Boolean network

A Boolean network consists of a set of Boolean variables whose state is determined by other variables in the network. They are a particular case of discrete dynamical networks, where time and states are discrete, i.e. they have a bijection onto an integer series. Boolean and elementary cellular automata are particular cases of Boolean networks, where the state of a variable is determined by its spatial neighbors.

Classical model
The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks (Kauffman 1969, 1993).

A Random Boolean network (RBN) is a system of N binary-state nodes (representing genes) with K inputs to each node representing regulatory mechanisms. The two states (on/off) represent respectively, the status of a gene being active or inactive. The variable K is typically held constant, but it can also be varied across all genes, making it a set of integers instead of a single integer. In the simplest case each gene is assigned, at random, K regulatory inputs from among the N genes, and one of the possible Boolean functions of K inputs. This gives a random sample of the possible ensembles of NK networks. The state of a network at any point in time is given by the current states of all N genes. Thus the state space of any such network is 2N.

Simulation of RBNs is done in discrete time steps. The state of a node at time t+1 is a function of the state of its input nodes and the boolean function associated with it. The behavior of specific RBNs and generalized classes of them has been the subject of much of Kauffman's (and others) research.

Such models are also known as NK models, or Kauffman networks.

Attractors
A Boolean network has 2N possible states. Since the dynamics are deterministic, sooner or later it will reach a previously visited state, thus falling into an attractor.

Dynamics
Order, chaos, and the edge

Topologies

 * homogeneous
 * normal
 * scale-free (Aldana, 2003)

Updating Schemes

 * synchronous
 * asynchronous (Harvey and Bossomaier, 1997)
 * semi-synchronous (Gershenson, 2002)
 * deterministic asynchronous
 * deterministic semi-synchronous

Applications

 * genetic regulatory networks