Physics:Boolean network

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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]

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] The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.[4]

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 22K 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 2N states of the N underlying nodes is itself connected essentially randomly.[5]

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.

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]


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]

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]

Author Year Mean attractor length Mean attractor number comment
Kauffmann [6] 1969 [math]\langle A\rangle\sim \sqrt{N}[/math] [math]\langle\nu\rangle\sim \sqrt{N}[/math]
Bastolla/ Parisi[11] 1998 faster than a power law, [math]\langle A\rangle \gt N^x \forall x[/math] faster than a power law, [math]\langle\nu\rangle \gt N^x \forall x[/math] first numerical evidences
Bilke/ Sjunnesson[12] 2002 linear with system size, [math]\langle\nu\rangle \sim N[/math]
Socolar/Kauffman[13] 2003 faster than linear, [math]\langle\nu\rangle \gt N^x[/math] with [math]x \gt 1[/math]
Samuelsson/Troein[14] 2003 superpolynomial growth, [math]\langle\nu\rangle \gt N^x \forall x[/math] mathematical proof
Mihaljev/Drossel[15] 2005 faster than a power law, [math]\langle A\rangle \gt N^x \forall x[/math] faster than a power law, [math]\langle\nu\rangle \gt N^x \forall x[/math]


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.

For N-K-model[16] the network is stable if [math]K\lt K_{c}[/math], critical if [math]K=K_{c}[/math], and unstable if [math]K\gt K_{c}[/math].

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

If [math] p_{i}=p=const. [/math] for every node, the transition between the stable and chaotic range depends on [math] p [/math]. The critical value of the average number of connections is [math] K_{c}=1/[2p(1-p)] [/math].[17]

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][18][19][20] The network is stable if [math]\langle K^{in}\rangle \lt K_{c}[/math], critical if [math]\langle K^{in}\rangle =K_{c}[/math], and unstable if [math]\langle K^{in}\rangle \gt 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] 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.[21]

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 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.[22] The network is stable if [math]\lambda_{Q}\lt 1[/math], critical if [math]\lambda_{Q}=1[/math], unstable if [math]\lambda_{Q}\gt 1[/math].

Variations of the model

Other 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.
  • 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:

  • 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


  • 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


  1. Naldi, A.; Monteiro, P. T.; Mussel, C.; Kestler, H. A.; Thieffry, D.; Xenarios, I.; Saez-Rodriguez, J.; Helikar, T. et al. (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. doi:10.1016/S0022-5193(03)00035-3. 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. doi:10.1073/pnas.0308195101. PMID 15123827. Bibcode2004PNAS..101.7341L. 
  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. 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. 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. doi:10.1038/224177a0. PMID 5343519. Bibcode1969Natur.224..177K. 
  7. Aldana, Maximo; Coppersmith, Susan; Kadanoff, Leo P. (2003). Boolean Dynamics with Random Couplings. 23–89. 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. Bibcode2004nlin......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. 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. PMID 22912642. 
  11. Bastolla, U.; Parisi, G. (May 1998). "The modular structure of Kauffman networks". Physica D: Nonlinear Phenomena 115 (3–4): 219–233. doi:10.1016/S0167-2789(97)00242-X. Bibcode1998PhyD..115..219B. 
  12. Bilke, Sven; Sjunnesson, Fredrik (December 2001). "Stability of the Kauffman model". Physical Review E 65 (1): 016129. doi:10.1103/PhysRevE.65.016129. PMID 11800758. Bibcode2002PhRvE..65a6129B. 
  13. Socolar, J.; Kauffman, S. (February 2003). "Scaling in Ordered and Critical Random Boolean Networks". Physical Review Letters 90 (6): 068702. doi:10.1103/PhysRevLett.90.068702. PMID 12633339. Bibcode2003PhRvL..90f8702S. 
  14. Samuelsson, Björn; Troein, Carl (March 2003). "Superpolynomial Growth in the Number of Attractors in Kauffman Networks". Physical Review Letters 90 (9): 098701. doi:10.1103/PhysRevLett.90.098701. PMID 12689263. Bibcode2003PhRvL..90i8701S. 
  15. Mihaljev, Tamara; Drossel, Barbara (October 2006). "Scaling in a general class of critical random Boolean networks". Physical Review E 74 (4): 046101. doi:10.1103/PhysRevE.74.046101. PMID 17155127. Bibcode2006PhRvE..74d6101M. 
  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. doi:10.1209/0295-5075/1/2/001. Bibcode1986EL......1...45D. 
  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. doi:10.1016/0375-9601(94)00876-Q. Bibcode1995PhLA..196..331S. 
  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. doi:10.1103/PhysRevE.55.257. Bibcode1997PhRvE..55..257L. 
  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. doi:10.1063/1.1414882. ISSN 1054-1500. PMID 12779520. Bibcode2001Chaos..11..809F. 
  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. doi:10.1073/pnas.1536783100. ISSN 0027-8424. PMID 12853565. Bibcode2003PNAS..100.8710A. 
  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. doi:10.1073/pnas.0900142106. ISSN 0027-8424. PMID 19416903. Bibcode2009PNAS..106.8209P. 
  23. Aldana, Maximino (October 2003). "Boolean dynamics of networks with scale-free topology". Physica D: Nonlinear Phenomena 185 (1): 45–66. doi:10.1016/s0167-2789(03)00174-x. Bibcode2003PhyD..185...45A. 
  24. Drossel, Barbara; Greil, Florian (4 August 2009). "Critical Boolean networks with scale-free in-degree distribution". Physical Review E 80 (2): 026102. doi:10.1103/PhysRevE.80.026102. PMID 19792195. Bibcode2009PhRvE..80b6102D. 
  25. Harvey, Imman; Bossomaier, Terry (1997). Husbands, Phil; Harvey, Imman. eds. Time out of joint: Attractors in asynchronous random Boolean networks. MIT Press. 67–75. ISBN 9780262581578. 
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  27. Gershenson, Carlos; Broekaert, Jan; Aerts, Diederik (14 September 2003). Contextual Random Boolean Networks. Lecture Notes in Computer Science. 2801. Dortmund, Germany. 615–624. 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. doi:10.1109/TSP.2016.2614798. ISSN 1053-587X. Bibcode2017ITSP...65..359I. 
  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. 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.

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