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<font color="#FF8000">布尔函数Boolean function</font>是一种可用于通过逻辑类型的计算来评估与其布尔输入有关的任何布尔输出的函数。这些功能在复杂性理论中起着基本作用。

{{Network Science}}

'''<font color="#FF8000">布尔网络 Boolean Network </font>'''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<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}}&lt;nowiki&gt;</ref>。

[[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]].]]

布尔网络在生物学中已被用于模拟'''<font color="#FF8000">调节网络 Regulatory Networks </font>'''。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<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}}<!--|accessdate=25 November 2014--></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>。

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.]]

直到2000年中期人们才完全理解看似简单的数学上的同步模型<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>。

 

==Classical model==

 

 

 

 

 

 

 

 

 

 

== Classical model ==

经典模型<br>

经典模型<br>

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{{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.

<nowiki>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.</nowiki>

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^k'' 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 ''2^k'' 种状态演化的网络本身基本上是随机连接的。

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^k'' 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 ''2^k'' 种状态演化的网络本身基本上是随机连接的。

A '''random Boolean network'''&nbsp;(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'''&nbsp;(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&nbsp;(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&nbsp;(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.

'''<font color="#FF8000">随机布尔网络 Random Boolean Network（RBN） </font>'''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'''<font color="#FF8000" > Average Connectivity 平均连通性 </font>'''的改变而改变。

'''<font color="#FF8000">随机布尔网络 Random Boolean Network（RBN） </font>'''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'''<font color="#FF8000"> Average Connectivity 平均连通性 </font>'''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 }}<!--|accessdate=25 November 2014--></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 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 }}<!--|accessdate=25 November 2014--></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.

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。

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 ==

==Attractors==

'''<font color="#FF8000">吸引子 Attractors </font>'''<br>

'''<font color="#FF8000">吸引子 Attractors </font>'''<br>

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 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 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 and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.

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 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 2^N 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'''<font color="#FF8000">伊甸园状态  

由于布尔网络只有 2^N 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'''<font color="#FF8000">伊甸园状态 '''

  garden-of-Eden states </font>'''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'''<font color="#FF8000">瞬时 transient time </font>'''。

  garden-of-Eden states </font>'''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'''<font color="#FF8000">瞬时 transient time </font>'''。'''

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}}<!--|accessdate=26 November 2014--></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.<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}}<!--|accessdate=26 November 2014--></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.

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.

{| class="wikitable sortable"

{| class="wikitable sortable"

{| class="wikitable sortable"

|}

{ | class = “ wikitable sortable”

{ | class = “ wikitable sortable”

|-

|-

|-

|-

! Author

!Author

! Author

!Author

!作者

!作者

! Year

!Year

! Year

!Year

!年份

!年份

! Mean attractor length

!Mean attractor length

! Mean attractor length

!Mean attractor length

!平均吸引长度

!平均吸引长度

! Mean attractor number

!Mean attractor number

! Mean attractor number

!Mean attractor number

!平均吸引子数

!平均吸引子数

! comment

!comment

! comment

!comment

!评论

! 评论

|-

|-

|-

|-

| Kauffmann <ref name=KauffmanOriginal/>

|Kauffmann <ref name="KauffmanOriginal" />

| Kauffmann  

|Kauffmann  

考夫曼

考夫曼

| 1969

|1969

| 1969

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

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

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

| <math>\langle\nu\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>

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

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

|

|

|-

|-

| 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 }}<!--|accessdate=26 November 2014--></ref>

|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 }}<!--|accessdate=26 November 2014--></ref>

| Bastolla/ Parisi

|Bastolla/ Parisi

| Bastolla/ Parisi

|Bastolla/ Parisi

| 1998

|1998

| 1998

|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>

| 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 >  

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

| 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>

| 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 >  

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

| first numerical evidences

|first numerical evidences

| first numerical evidences

|first numerical evidences

第一个数字证据

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 }}<!--|accessdate=26 November 2014--></ref>

|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 }}<!--|accessdate=26 November 2014--></ref>

| Bilke/ Sjunnesson

|Bilke/ Sjunnesson

| Bilke/Sjunnesson

|Bilke/Sjunnesson

| 2002

|2002

| 2002

|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>

| 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

|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<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

| Socolar/Kauffman

|Socolar/Kauffman

| 2003

|2003

| 2003

|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>

| 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 >  

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

|

|

|-

|-

| 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}}<!--|accessdate=26 November 2014--></ref>

|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}}<!--|accessdate=26 November 2014--></ref>

| Samuelsson/Troein

|Samuelsson/Troein

| Samuelsson/Troein

|Samuelsson/Troein

| 2003

|2003

| 2003

|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>

| 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 >  

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

| mathematical proof

|mathematical proof

| mathematical proof

|mathematical proof

数学证明

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 }}<!--|accessdate=26 November 2014--></ref>

|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 }}<!--|accessdate=26 November 2014--></ref>

| Mihaljev/Drossel

|Mihaljev/Drossel

| Mihaljev/Drossel

|Mihaljev/Drossel

| 2005

|2005

| 2005

|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>

| 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 >  

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

| 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>

| 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 >  

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

|  

|

|  

|

|

|

|}

|}

==Stability==

 

 

== Stability ==

'''<font color="#FF8000">稳定性 Stability </font>'''<br>

'''<font color="#FF8000">稳定性 Stability </font>'''<br>

<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。

<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 ==

==Variations of the model==

模型的变化<br>

模型的变化<br>

== Other topologies ==

==Other topologies==

其他拓扑性质<br>

其他拓扑性质<br>

一个主题是研究不同的基础图拓扑结构。

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]].  

*The homogeneous case simply refers to a grid which is simply the reduction to the famous [[Ising model]].

*'''<font color="#FF8000">同质情况 Homogeneous Case </font>'''只是指网格，它只是对著名的'''<font color="#FF8000">伊辛模型 lsing model </font>'''的还原。

*'''<font color="#FF8000">同质情况 Homogeneous Case </font>'''只是指网格，它只是对著名的'''<font color="#FF8000">伊辛模型 lsing model </font>'''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.

*[[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 ==

==Other updating schemes==

其他更新方案<br>

其他更新方案<br>

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,<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:

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:

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 ≡ Q_i (''mod'' P_i)'' 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>

*'''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 ≡ Q_i (''mod'' P_i)'' 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≡Q_i(''mod''P_i)'' 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≡Q_i(''mod''P_i)'' 时，''i'' 节点将被更新，其中 ''t'' 是时间步长。

当 ''t≡Q_i(''mod''P_i)'' 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≡Q_i(''mod''P_i)'' 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.

*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。

*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.

*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 ==

==Application of Boolean Networks==

布尔网络的应用<br>

布尔网络的应用<br>

 

==Classification==

== Classification ==

分类<br>

分类<br>

 

*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.

* 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.

'''<font color="#FF8000">可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification </font>''' <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。

'''<font color="#FF8000">可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification </font>''' <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 ==

==See also==

* [[NK model]] <!-- to be merged here -->

*[[NK model]]<!-- to be merged here -->

'''<font color="#FF8000">NK模型 NK Model </font>'''

'''<font color="#FF8000">NK模型 NK Model </font>'''

== References ==

==References==

{{Reflist|30em}}

{{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. <!-- to be cited or not -->

*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.<!-- to be cited or not -->

 

== External links ==

==External links==

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

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

{{Stochastic processes}}

{{Stochastic processes}}

[[Category:Bioinformatics]]

[[Category:Bioinformatics]]