# Boolean network

State space of a Boolean Network with N=4 nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.

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.

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.[1][2] The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000's.[3]

## 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. Boolean networks are related to cellular automata. Usually, cellular automata are defined with an homogeneous topology, i.e. a single line of nodes, a square or hexagonal grid of nodes or an even higher-dimensional structure, but each variable (node in the grid) may take on more than two values (and hence not be boolean).

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.[4]

Random Boolean networks (RBNs) are known as NK networks or Kauffman networks (Dubrova 2005). An 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. 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 single random sample from the ensemble of possible networks of size N and either connectivity =k or with connectivities with some deviation around k. The state of a network at any point in time is given by the current states of all N genes. Thus the size of 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 computed by applying the boolean function associated with the node to the state of its input nodes at time t. The sequence of states of the whole network starting from some initial state is called the trajectory of that state.

The behavior of specific RBNs and generalized classes of them has been the subject of much of Kauffman's (and others) research.

### Attractors

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 cycle. In the literature in this field, each cycle is also 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. (Gershenson 2004)

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.[5]

Author Year Mean attractor length Mean attractor number comment
Kauffmann [4] 1969 $\langle A\rangle\sim \sqrt{N}$ $\langle\nu\rangle\sim \sqrt{N}$
Bastolla/ Parisi[6] 1998 faster than a power law, $\langle A\rangle > N^x \forall x$ faster than a power law, $\langle\nu\rangle > N^x \forall x$ first numerical evidences
Bilke/ Sjunnesson[7] 2002 linear with system size, $\langle\nu\rangle \sim N$
Socolar/Kauffman[8] 2003 faster than linear, $\langle\nu\rangle > N^x$ with $x > 1$
Samuelsson/Troein[9] 2003 superpolynomial growth, $\langle\nu\rangle > N^x \forall x$ mathematical proof
Mihaljev/Drossel[10] 2005 faster than a power law, $\langle A\rangle > N^x \forall x$ faster than a power law, $\langle\nu\rangle > N^x \forall x$

## Updating Schemes

Classical RBNs (CRBNs) use a synchronous updating scheme and a criticism of CRBNs as models of genetic regulatory networks is that genes do not change their states all at the same moment. Harvey and Bossomaier introduced this criticism and defined asynchronous RBNs (ARBNs) where a random node is selected at each time step and updated (Harvey and Bossomaier, 1997). Since a random node is updated ARBNs are non-deterministic and do not have the cycle attractors found in CRBNs (Gershenson, 2004).

Deterministic asynchronous RBNs (DARBNs) were introduced by Gershenson as a way to have RBNs that do not have asynchronous updating but still are deterministic. In DARBNs each node has two randomly generated parameters Pi and Qi (Pi, Qi ∈ ℕ, Pi > Qi). These parameters remain fixed. A node i will be updated when t ≡ Qi (mod Pi) where t is the time step. If more than one node is to be updated at a time step the nodes are updated in a pre-defined order, e.g. from lowest to highest i. Another way to do this is to synchronously update all nodes that fulfill the updating condition. The latter scheme is called deterministic semi-synchronous or deterministic generalized asynchronous RBNs (DGARBNs) (Gershenson, 2004).

RBNs where one or more nodes are selected for updating at each time step and the selected nodes are then synchronously updated are called generalized asynchronous RBNs (GARBNs). GARBNs are semi-synchronous, but non-deterministic (Gershenson, 2002).