Boolean network

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. 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. 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. The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s. 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. 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 but their mathematical understanding only started in the 2000s. 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. 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. 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 {displaystyle 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 {displaystyle N} ) in the network. For N-K-model the network is stable if K < K c {displaystyle K<K_{c}} , critical if K = K c {displaystyle K=K_{c}} , and unstable if K > K c {displaystyle K>K_{c}} .