更改

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.]]

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。]]

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 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.

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。

'''<font color="#FF8000">布尔网络 Boolean Network </font>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.  

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.  

 

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.<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.

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

 

== Classical model ==

== 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{{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.

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。

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&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.

 

=== Attractors ===

== Attractors ==

'''<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 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.

 

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"

== Stability ==

== 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 [[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.

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 >)。

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'''<font color="#FF8000">海明距离 Hamming Distance </font>'''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 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 > 。

对于'''<font color="#FF8000">N-K模型 N-K-Model </font>'''，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。

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

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

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>.

, 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 > 。

，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> 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 > 。

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 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.

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。

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,

 

<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 [[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>. 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 > 。

<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 ==

== Variations of the model ==

=== Other topologies ===

== Other topologies ==

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

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

One theme is to study different underlying graph topologies.

One theme is to study different underlying graph topologies.

 

* 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]].  

 

* [[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 ==

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:

 

* '''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>

 

* 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.

 

* 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 ==

=== Classification ===

== 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.

* 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 ==

== See also ==