ABoolean network consists of a discrete set ofBoolean variables each of which has aBoolean 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 anetwork. Usually, the dynamics of the system is taken as a discretetime series where the state of the entire network at timet+1 is determined by evaluating each variable's function on the state of the network at timet. This may be donesynchronously orasynchronously.^[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 convey 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]

A Boolean network is a particular kind ofsequential 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 abijection onto an integer series.

Arandom 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 byStuart A. Kauffman in 1969, asrandom models ofgenetic regulatory networks^[5] but their mathematical understanding only started in the 2000s.^[6][7]

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 anattractor (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 apoint attractor, and if the attractor consists of more than one state it is called acycle attractor. The set of states that lead to an attractor is called thebasin of the attractor. States which occur only at the beginning of trajectories (no trajectories leadto them), are calledgarden-of-Eden states^[8] and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is calledtransient 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.^[9]

Author	Year	Mean attractor length	Mean attractor number	comment
Kauffman[5]	1969	${\displaystyle ⟨A⟩\sim \sqrt{N}}$	${\displaystyle ⟨\nu ⟩\sim \sqrt{N}}$
Bastolla/ Parisi[10]	1998	faster than a power law,${\displaystyle ⟨A⟩>N^x\mathrm{\forall }x}$	faster than a power law,${\displaystyle ⟨\nu ⟩>N^x\mathrm{\forall }x}$	first numerical evidences
Bilke/ Sjunnesson[11]	2002		linear with system size,${\displaystyle ⟨\nu ⟩\sim N}$
Socolar/Kauffman[12]	2003		faster than linear,${\displaystyle ⟨\nu ⟩>N^x}$ with${\displaystyle x>1}$
Samuelsson/Troein[13]	2003		superpolynomial growth,${\displaystyle ⟨\nu ⟩>N^x\mathrm{\forall }x}$	mathematical proof
Mihaljev/Drossel[14]	2005	faster than a power law,${\displaystyle ⟨A⟩>N^x\mathrm{\forall }x}$	faster than a power law,${\displaystyle ⟨\nu ⟩>N^x\mathrm{\forall }x}$
Fink/Sheldon[15]	2023		exponential growth,${\displaystyle ⟨\nu ⟩=(2/\sqrt{e}{)}^N}$	mathematical proof

In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. Thestability of Boolean networks depends on the connections of theirnodes. A Boolean network can exhibit stable, critical orchaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes (${\displaystyle K_c}$), and can be characterized by theHamming 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 (${\displaystyle N}$) in the network.

ForN-K-model^[16] the network is stable if, critical if${\displaystyle K=K_c}$, and unstable if${\displaystyle K>K_c}$.

The state of a given node${\displaystyle n_i}$ is updated according to itstruth table, whose outputs are randomly populated.${\displaystyle p_i}$ denotes the probability of assigning an off output to a given series of input signals.

If${\displaystyle p_i=p=const.}$ for every node, the transition between the stable and chaotic range depends on${\displaystyle p}$. According toBernard Derrida andYves Pomeau^[17], the critical value of the average number of connections is${\displaystyle K_c=1/[2p(1-p)]}$.

If${\displaystyle K}$ is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by${\displaystyle ⟨K^{in}⟩}$^[18][19][20] The network is stable if, critical if${\displaystyle ⟨K^{in}⟩=K_c}$, and unstable if${\displaystyle ⟨K^{in}⟩>K_c}$.

The conditions of stability are the same in the case of networks withscale-freetopology where the in-and out-degree distribution is a power-law distribution:${\displaystyle P(K)\propto K^{-\gamma }}$, and${\displaystyle ⟨K^{in}⟩=⟨K^{out}⟩}$, 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,${\displaystyle q_i=2p_i(1-p_i)}$. In the general case, stability of the network is governed by the largesteigenvalue${\displaystyle {\lambda }_Q}$ of matrix${\displaystyle Q}$, where${\displaystyle Q_{ij}=q_i{A}_{ij}}$, and${\displaystyle A}$ is theadjacency matrix of the network.^[22] The network is stable if, critical if${\displaystyle {\lambda }_Q=1}$, unstable if${\displaystyle {\lambda }_Q>1}$.

One theme is to studydifferent underlyinggraph topologies.

• The homogeneous case simply refers to a grid which is simply the reduction to the famousIsing 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.

Classical Boolean networks (sometimes calledCRBN, 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 nodei will be updated whent ≡ Q_i (mod P_i) wheret 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.
• ThePartially-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.
• 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.^[32][33]

• TheScalable Optimal Bayesian Classification^[34] 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

• Dubrova, E., Teslenko, M., Martinelli, A., (2005). *Kauffman Networks: Analysis and Applications, in "Proceedings of International Conference on Computer-Aided Design", pages 479-484.

• Analysis of Dynamic Algebraic Models (ADAM) v1.1
• bioasp/bonesis: Synthesis of Most Permissive Boolean Networks from network architecture and dynamical properties
• CoLoMoTo (Consortium for Logical Models and Tools)
• DDLab
• NetBuilder Boolean Networks Simulator
• Open Source Boolean Network Simulator
• JavaScript Kauffman Network
• Probabilistic Boolean Networks (PBN)
• RBNLab
• A SAT-based tool for computing attractors in Boolean Networks

• Bioinformatics
• Logic
• Spin models
• Exactly solvable models
• Statistical mechanics

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