布尔网络

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文件:Hou710 BooleanNetwork.svg
State space of a Boolean Network with N=4 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.]]

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 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.[1]

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.[2][3]

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.[4]

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模板:Sup 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模板:Sup states of the N underlying nodes is itself connected essentially randomly.[5]

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 networks[6] but their mathematical understanding only started in the 2000s.[7][8]

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 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[9] and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.[4]

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.[10]

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”
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 [6] Kauffmann

考夫曼

1969 1969 1969 [math]\displaystyle{ \langle A\rangle\sim \sqrt{N} }[/math] [math]\displaystyle{ \langle A\rangle\sim \sqrt{N} }[/math] < math > langle a rangle sim sqrt { n } </math > [math]\displaystyle{ \langle\nu\rangle\sim \sqrt{N} }[/math] [math]\displaystyle{ \langle\nu\rangle\sim \sqrt{N} }[/math] < math > langle nu rangle sim sqrt { n } </math >
Bastolla/ Parisi[11] Bastolla/ Parisi Bastolla/ Parisi 1998 1998 1998 faster than a power law, [math]\displaystyle{ \langle A\rangle \gt N^x \forall x }[/math] faster than a power law, [math]\displaystyle{ \langle A\rangle \gt 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]\displaystyle{ \langle\nu\rangle \gt N^x \forall x }[/math] faster than a power law, [math]\displaystyle{ \langle\nu\rangle \gt 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[12] Bilke/ Sjunnesson Bilke/Sjunnesson 2002 2002 2002 linear with system size, [math]\displaystyle{ \langle\nu\rangle \sim N }[/math] linear with system size, [math]\displaystyle{ \langle\nu\rangle \sim N }[/math] 与系统大小成线性关系
Socolar/Kauffman[13] Socolar/Kauffman Socolar/Kauffman 2003 2003 2003 faster than linear, [math]\displaystyle{ \langle\nu\rangle \gt N^x }[/math] with [math]\displaystyle{ x \gt 1 }[/math] faster than linear, [math]\displaystyle{ \langle\nu\rangle \gt N^x }[/math] with [math]\displaystyle{ x \gt 1 }[/math]

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

Samuelsson/Troein[14] Samuelsson/Troein Samuelsson/Troein 2003 2003 2003 superpolynomial growth, [math]\displaystyle{ \langle\nu\rangle \gt N^x \forall x }[/math] superpolynomial growth, [math]\displaystyle{ \langle\nu\rangle \gt 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[15] Mihaljev/Drossel Mihaljev/Drossel 2005 2005 2005 faster than a power law, [math]\displaystyle{ \langle A\rangle \gt N^x \forall x }[/math] faster than a power law, [math]\displaystyle{ \langle A\rangle \gt 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]\displaystyle{ \langle\nu\rangle \gt N^x \forall x }[/math] faster than a power law, [math]\displaystyle{ \langle\nu\rangle \gt 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 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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[16] the network is stable if [math]\displaystyle{ K\lt K_{c} }[/math], critical if [math]\displaystyle{ K=K_{c} }[/math], and unstable if [math]\displaystyle{ K\gt K_{c} }[/math].

For N-K-model the network is stable if [math]\displaystyle{ K\lt K_{c} }[/math], critical if [math]\displaystyle{ K=K_{c} }[/math], and unstable if [math]\displaystyle{ K\gt 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]\displaystyle{ n_{i} }[/math] is updated according to its truth table, whose outputs are randomly populated. [math]\displaystyle{ 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]\displaystyle{ n_{i} }[/math] is updated according to its truth table, whose outputs are randomly populated. [math]\displaystyle{ 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]\displaystyle{ p_{i}=p=const. }[/math] for every node, the transition between the stable and chaotic range depends on [math]\displaystyle{ p }[/math]. According to Bernard Derrida and Yves Pomeau[17]

If [math]\displaystyle{ p_{i}=p=const. }[/math] for every node, the transition between the stable and chaotic range depends on [math]\displaystyle{ 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]\displaystyle{ K_{c}=1/[2p(1-p)] }[/math].

, the critical value of the average number of connections is [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \langle K^{in}\rangle }[/math][18][19][20] The network is stable if [math]\displaystyle{ \langle K^{in}\rangle \lt K_{c} }[/math], critical if [math]\displaystyle{ \langle K^{in}\rangle =K_{c} }[/math], and unstable if [math]\displaystyle{ \langle K^{in}\rangle \gt K_{c} }[/math].

If [math]\displaystyle{ 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]\displaystyle{ \langle K^{in}\rangle }[/math] The network is stable if [math]\displaystyle{ \langle K^{in}\rangle \lt K_{c} }[/math], critical if [math]\displaystyle{ \langle K^{in}\rangle =K_{c} }[/math], and unstable if [math]\displaystyle{ \langle K^{in}\rangle \gt 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 topology where the in-and out-degree distribution is a power-law distribution: [math]\displaystyle{ P(K) \propto K^{-\gamma} }[/math], and [math]\displaystyle{ \langle K^{in} \rangle=\langle K^{out} \rangle }[/math], since every out-link from a node is an in-link to another.[21]

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]\displaystyle{ P(K) \propto K^{-\gamma} }[/math], and [math]\displaystyle{ \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]\displaystyle{ q_{i}=2p_{i}(1-p_{i}) }[/math]. In the general case, stability of the network is governed by the largest eigenvalue [math]\displaystyle{ \lambda_{Q} }[/math] of matrix [math]\displaystyle{ Q }[/math], where [math]\displaystyle{ Q_{ij}=q_{i}A_{ij} }[/math], and [math]\displaystyle{ A }[/math] is the adjacency matrix of the network.[22] The network is stable if [math]\displaystyle{ \lambda_{Q}\lt 1 }[/math], critical if [math]\displaystyle{ \lambda_{Q}=1 }[/math], unstable if [math]\displaystyle{ \lambda_{Q}\gt 1 }[/math].

[math]\displaystyle{ q_{i}=2p_{i}(1-p_{i}) }[/math]. In the general case, stability of the network is governed by the largest eigenvalue [math]\displaystyle{ \lambda_{Q} }[/math] of matrix [math]\displaystyle{ Q }[/math], where [math]\displaystyle{ Q_{ij}=q_{i}A_{ij} }[/math], and [math]\displaystyle{ A }[/math] is the adjacency matrix of the network. The network is stable if [math]\displaystyle{ \lambda_{Q}\lt 1 }[/math], critical if [math]\displaystyle{ \lambda_{Q}=1 }[/math], unstable if [math]\displaystyle{ \lambda_{Q}\gt 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 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 topologies may be chosen for Boolean networks.[23] One can distinguish the case where only in-degree distribution in power-law distributed,[24] 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,[25] different alternatives have been introduced. A common classification[26] 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 (DRBNs) 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.[27]
  • 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)[28][29][30][31] 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[32] 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


References

  1. ↑ Naldi, A.; Monteiro, P. T.; Mussel, C.; Kestler, H. A.; Thieffry, D.; Xenarios, I.; Saez-Rodriguez, J.; Helikar, T.; Chaouiya, C. (25 January 2015). "Cooperative development of logical modelling standards and tools with CoLoMoTo". Bioinformatics. 31 (7): 1154–1159. doi:10.1093/bioinformatics/btv013. PMID 25619997.
  2. ↑ Albert, Réka; Othmer, Hans G (July 2003). "The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster". Journal of Theoretical Biology. 223 (1): 1–18. CiteSeerX 10.1.1.13.3370. doi:10.1016/S0022-5193(03)00035-3. PMC 6388622. PMID 12782112.
  3. ↑ Li, J.; Bench, A. J.; Vassiliou, G. S.; Fourouclas, N.; Ferguson-Smith, A. C.; Green, A. R. (30 April 2004). "Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies". Proceedings of the National Academy of Sciences. 101 (19): 7341–7346. Bibcode:2004PNAS..101.7341L. doi:10.1073/pnas.0308195101. PMC 409920. PMID 15123827.
  4. ↑ 4.0 4.1 Drossel, Barbara (December 2009). "Random Boolean Networks". In Schuster, Heinz Georg. Chapter 3. Random Boolean Networks. Reviews of Nonlinear Dynamics and Complexity. Wiley. pp. 69–110. arXiv:0706.3351. doi:10.1002/9783527626359.ch3. ISBN 9783527626359. 
  5. ↑ Wolfram, Stephen (2002). A New Kind of Science. Champaign, Illinois: Wolfram Media, Inc.. p. 936. ISBN 978-1579550080. https://archive.org/details/newkindofscience00wolf/page/936. Retrieved 15 March 2018. 
  6. ↑ 6.0 6.1 Kauffman, Stuart (11 October 1969). "Homeostasis and Differentiation in Random Genetic Control Networks". Nature. 224 (5215): 177–178. Bibcode:1969Natur.224..177K. doi:10.1038/224177a0. PMID 5343519.
  7. ↑ Aldana, Maximo; Coppersmith, Susan; Kadanoff, Leo P. (2003). Boolean Dynamics with Random Couplings. pp. 23–89. arXiv:nlin/0204062. doi:10.1007/978-0-387-21789-5_2. ISBN 978-1-4684-9566-9. 
  8. ↑ Gershenson, Carlos (2004). "Introduction to Random Boolean Networks". 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. 2004: 160–173. arXiv:nlin.AO/0408006. Bibcode:2004nlin......8006G.
  9. ↑ Wuensche, Andrew (2011). Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic and beyond]]. Frome, England: Luniver Press. p. 16. ISBN 9781905986316. https://books.google.de/books?id=qsktzY_Vg8QC&pg=PA16. Retrieved 12 January 2016. 
  10. ↑ Greil, Florian (2012). "Boolean Networks as Modeling Framework". Frontiers in Plant Science. 3: 178. doi:10.3389/fpls.2012.00178. PMC 3419389. PMID 22912642.
  11. ↑ Bastolla, U.; Parisi, G. (May 1998). "The modular structure of Kauffman networks". Physica D: Nonlinear Phenomena. 115 (3–4): 219–233. arXiv:cond-mat/9708214. Bibcode:1998PhyD..115..219B. doi:10.1016/S0167-2789(97)00242-X.
  12. ↑ Bilke, Sven; Sjunnesson, Fredrik (December 2001). "Stability of the Kauffman model". Physical Review E. 65 (1): 016129. arXiv:cond-mat/0107035. Bibcode:2002PhRvE..65a6129B. doi:10.1103/PhysRevE.65.016129. PMID 11800758.
  13. ↑ Socolar, J.; Kauffman, S. (February 2003). "Scaling in Ordered and Critical Random Boolean Networks". Physical Review Letters. 90 (6): 068702. arXiv:cond-mat/0212306. Bibcode:2003PhRvL..90f8702S. doi:10.1103/PhysRevLett.90.068702. PMID 12633339.
  14. ↑ Samuelsson, Björn; Troein, Carl (March 2003). "Superpolynomial Growth in the Number of Attractors in Kauffman Networks". Physical Review Letters. 90 (9): 098701. Bibcode:2003PhRvL..90i8701S. doi:10.1103/PhysRevLett.90.098701. PMID 12689263.
  15. ↑ Mihaljev, Tamara; Drossel, Barbara (October 2006). "Scaling in a general class of critical random Boolean networks". Physical Review E. 74 (4): 046101. arXiv:cond-mat/0606612. Bibcode:2006PhRvE..74d6101M. doi:10.1103/PhysRevE.74.046101. PMID 17155127.
  16. ↑ Kauffman, S. A. (1969). "Metabolic stability and epigenesis in randomly constructed genetic nets". Journal of Theoretical Biology. 22 (3): 437–467. doi:10.1016/0022-5193(69)90015-0. PMID 5803332.
  17. ↑ Derrida, B; Pomeau, Y (1986-01-15). "Random Networks of Automata: A Simple Annealed Approximation". Europhysics Letters (EPL). 1 (2): 45–49. Bibcode:1986EL......1...45D. doi:10.1209/0295-5075/1/2/001.
  18. ↑ Solé, Ricard V.; Luque, Bartolo (1995-01-02). "Phase transitions and antichaos in generalized Kauffman networks". Physics Letters A. 196 (5–6): 331–334. Bibcode:1995PhLA..196..331S. doi:10.1016/0375-9601(94)00876-Q.
  19. ↑ Luque, Bartolo; Solé, Ricard V. (1997-01-01). "Phase transitions in random networks: Simple analytic determination of critical points". Physical Review E. 55 (1): 257–260. Bibcode:1997PhRvE..55..257L. doi:10.1103/PhysRevE.55.257.
  20. ↑ Fox, Jeffrey J.; Hill, Colin C. (2001-12-01). "From topology to dynamics in biochemical networks". Chaos: An Interdisciplinary Journal of Nonlinear Science. 11 (4): 809–815. Bibcode:2001Chaos..11..809F. doi:10.1063/1.1414882. ISSN 1054-1500. PMID 12779520.
  21. ↑ Aldana, Maximino; Cluzel, Philippe (2003-07-22). "A natural class of robust networks". Proceedings of the National Academy of Sciences. 100 (15): 8710–8714. Bibcode:2003PNAS..100.8710A. doi:10.1073/pnas.1536783100. ISSN 0027-8424. PMC 166377. PMID 12853565.
  22. ↑ Pomerance, Andrew; Ott, Edward; Girvan, Michelle; Losert, Wolfgang (2009-05-19). "The effect of network topology on the stability of discrete state models of genetic control". Proceedings of the National Academy of Sciences. 106 (20): 8209–8214. arXiv:0901.4362. Bibcode:2009PNAS..106.8209P. doi:10.1073/pnas.0900142106. ISSN 0027-8424. PMC 2688895. PMID 19416903.
  23. ↑ Aldana, Maximino (October 2003). "Boolean dynamics of networks with scale-free topology". Physica D: Nonlinear Phenomena. 185 (1): 45–66. arXiv:cond-mat/0209571. Bibcode:2003PhyD..185...45A. doi:10.1016/s0167-2789(03)00174-x.
  24. ↑ Drossel, Barbara; Greil, Florian (4 August 2009). "Critical Boolean networks with scale-free in-degree distribution". Physical Review E. 80 (2): 026102. arXiv:0901.0387. Bibcode:2009PhRvE..80b6102D. doi:10.1103/PhysRevE.80.026102. PMID 19792195.
  25. ↑ Harvey, Imman; Bossomaier, Terry (1997). Husbands, Phil; Harvey, Imman. eds. Time out of joint: Attractors in asynchronous random Boolean networks. MIT Press. pp. 67–75. ISBN 9780262581578. https://books.google.de/books?id=ccp8fzlyorAC&pg=PA67. 
  26. ↑ Gershenson, Carlos (2002). Standish, Russell K; Bedau, Mark A. eds. Classification of Random Boolean Networks. Artificial Life. 8. Cambridge, Massachusetts, USA. pp. 1–8. arXiv:cs/0208001. Bibcode 2002cs........8001G. ISBN 9780262692816. https://books.google.de/books?id=si_KlRbL1XoC&pg=PA1. Retrieved 12 January 2016. 
  27. ↑ Gershenson, Carlos; Broekaert, Jan; Aerts, Diederik (14 September 2003). Contextual Random Boolean Networks. Lecture Notes in Computer Science. 2801. Dortmund, Germany. pp. 615–624. arXiv:nlin/0303021. doi:10.1007/978-3-540-39432-7_66. ISBN 978-3-540-39432-7. 
  28. ↑ Imani, M.; Braga-Neto, U. M. (2017-01-01). "Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems". IEEE Transactions on Signal Processing. 65 (2): 359–371. arXiv:1702.07269. Bibcode:2017ITSP...65..359I. doi:10.1109/TSP.2016.2614798. ISSN 1053-587X.
  29. ↑ Imani, M.; Braga-Neto, U. M. (2015). "Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother" (in en-US). 2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP). pp. 972–976. doi:10.1109/GlobalSIP.2015.7418342. ISBN 978-1-4799-7591-4. 
  30. ↑ Imani, M.; Braga-Neto, U. M. (2016) (in en-US). 2016 American Control Conference (ACC). pp. 227–232. doi:10.1109/ACC.2016.7524920. ISBN 978-1-4673-8682-1. 
  31. ↑ Imani, M.; Braga-Neto, U. (2016-12-01). Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space. pp. 4208–4213. doi:10.1109/CDC.2016.7798908. ISBN 978-1-5090-1837-6. 
  32. ↑ 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


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