Boolean network

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.

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{{cite journal|last1=Kauffman|first1=Stuart|title=Homeostasis and Differentiation in Random Genetic Control Networks|journal=Nature|date=11 October 1969|volume=224|issue=5215|pages=177–178|doi=10.1038/224177a0|pmid=5343519|bibcode = 1969Natur.224..177K |s2cid=4179318}} but their mathematical understanding only started in the 2000s.{{cite book|last1=Aldana|first1=Maximo|last2=Coppersmith|first2=Susan|author2-link= Susan Coppersmith |last3=Kadanoff|first3=Leo P.|title=Perspectives and Problems in Nonlinear Science: A Celebratory Volume in Honor of Lawrence Sirovich |chapter=Boolean Dynamics with Random Couplings|date=2003|pages=23–89|doi=10.1007/978-0-387-21789-5_2|arxiv=nlin/0204062|isbn=978-1-4684-9566-9|s2cid=15024306}}{{Cite journal|arxiv=nlin.AO/0408006|last1=Gershenson|first1=Carlos|title=Introduction to Random Boolean Networks|journal=In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (Eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp|volume=2004|pages=160–173|year=2004|bibcode=2004nlin......8006G}}

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{{cite book|last1=Wuensche|first1=Andrew|title=Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]|date=2011|publisher=Luniver Press|location=Frome, England|isbn=9781905986316|page=16|url=https://books.google.com/books?id=qsktzY_Vg8QC&pg=PA16|access-date=12 January 2016|archive-date=4 February 2023|archive-url=https://web.archive.org/web/20230204160659/https://books.google.com/books?id=qsktzY_Vg8QC&pg=PA16|url-status=live}} 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.{{cite journal|last1=Greil|first1=Florian|title=Boolean Networks as Modeling Framework|journal=Frontiers in Plant Science|date=2012|volume=3|pages=178|doi=10.3389/fpls.2012.00178|pmid=22912642|pmc=3419389|doi-access=free}}

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 ($K\_\{c\}$), 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 ($N$) in the network.

For N-K-model{{cite journal |last=Kauffman |first=S. A. |date=1969 |title=Metabolic stability and epigenesis in randomly constructed genetic nets |journal=Journal of Theoretical Biology |volume=22 |issue=3 |pages=437–467 |doi=10.1016/0022-5193(69)90015-0|pmid=5803332 |bibcode=1969JThBi..22..437K }} the network is stable if

The state of a given node $n\_\{i\}$ is updated according to its truth table, whose outputs are randomly populated. $p\_\{i\}$ denotes the probability of assigning an off output to a given series of input signals.

If $p\_\{i\}=p=const.$ for every node, the transition between the stable and chaotic range depends on $p$. According to Bernard Derrida and Yves Pomeau{{Cite journal|title = Random Networks of Automata: A Simple Annealed Approximation|url = http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09|journal = Europhysics Letters|date = 1986-01-15|pages = 45–49|volume = 1|issue = 2|doi = 10.1209/0295-5075/1/2/001|first1 = B|last1 = Derrida|first2 = Y|last2 = Pomeau|bibcode = 1986EL......1...45D|s2cid = 160018158|access-date = 2016-01-12|archive-date = 2020-05-17|archive-url = https://web.archive.org/web/20200517203049/http://stacks.iop.org/0295-5075/1/i=2/a=001?key=crossref.8cf81041d96c6144f3a397d9cf72cf09%2F|url-status = live}}

, the critical value of the average number of connections is $K\_\{c\}=1/[2p(1-p)]$.

If $K$ is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by $\backslash langle\; K^\{in\}\backslash rangle${{Cite journal|title = Phase transitions and antichaos in generalized Kauffman networks|journal = Physics Letters A|date = 1995-01-02|pages = 331–334|volume = 196|issue = 5–6|doi = 10.1016/0375-9601(94)00876-Q|first1 = Ricard V.|last1 = Solé|first2 = Bartolo|last2 = Luque|bibcode = 1995PhLA..196..331S}}{{Cite journal|title = Phase transitions in random networks: Simple analytic determination of critical points|journal = Physical Review E|date = 1997-01-01|pages = 257–260|volume = 55|issue = 1|doi = 10.1103/PhysRevE.55.257|first1 = Bartolo|last1 = Luque|first2 = Ricard V.|last2 = Solé|bibcode = 1997PhRvE..55..257L}}{{Cite journal|title = From topology to dynamics in biochemical networks |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science|date = 2001-12-01|issn = 1054-1500|pages = 809–815|volume = 11|issue = 4|doi = 10.1063/1.1414882|pmid = 12779520 |first1 = Jeffrey J.|last1 = Fox|first2 = Colin C.|last2 = Hill|bibcode = 2001Chaos..11..809F|doi-access = free}} The network is stable if

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: $P(K)\; \backslash propto\; K^\{-\backslash gamma\}$, and $\backslash langle\; K^\{in\}\; \backslash rangle=\backslash langle\; K^\{out\}\; \backslash rangle$, since every out-link from a node is an in-link to another.{{Cite journal|title = A natural class of robust networks|journal = Proceedings of the National Academy of Sciences|date = 2003-07-22|issn = 0027-8424|pmc = 166377|pmid = 12853565|pages = 8710–8714|volume = 100|issue = 15|doi = 10.1073/pnas.1536783100|first1 = Maximino|last1 = Aldana|first2 = Philippe|last2 = Cluzel|bibcode = 2003PNAS..100.8710A|doi-access = free}}

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

$q\_\{i\}=2p\_\{i\}(1-p\_\{i\})$. In the general case, stability of the network is governed by the largest eigenvalue $\backslash lambda\_\{Q\}$ of matrix $Q$, where $Q\_\{ij\}=q\_\{i\}A\_\{ij\}$, and $A$ is the adjacency matrix of the network.{{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|first1 = Andrew|last1 = Pomerance|first2 = Edward|last2 = Ott|first3 = Michelle|last3 = Girvan|author3-link= Michelle Girvan |first4 = Wolfgang|last4 = Losert|arxiv = 0901.4362|bibcode = 2009PNAS..106.8209P|doi-access = free}} The network is stable if , critical if $\backslash lambda\_\{Q\}=1$, unstable if $\backslash lambda\_\{Q\}>1$.

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.{{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}} One can distinguish the case where only in-degree distribution in power-law distributed,{{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|s2cid=2487442}} or only the out-degree-distribution or both.

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,{{cite book|last1=Harvey|first1=Imman|last2=Bossomaier|first2=Terry|editor1-last=Husbands|editor1-first=Phil|editor2-last=Harvey|editor2-first=Imman|contribution=Time out of joint: Attractors in asynchronous random Boolean networks|title=Proceedings of the Fourth European Conference on Artificial Life (ECAL97)|date=1997|pages=67–75|url=https://books.google.com/books?id=ccp8fzlyorAC&pg=PA67|publisher=MIT Press|isbn=9780262581578|access-date=2020-09-16|archive-date=2023-02-04|archive-url=https://web.archive.org/web/20230204160659/https://books.google.com/books?id=ccp8fzlyorAC&pg=PA67|url-status=live}} different alternatives have been introduced. A common classification{{cite book|last1=Gershenson|first1=Carlos|editor1-last=Standish|editor1-first=Russell K|editor2-last=Bedau|editor2-first=Mark A|contribution=Classification of Random Boolean Networks|title=Proceedings of the Eighth International Conference on Artificial Life|date=2002|volume=8|pages=1–8|url=https://books.google.com/books?id=si_KlRbL1XoC&pg=PA1|access-date=12 January 2016|arxiv=cs/0208001|series=Artificial Life|location=Cambridge, Massachusetts, USA|isbn=9780262692816|bibcode=2002cs........8001G|archive-date=4 February 2023|archive-url=https://web.archive.org/web/20230204160659/https://books.google.com/books?id=si_KlRbL1XoC&pg=PA1|url-status=live}} 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 ≡ Q_i (mod P_i) where t is the time step.{{cite book|last1=Gershenson|first1=Carlos|last2=Broekaert|first2=Jan|last3=Aerts|first3=Diederik|title=Advances in Artificial Life |chapter=Contextual Random Boolean Networks |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|s2cid=4309400}}
• 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){{Cite journal|last1=Imani|first1=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|s2cid=178376}}{{Cite book|pages=972–976|last1=Imani|first1=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)|s2cid=8672734}}{{Cite book|last1=Imani|first1=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|s2cid=7210088}}{{Cite book|last1=Imani|first1=M.|last2=Braga-Neto|first2=U.|title=2016 IEEE 55th Conference on Decision and Control (CDC) |chapter=Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space |date=2016-12-01|pages=4208–4213|doi=10.1109/CDC.2016.7798908|isbn=978-1-5090-1837-6|s2cid=11341805}} 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.
• Autonomous Boolean networks (ABNs) are updated in continuous time (t is a real number, not an integer), which leads to race conditions and complex dynamical behavior such as deterministic chaos.{{cite journal|last1=Zhang|first1=Rui|last2=Cavalcante|first2=Hugo L. D. de S.|last3=Gao|first3=Zheng|last4=Gauthier|first4=Daniel J.|last5=Socolar|first5=Joshua E. S.|last6=Adams|first6=Matthew M.|last7=Lathrop|first7=Daniel P.|title=Boolean chaos|journal=Physical Review E|volume=80|issue=4|year=2009|page=045202|issn=1539-3755|doi=10.1103/PhysRevE.80.045202|pmid=19905381|arxiv=0906.4124|bibcode=2009PhRvE..80d5202Z|s2cid=43022955}}{{cite journal|last1=Cavalcante|first1=Hugo L. D. de S.|last2=Gauthier|first2=Daniel J.|last3=Socolar|first3=Joshua E. S.|last4=Zhang|first4=Rui|title=On the origin of chaos in autonomous Boolean networks|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=368|issue=1911|year=2010|pages=495–513|issn=1364-503X|doi=10.1098/rsta.2009.0235|pmid=20008414|arxiv=0909.2269|bibcode=2010RSPTA.368..495C|s2cid=426841}}

• The Scalable Optimal Bayesian ClassificationHajiramezanali, 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 {{Webarchive|url=https://web.archive.org/web/20210322211733/https://dl.acm.org/doi/10.1145/3233547.3233689 |date=2021-03-22 }} 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.

• NK model

{{Reflist|30em}}

• Dubrova, E., Teslenko, M., Martinelli, A., (2005). *[http://dl.acm.org/citation.cfm?id=1129670 Kauffman Networks: Analysis and Applications], in "Proceedings of International Conference on Computer-Aided Design", pages 479-484.

• [https://web.archive.org/web/20150925155852/http://adam.plantsimlab.org/ Analysis of Dynamic Algebraic Models (ADAM) v1.1]
• [https://github.com/bioasp/bonesis bioasp/bonesis]: Synthesis of Most Permissive Boolean Networks from network architecture and dynamical properties
• [http://www.colomoto.org/ CoLoMoTo (Consortium for Logical Models and Tools)]
• [http://www.ddlab.com/ DDLab]
• [https://web.archive.org/web/20160304053726/http://homepages.stca.herts.ac.uk/~erdqmjs/NetBuilder%20home/NetBuilder/index.html NetBuilder Boolean Networks Simulator]
• [https://web.archive.org/web/20090214202740/http://www.rustyspigot.com/software/BooleanNetwork/?url=%2Fsoftware%2FBooleanNetwork Open Source Boolean Network Simulator]
• [http://www.beteredingen.nl/?e=179&w=neuroscience JavaScript Kauffman Network]
• [https://web.archive.org/web/20110724090835/http://personal.systemsbiology.net/ilya/PBN/PBN.htm Probabilistic Boolean Networks (PBN)]
• [https://sourceforge.net/projects/rbn/ RBNLab]
• [https://web.archive.org/web/20120813214739/http://web.it.kth.se/~dubrova/bns.html A SAT-based tool for computing attractors in Boolean Networks]

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