver nodes needed to maintain full control of the network is determined by the 'maximum matching’ in the network, that is, the maximum set of links that do not share start or end nodes. From this result, an analytical framework, based on the in-out degree distribution, was developed to predict <math>n_\mathrm{D} =N_\mathrm{D}/N </math> for scale-free and Erdős–Rényi Graphs.<ref name="Liu-Nature-11"/> However, more recently it has been demonstrated that network controllability (and other structure-only methods which use exclusively the connectivity of a graph, <math>\mathbf{A}</math>, to simplify the underlying dynamics), both undershoot and overshoot the number and which sets of driver nodes best control network dynamics, highlighting the importance of redundancy (e.g. canalization) and non-linear dynamics in determining control.<ref name="gates_rocha_scirep">{{cite journal | last=Gates | first=Alexander J. | last2=Rocha | first2=Luis M. | title=Control of complex networks requires both structure and dynamics | journal=Scientific Reports | publisher=Springer Science and Business Media LLC | volume=6 | issue=1 | date=2016-04-18 | issn=2045-2322 | doi=10.1038/srep24456 | page=24456|doi-access=free}}</ref>

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the minimum number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al. attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed

<math>N \times N</math> 矩阵 <math>\mathbf{A}</math> 描述了系统的接线图和元件之间的交互强度。<math>N \times M</math> 矩阵 <math>\mathbf{B}</math> 识别由外部控制器控制的节点。系统通过控制器强加给系统的时间相关向量 <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> 来控制。为了确定驱动节点的最小数目，用<math>N_\mathrm{D}</math>来表示，其控制足以完全控制系统的动力学进程，在这方面，Liu等人成功做到了'''将结构控制理论 Structural Control Theory '''、'''图论 Graph Theory '''和'''统计物理 Statistical Physics '''的工具的结合。

It is also notable, that Liu's et al. formulation<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math> for a chain graph and for a weak densely connected graph. Obviously, both these graphs have very different in and out degree distributions. A recent unpublished work,<ref name = "Arxiv_Close_Betw">{{cite arXiv|first=SJ |last=Banerjee |first2=S |last2=Roy|title=Key to Network Controllability|eprint=1209.3737}}</ref> questions whether [[Degree (graph theory)|degree]], which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued <ref name="Liu-Nature-11"/> (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown<ref name = "Arxiv_Close_Betw"/> that network controllability can be altered only by using [[betweenness centrality]] and [[closeness centrality]], without using [[degree (graph theory)]] or degree correlations at all.

It is also notable, that Liu's et al. formulation questions whether degree, which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown that network controllability can be altered only by using betweenness centrality and closeness centrality, without using degree (graph theory) or degree correlations at all.

同样值得关注的是，刘等人的发现。他们提出'''度 Degree '''是网络中一种纯粹的局部度量，能够完全描述网络的可控性，即便是稍微远一点的节点也能确定它对网络的可控性是否有影响。事实上，对于许多'''实词网络 Real-Word Networks '''，像'''食物网络 Food Webs '''、'''神经元网络 Neuronal Network ''' 和'''代谢网络 Metabolic Network '''，Liu等人计算的<math>{n_\mathrm{D}}^{real}</math><math> 和 {n_\mathrm{D}}^\mathrm{rand\_degree}</math> 的值并不匹配。值得注意的是。如果可控性主要是由度决定，那么为什么对于许多现实世界的网络来说，<math>{n_\mathrm{D}}^{real}</math> 和 <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> 如此不同？他们认为，这可能是由于度相关性的影响。然而，已有的研究表明，网络的可控性只能通过中间性和封闭性来改变，而完全不需要使用度（图论）或'''度关联 Degree Correlations '''。(arXiv:1203.5161v1)

[[File:YYL2.pdf|right|thumb|A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem,<ref name="Lin-74"/> since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.]]

A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem, since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.

示意图显示了定向网络的控制。对于给定的定向网络（图a），可以计算其最大匹配：没有共同头部或尾部的最大边缘集。最大匹配将由一组顶点不相交的有向路径和有向循环组成（请参见图b中的红色边缘）。如果一个节点是匹配边的头部，则该节点是匹配的（图b中的绿色节点）。否则，它是不匹配的（图b中的白色节点）。那些不匹配的节点是需要控制的节点，即驱动节点。通过向这些驱动节点注入信号，可以得到一组以起点为输入的有向路径（见图c）。这些路径被称为“茎”。得到的有向图称为U根因子连接。通过将定向周期“嫁接”到那些“茎”，人们就会得到“芽”。得到的有向图称为cacti（见图d）。根据结构可控性定理，由于存在跨越受控网络的cacti结构（参见图e），因此该系统是可控的。受控网络（图e）下的cacti结构（图d）是维持可控性的“骨架”。

==Structural Controllability==
'''结构可控性 Structural Controllability 

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, IEEE Trans. Auto. Contr. 19

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, ''IEEE Trans. Auto. Contr.'' '''21'''

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, IEEE Trans. Auto. Contr. 21

(1974) .</ref>and then extended by Shields and Pearson (1976) ref name"Shields-76"r.希尔兹和 j。培生，IEEE Trans。自动。女名女子名。21

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, IEEE Trans. Auto. Contr. 21

结构性质的概念最早是由 Lin (1974) <ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''(1974).</ref>
和另外由 Glover 和 Silverman (1976)派生。 <ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''(1976).</ref>

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976年)。 / ref '''主要的问题是，对于可变系统参数来说，缺乏可控性或可观测性是否是一般性的。在结构控制的框架下，系统参数可以是独立自由变量，也可以是固定零点。这对于物理系统的模型是一致的，因为参数值永远不会准确地知道，零值除外，零值表示交互或连接的缺失。

==Maximum Matching==
'''最大匹配 Maximum Matching ''' 

In graph theory, a [[matching (graph theory)|matching]] is a set of edges without common vertices. Liu et al.<ref name="Liu-Nature-11"/> extended this definition to directed graph, where a matching is a set of directed edges that do not share start or end vertices. It is easy to check that a matching of a directed graph composes of a set of vertex-disjoint simple paths and cycles. The maximum matching of a directed network can be efficiently calculated by working in the bipartite representation using the classical [[Hopcroft–Karp algorithm]], which runs in O(''E''{{radic|''N''}}) time in the worst case. For undirected graph, analytical solutions of the size and number of maximum matchings have been studied using the [[cavity method]] developed in statistical physics.<ref name="Zdeborova-06">[[Lenka Zdeborová|L. Zdeborová]] and M. Mezard, ''J. Stat. Mech.'' '''05''' (2006).</ref> Liu et al.<ref name="Liu-Nature-11"/> extended the calculations for directed graph.
在图论中，匹配（图论）是一组没有共同顶点的边。Liu等人将此定义扩展到有向图，其中匹配是一组不共享起始或终止顶点的有向边。很容易检查有向图的匹配是否由一组顶点不相交的简单路径和周期组成。定向网络的最大匹配可以通过使用经典的'''Hopcroft-Karp算法 '''在'''二分表示 Bipartite Representation '''中工作来有效地计算，该算法在最坏情况下的时间复杂度是O(''E''{{radic|''N''}})。对于无向图，使用统计物理中开发的'''腔法 Cavity Method '''研究了最大匹配的大小和数量的分析解。<ref name=“Zdeborova-06”>[[Lenkazdeborová| L Zdeborová]]和M.Mezard，''J.Stat.Mech'' '''05'''（2006）。</ref>Liu等人<ref name=“Liu-Nature-11”/>扩展了有向图的计算范围。
In graph theory, a matching is a set of edges without common vertices. Liu et al. Liu et al. extended the calculations for directed graph.

By calculating the maximum matchings of a wide range of real networks, Liu et al.<ref name="Liu-Nature-11"/> asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the [[cavity method]]. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al.<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether ''control robustness" in networks is influenced more by using [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> over [[degree (graph theory)]] based metrics.

By calculating the maximum matchings of a wide range of real networks, Liu et al. asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the cavity method. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al. would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether control robustness" in networks is influenced more by using betweenness centrality and closeness centrality over degree (graph theory) based metrics.

通过计算各种实际网络的最大匹配，刘等人提出了一种新的网络匹配算法。断言驱动节点数主要由网络度分布数学 p (k，k，k) / 数学决定。他们还利用空腔法计算了任意度分布的网络集成的平均驱动节点数。有趣的是，对于链图和弱紧连通图，它们的内外度分布都有很大的不同。可以预测相同的数学数值。此外，对于许多实词网络，即食物网络、神经元网络和代谢网络，Liu 等人计算出的数学函数 d ^ 实数 / 数学和数学函数 d ^ 实数 / 数学的值不匹配。值得注意的是。如果可控性纯粹是由程度决定的，那么为什么对于许多现实世界的网络来说，数学、数学和数学如此不同？网络中的控制鲁棒性是否受基于度的度量中介中心性和贴近中心性的影响仍有待进一步研究。

While sparser graphs are more difficult to control,<ref name="Liu-Nature-11"/><ref name = "Arxiv_Close_Betw"/> it would obviously be interesting to find whether [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> or degree heterogeneity<ref name="Liu-Nature-11"/> plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

While sparser graphs are more difficult to control, it would obviously be interesting to find whether betweenness centrality and closeness centrality or degree heterogeneity plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

尽管稀疏图的可控性难以控制，但是研究具有几乎相似度分布的稀疏图的可控性中，是否存在介于介于中间中心性和封闭中心性或度异质性起着更重要的作用显然是很有意义的。

==Control of composite quantum systems and algebraic graph theory==

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively.<ref name="BG-07">{{cite journal | last=Burgarth | first=Daniel | last2=Giovannetti | first2=Vittorio | title=Full Control by Locally Induced Relaxation | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=99 | issue=10 | date=2007-09-05 | issn=0031-9007 | doi=10.1103/physrevlett.99.100501 | page=100501| arxiv=0704.3027 }}</ref> This framework permits to formulate Kalman's criterion with tools from [[algebraic graph theory]] via the [[minimum rank of a graph]] and related notions.<ref name="BDHSY-13">{{cite journal|last1=Burgarth|first1=Daniel|last2=D'Alessandro|first2=Domenico|last3=Hogben|first3=Leslie|author3-link=Leslie Hogben|last4=Severini|first4=Simone|last5=Young|first5=Michael|doi=10.1109/TAC.2013.2250075|issue=9|journal=IEEE Transactions on Automatic Control|mr=3101617|pages=2349–2354|title=Zero forcing, linear and quantum controllability for systems evolving on networks|volume=58|year=2013}}</ref><ref name="OT-15">S. O'Rourke, B. Touri, https://arxiv.org/abs/1511.05080.</ref>

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively. This framework permits to formulate Kalman's criterion with tools from algebraic graph theory via the minimum rank of a graph and related notions.

在复合量子系统的通用控制的背景下，也发展了网络控制理论，其中子系统及其相互作用分别与节点和链路相关联。这个框架允许用代数图论的工具通过图的最小秩和相关概念来建立卡尔曼准则。

== See also ==

* [[Controllability Gramian]]

==References==

<references/>

== External links ==

* [http://barabasilab.neu.edu/projects/controllability/ The network controllability project website]

* [https://www.youtube.com/watch?v=9-q2qpOJfkg The video showing network controllability]





!-类别-

[[Category:Network theory]]

Category:Network theory

范畴: 网络理论

<noinclude>

This page was moved from [[wikipedia:en:Network controllability]]. Its edit history can be viewed at [[网络可控性/edithistory]]</noinclude>

[[Category:待整理页面]]

网络可控性

2020-09-25T09:43:09Z

923397935：/* Maximum Matching */

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 

本词条由信白初步翻译 

[[Image:YYL1.png|right|thumb|Controlling a simple network.]]

Controlling a simple network.

图1：控制一个简单的网络。

'''Network Controllability''' is concerned about the structural [[controllability]] of a [[Graph (discrete mathematics)|network]]. Controllability describes our ability to guide a [[dynamical system]] from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al.<ref>{{Cite journal|last=Sharma|first=Ankush|last2=Cinti|first2=Caterina|last3=Capobianco|first3=Enrico|date=2017|title=Multitype Network-Guided Target Controllability in Phenotypically Characterized Osteosarcoma: Role of Tumor Microenvironment|journal=Frontiers in Immunology|language=English|volume=8|pages=918|doi=10.3389/fimmu.2017.00918|pmid=28824643|pmc=5536125|issn=1664-3224}}</ref> on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

Network Controllability is concerned about the structural controllability of a network. Controllability describes our ability to guide a dynamical system from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al. on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

'''网络可控性 Network Controllability '''研究的是网络的结构可控性。可控性描述了我们在有限时间内，通过合适的输入选择，引导动力系统从任何初始状态到任何期望的最终状态的能力。这个定义与我们直观的控制概念非常吻合。一般有向加权复杂网络的可控性是近年来世界范围内许多研究小组的重点研究课题。夏尔马等人最近在多种生物学网络(基因-基因、 miRNA-基因、蛋白质-蛋白质相互作用网络)上的研究，确定了骨肉瘤表型特征的控制目标，显示了基因和蛋白质在维持肿瘤微环境中的重要作用。

==Background==
背景 

Consider the canonical linear time-invariant dynamics on a complex network

Consider the canonical linear time-invariant dynamics on a complex network

考虑'''复杂网络 Complex Network '''上的正则线性动力学

<math>
\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)

\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)
</math>

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>.

其中，向量 <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> 表示用 <math>t</math> 时间捕获 <math>N</math> 个节点系统。

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the ''minimum'' number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al.<ref name="Liu-Nature-11">{{cite journal | last=Liu | first=Yang-Yu | last2=Slotine | first2=Jean-Jacques | last3=Barabási | first3=Albert-László | title=Controllability of complex networks | journal=Nature | publisher=Springer Science and Business Media LLC | volume=473 | issue=7346 | year=2011 | issn=0028-0836 | doi=10.1038/nature10011 | pages=167–173}}</ref> attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed<ref name="Liu-Nature-11"/> that the minimum number of inputs or driver nodes needed to maintain full control of the network is determined by the 'maximum matching’ in the network, that is, the maximum set of links that do not share start or end nodes. From this result, an analytical framework, based on the in-out degree distribution, was developed to predict <math>n_\mathrm{D} =N_\mathrm{D}/N </math> for scale-free and Erdős–Rényi Graphs.<ref name="Liu-Nature-11"/> However, more recently it has been demonstrated that network controllability (and other structure-only methods which use exclusively the connectivity of a graph, <math>\mathbf{A}</math>, to simplify the underlying dynamics), both undershoot and overshoot the number and which sets of driver nodes best control network dynamics, highlighting the importance of redundancy (e.g. canalization) and non-linear dynamics in determining control.<ref name="gates_rocha_scirep">{{cite journal | last=Gates | first=Alexander J. | last2=Rocha | first2=Luis M. | title=Control of complex networks requires both structure and dynamics | journal=Scientific Reports | publisher=Springer Science and Business Media LLC | volume=6 | issue=1 | date=2016-04-18 | issn=2045-2322 | doi=10.1038/srep24456 | page=24456|doi-access=free}}</ref>

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the minimum number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al. attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed

<math>N \times N</math> 矩阵 <math>\mathbf{A}</math> 描述了系统的接线图和元件之间的交互强度。<math>N \times M</math> 矩阵 <math>\mathbf{B}</math> 识别由外部控制器控制的节点。系统通过控制器强加给系统的时间相关向量 <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> 来控制。为了确定驱动节点的最小数目，用<math>N_\mathrm{D}</math>来表示，其控制足以完全控制系统的动力学进程，在这方面，Liu等人成功做到了'''将结构控制理论 Structural Control Theory '''、'''图论 Graph Theory '''和'''统计物理 Statistical Physics '''的工具的结合。

It is also notable, that Liu's et al. formulation<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math> for a chain graph and for a weak densely connected graph. Obviously, both these graphs have very different in and out degree distributions. A recent unpublished work,<ref name = "Arxiv_Close_Betw">{{cite arXiv|first=SJ |last=Banerjee |first2=S |last2=Roy|title=Key to Network Controllability|eprint=1209.3737}}</ref> questions whether [[Degree (graph theory)|degree]], which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued <ref name="Liu-Nature-11"/> (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown<ref name = "Arxiv_Close_Betw"/> that network controllability can be altered only by using [[betweenness centrality]] and [[closeness centrality]], without using [[degree (graph theory)]] or degree correlations at all.

It is also notable, that Liu's et al. formulation questions whether degree, which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown that network controllability can be altered only by using betweenness centrality and closeness centrality, without using degree (graph theory) or degree correlations at all.

同样值得关注的是，刘等人的发现。他们提出'''度 Degree '''是网络中一种纯粹的局部度量，能够完全描述网络的可控性，即便是稍微远一点的节点也能确定它对网络的可控性是否有影响。事实上，对于许多'''实词网络 Real-Word Networks '''，像'''食物网络 Food Webs '''、'''神经元网络 Neuronal Network ''' 和'''代谢网络 Metabolic Network '''，Liu等人计算的<math>{n_\mathrm{D}}^{real}</math><math> 和 {n_\mathrm{D}}^\mathrm{rand\_degree}</math> 的值并不匹配。值得注意的是。如果可控性主要是由度决定，那么为什么对于许多现实世界的网络来说，<math>{n_\mathrm{D}}^{real}</math> 和 <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> 如此不同？他们认为，这可能是由于度相关性的影响。然而，已有的研究表明，网络的可控性只能通过中间性和封闭性来改变，而完全不需要使用度（图论）或'''度关联 Degree Correlations '''。(arXiv:1203.5161v1)

[[File:YYL2.pdf|right|thumb|A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem,<ref name="Lin-74"/> since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.]]

A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem, since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.

示意图显示了定向网络的控制。对于给定的定向网络（图a），可以计算其最大匹配：没有共同头部或尾部的最大边缘集。最大匹配将由一组顶点不相交的有向路径和有向循环组成（请参见图b中的红色边缘）。如果一个节点是匹配边的头部，则该节点是匹配的（图b中的绿色节点）。否则，它是不匹配的（图b中的白色节点）。那些不匹配的节点是需要控制的节点，即驱动节点。通过向这些驱动节点注入信号，可以得到一组以起点为输入的有向路径（见图c）。这些路径被称为“茎”。得到的有向图称为U根因子连接。通过将定向周期“嫁接”到那些“茎”，人们就会得到“芽”。得到的有向图称为cacti（见图d）。根据结构可控性定理，由于存在跨越受控网络的cacti结构（参见图e），因此该系统是可控的。受控网络（图e）下的cacti结构（图d）是维持可控性的“骨架”。

==Structural Controllability==
'''结构可控性 Structural Controllability 

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, IEEE Trans. Auto. Contr. 19

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, ''IEEE Trans. Auto. Contr.'' '''21'''

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, IEEE Trans. Auto. Contr. 21

(1974) .</ref>and then extended by Shields and Pearson (1976) ref name"Shields-76"r.希尔兹和 j。培生，IEEE Trans。自动。女名女子名。21

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, IEEE Trans. Auto. Contr. 21

结构性质的概念最早是由 Lin (1974) <ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''(1974).</ref>
和另外由 Glover 和 Silverman (1976)派生。 <ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''(1976).</ref>

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976年)。 / ref '''主要的问题是，对于可变系统参数来说，缺乏可控性或可观测性是否是一般性的。在结构控制的框架下，系统参数可以是独立自由变量，也可以是固定零点。这对于物理系统的模型是一致的，因为参数值永远不会准确地知道，零值除外，零值表示交互或连接的缺失。

==Maximum Matching==
'''最大匹配 Maximum Matching ''' 

In graph theory, a [[matching (graph theory)|matching]] is a set of edges without common vertices. Liu et al.<ref name="Liu-Nature-11"/> extended this definition to directed graph, where a matching is a set of directed edges that do not share start or end vertices. It is easy to check that a matching of a directed graph composes of a set of vertex-disjoint simple paths and cycles. The maximum matching of a directed network can be efficiently calculated by working in the bipartite representation using the classical [[Hopcroft–Karp algorithm]], which runs in O(''E''{{radic|''N''}}) time in the worst case. For undirected graph, analytical solutions of the size and number of maximum matchings have been studied using the [[cavity method]] developed in statistical physics.<ref name="Zdeborova-06">[[Lenka Zdeborová|L. Zdeborová]] and M. Mezard, ''J. Stat. Mech.'' '''05''' (2006).</ref> Liu et al.<ref name="Liu-Nature-11"/> extended the calculations for directed graph.
在图论中，匹配（图论）是一组没有共同顶点的边。Liu等人将此定义扩展到有向图，其中匹配是一组不共享起始或终止顶点的有向边。很容易检查有向图的匹配是否由一组顶点不相交的简单路径和周期组成。定向网络的最大匹配可以通过使用经典的'''Hopcroft-Karp算法 '''在'''二分表示 Bipartite Representation '''中工作来有效地计算，该算法在最坏情况下的时间复杂度是O(''E'{{radic |''N''}})。对于无向图，使用统计物理中开发的[[腔法]]研究了最大匹配的大小和数量的分析解。<ref name=“Zdeborova-06”>[[Lenkazdeborová| L Zdeborová]]和M.Mezard，''J.Stat.Mech''''05''（2006）。</ref>Liu等人<ref name=“Liu-Nature-11”/>扩展了有向图的计算范围。
In graph theory, a matching is a set of edges without common vertices. Liu et al. Liu et al. extended the calculations for directed graph.

By calculating the maximum matchings of a wide range of real networks, Liu et al.<ref name="Liu-Nature-11"/> asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the [[cavity method]]. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al.<ref name="Liu-Nature-11"/> would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al.<ref name="Liu-Nature-11"/> is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether ''control robustness" in networks is influenced more by using [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> over [[degree (graph theory)]] based metrics.

By calculating the maximum matchings of a wide range of real networks, Liu et al. asserted that the number of driver nodes is determined mainly by the networks degree distribution <math>P(k_\mathrm{in}, k_\mathrm{out})</math>. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the cavity method. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al. would predict same values of <math>{n_\mathrm{D}}</math>. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided purely by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? It remains open to scrutiny whether control robustness" in networks is influenced more by using betweenness centrality and closeness centrality over degree (graph theory) based metrics.

通过计算各种实际网络的最大匹配，刘等人提出了一种新的网络匹配算法。断言驱动节点数主要由网络度分布数学 p (k，k，k) / 数学决定。他们还利用空腔法计算了任意度分布的网络集成的平均驱动节点数。有趣的是，对于链图和弱紧连通图，它们的内外度分布都有很大的不同。可以预测相同的数学数值。此外，对于许多实词网络，即食物网络、神经元网络和代谢网络，Liu 等人计算出的数学函数 d ^ 实数 / 数学和数学函数 d ^ 实数 / 数学的值不匹配。值得注意的是。如果可控性纯粹是由程度决定的，那么为什么对于许多现实世界的网络来说，数学、数学和数学如此不同？网络中的控制鲁棒性是否受基于度的度量中介中心性和贴近中心性的影响仍有待进一步研究。

While sparser graphs are more difficult to control,<ref name="Liu-Nature-11"/><ref name = "Arxiv_Close_Betw"/> it would obviously be interesting to find whether [[betweenness centrality]] and [[closeness centrality]]<ref name = "Arxiv_Close_Betw"/> or degree heterogeneity<ref name="Liu-Nature-11"/> plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

While sparser graphs are more difficult to control, it would obviously be interesting to find whether betweenness centrality and closeness centrality or degree heterogeneity plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

尽管稀疏图的可控性难以控制，但是研究具有几乎相似度分布的稀疏图的可控性中，是否存在介于介于中间中心性和封闭中心性或度异质性起着更重要的作用显然是很有意义的。

==Control of composite quantum systems and algebraic graph theory==

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively.<ref name="BG-07">{{cite journal | last=Burgarth | first=Daniel | last2=Giovannetti | first2=Vittorio | title=Full Control by Locally Induced Relaxation | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=99 | issue=10 | date=2007-09-05 | issn=0031-9007 | doi=10.1103/physrevlett.99.100501 | page=100501| arxiv=0704.3027 }}</ref> This framework permits to formulate Kalman's criterion with tools from [[algebraic graph theory]] via the [[minimum rank of a graph]] and related notions.<ref name="BDHSY-13">{{cite journal|last1=Burgarth|first1=Daniel|last2=D'Alessandro|first2=Domenico|last3=Hogben|first3=Leslie|author3-link=Leslie Hogben|last4=Severini|first4=Simone|last5=Young|first5=Michael|doi=10.1109/TAC.2013.2250075|issue=9|journal=IEEE Transactions on Automatic Control|mr=3101617|pages=2349–2354|title=Zero forcing, linear and quantum controllability for systems evolving on networks|volume=58|year=2013}}</ref><ref name="OT-15">S. O'Rourke, B. Touri, https://arxiv.org/abs/1511.05080.</ref>

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively. This framework permits to formulate Kalman's criterion with tools from algebraic graph theory via the minimum rank of a graph and related notions.

在复合量子系统的通用控制的背景下，也发展了网络控制理论，其中子系统及其相互作用分别与节点和链路相关联。这个框架允许用代数图论的工具通过图的最小秩和相关概念来建立卡尔曼准则。

== See also ==

* [[Controllability Gramian]]

==References==

<references/>

== External links ==

* [http://barabasilab.neu.edu/projects/controllability/ The network controllability project website]

* [https://www.youtube.com/watch?v=9-q2qpOJfkg The video showing network controllability]





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[[Category:Network theory]]

Category:Network theory

范畴: 网络理论

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