# Random Boolean Network

### From CasGroup

A Boolean network consists of a set of Boolean variables whose state is determined by other variables in the network. In a random boolean network the connections are wired randomly and the output of nodes are determined by randomly generated logic functions.

**Random Boolean Networks** are therefore networks of boolean nodes that can
be in one of two possible states, 0 or 1, and whose evolution
from one time point to another is governed by simple boolean
transition rules. The nodes change their states according to
this boolean transition rules that depend on their current states
and those of their neighbors.
RBNs are closely related to Cellular Automata (CA) and are
used to study complex systems. Both are usually
based on local boolean transition functions.
In RBNs we have nodes instead of Cells in CA, and connections
to remote neighbors instead of a local neighborhood grid as in CA.
RBNs and NK Networks have been proposed as a biological
model by Stuart Kauffman, see his book "The Origins of Order".

## NK Network

An NK-Boolean network is defined as a network of N nodes with connectivity K, where K refers to the maximum number of nodes that regulate some othernode, i.e. each of the N nodes has K inputs and one output. It can be considered as a network of N light bulbs. At every timestep, each bulb changes the state: the light bulbs can only be on or off, and each of the bulbs influences K other bulbs in the network.

## Links

Tutorials

- RBN Tutorial from Kai Willadsen

- RBN Tutorial from Carlos Gershenson

arXiv papers

- Introduction to Random Boolean Networks from Carlos Gershenson

- Boolean Dynamics with Random Couplings from Leo Kadanoff et al.

Wikipedia Pages

- Wikipedia page for Boolean Network

## Examples

The Finite State Machine (FSM) for the whole boolean network reveals the attractor structures and the basins of attraction. The attractor - if one exists - is a fixed point or a discrete limit cycle. The limit cycle is of course shorter than the total number of states, which is 2^N for N nodes (2^3=8 for 3 nodes).