布尔网络

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布尔函数Boolean function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。当布尔函数应用于复杂网络中时，我们定义了布尔网络 Boolean Network 的概念：布尔网络是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。每个变量的状态都由二进制1（开）和0（关）表示，每个模型都有着对应的逻辑规则表，每个变量的邻接变量可以在逻辑规则表的作用下得到自己的状态。由布尔表达式即可看出各个变量之间的逻辑关系。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[1]。

布尔网络在生物学中已被用于模拟调节网络 Regulatory Networks 。虽然布尔网络是对遗传现实的粗略简化，基因不是简单的二进制开关，但在一些情况下，它们正确地捕捉了表达和抑制基因的正确模式[2][3]。

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[4]。


经典模型

布尔网络是一种有着特殊的顺序动力学的系统，其中时间和状态都是离散的。也就是说，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元胞自动机 cellular automata（CA）一样，只是当它们被建立起来的时候，每个节点都遵从着一个规则，这个规则是从所有 2k 个可能的规则中随机选择的，有 K 个输入。在 K=2 时，第二类行为往往占主导地位。但对于 K>2 ，人们看到的行为很快就会接近随机映射的典型特征，其中代表 N 个底层节点的 2k 种状态演化的网络本身基本上是随机连接的[5]。

为了结构化分析布尔网络，我们先介绍一下状态迁移图State Transition Diagram (STD)的概念。状态迁移图也称为状态转移图，被用来描述系统或对象的状态，以及导致系统或对象的状态改变的事件，从而描述系统的行为。属于结构化分析方法使用工具。在状态迁移图中，由一个状态和一个事件所确定的下一状态可能会有多个。实际系统在下一步会迁移到哪一个状态，是由更详细的内部状态和更详细的事件信息来决定的，此时在状态迁移图中可能需要使用加进判断框和处理框的记法。状态迁移图在结构化分析中具有如下优点：第一，状态之间的关系能够直观地捕捉到，这样很直观地就能看到是否所有可能的状态迁移都已纳入图中，是否存在不必要的状态等。第二，由于状态迁移图的单纯性，能够简单机械地分析许多状态转移的情况，可以很容易地建立分析工具。

那么对于布尔网络的结构分析，其状态迁移图的一个重要性质是图中的每个节点都有一条出边，因为布尔网络的下一个状态是由布尔网络的当前状态唯一确定的。从这个属性可以看出，状态迁移图是树状结构的集合，每个树状结构由树和循环组成，其中树和/或循环可以由单个节点和一个自循环组成。在这些树状结构中，每条边都是从叶指向根的，循环对应于树的根。

接下来我们了解布尔网络中的另一种随机型网络：随机布尔网络 Random Boolean Network（RBN） 是指从所有具有特定大小N 的可能布尔网络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，人们可以研究随机布尔网络的行为如何随着网络集合属性中 平均连通性 Average Connectivity 的改变而改变。

对于布尔网络经典模型的研究历史，最早可以追溯到1969年，Stuart A. Kauffman 就提出了第一个布尔网络，作为遗传调控网络的随机模型[6]，但直到2000年之后，人们对于布尔网络模型的数学理解才逐步开始[7][8]。

吸引子


由于布尔网络只有 2N 种有限可能的状态，所以一个系统历经的状态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，系统的状态轨迹将落入一个稳定的状态或状态周期中，这种稳定状态或者周期即被称为吸引子 Attractors 。这里说明一下，对于在更广泛的动力学系统领域的情况，一个稳定状态或者周期只有当对于系统的扰动导致系统状态回到这个稳定状态或者周期状态时才称其为吸引子。如果吸引子只有一个孤立的状态，则称为点吸引子point attractor，如果吸引子由一个以上的多个状态组成，则称为周期吸引子cycle attractor或者极限环limit cycle。导致吸引子的状态集称为吸引子的吸引域basin of the attractor。

只发生在系统状态轨迹开始时的状态（也就是没有其他轨迹通向它们）被称为伊甸园状态garden-of-Eden states[9]，网络的动态从这些状态流向吸引子，而它们到达吸引子所需的时间也被称为瞬态时间transient time[4]。 

garden-of-Eden states 网络的动态从这些状态流向吸引子。到达吸引子所需的时间称为瞬时 transient time 。


随着计算机能力的不断提高，人们对看似简单的模型的理解也越来越深刻，在网络研究中不同的作者对吸引子的平均数量和长度给出了不同的估计，这里简单地总结一下在主要出版物上发表的研究成果[10]。

{ | 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

稳定性 Stability

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their 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.

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


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，如果 [math]\displaystyle{ K\lt K_{c} }[/math] ，网络是稳定的；如果 [math]\displaystyle{ K=K_{c} }[/math] ，网络是临界的；如果 [math]\displaystyle{ K\gt K_{c} }[/math] ，网络是不稳定的。

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.

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


If [math]\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]\displaystyle{ p_{i}=p=const. }[/math] 对于每一个节点，稳定和混沌范围之间的转换取决于 [math]\displaystyle{ p }[/math] 。根据伯纳德-德里达和伊夫-波莫的观点 , the critical value of the average number of connections is [math]\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为 [math]\displaystyle{ 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]\displaystyle{ K }[/math] 不是常数，且内度和外度之间没有相关性，则稳定性的条件由 [math]\displaystyle{ \langle K^{in}\rangle }[/math] 决定，如果 [math]\displaystyle{ \langle K^{in}\rangle \lt K_{c} }[/math] ，网络是稳定的。如果 [math]\displaystyle{ \langle K^{in}\rangle =K_{c} }[/math] ，则为临界；如果[math]\displaystyle{ \langle K^{in}\rangle \gt K_{c} }[/math] ，则为不稳定。

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

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

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

灵敏度显示了给定节点的布尔函数的输出在其输入变化时发生变化的概率。对于随机布尔网络。 [math]\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].

[math]\displaystyle{ 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]\displaystyle{ \lambda_{Q} }[/math]来控制。的矩阵 [math]\displaystyle{ Q }[/math]，其中[math]\displaystyle{ Q_{ij}=q_{i}A_{ij} }[/math]，[math]\displaystyle{ A }[/math] 是网络的邻接矩阵。如果 [math]\displaystyle{ \lambda_{Q}\lt 1 }[/math]，网络是稳定的；如果 [math]\displaystyle{ \lambda_{Q}=1 }[/math]，网络是临界的；如果 [math]\displaystyle{ \lambda_{Q}\gt 1 }[/math]，网络是不稳定的。

Variations of the model

模型的变化

Other topologies

其他拓扑性质

One theme is to study different underlying graph topologies.

One theme is to study different underlying graph topologies.

一个主题是研究不同的基础图拓扑结构。

  • The homogeneous case simply refers to a grid which is simply the reduction to the famous Ising model.
  • 同质情况 Homogeneous Case 只是指网格，它只是对著名的伊辛模型 lsing model 的还原。
  • Scale-free 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

其他更新方案

Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,[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 (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]

当 t≡Qi(modPi) 其中 t 是时间步长时，i 节点将被更新。确定性异步更新布尔网络(DRBNs)不是同步更新，但确定性解仍然存在。当 t≡Qi(modPi) 时，i 节点将被更新，其中 t 是时间步长。

  • The most general case is full stochastic updating (GARBN, general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
  • 最一般的情况是完全随机更新（GARBN，一般异步随机布尔网络）。在这里，在每个计算步骤中选择一个（或多个）节点进行更新。
  • The Partially-Observed Boolean Dynamical System (POBDS)[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

布尔网络的应用


Classification

分类


  • 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.

可伸缩的最佳贝叶斯分类 Scalable Optimal Bayesian Classification [32]开发了一种考虑潜在模型不确定性的轨迹最优分类，还提出了一种基于粒子的轨迹分类，对于大型网络具有高度的可扩展性，复杂度比最优解低得多。

See also

NK模型 NK Model

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.<nowiki>
  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. ↑ 32.0 32.1 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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