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# 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. Sooner or later it will reach a previously visited state, and thus, since the dynamics are deterministic, fall 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

## References

• Aldana, M. (2003). *Boolean dynamics of networks with scale-free topology. Physica D 185:45–66
• Aldana , M., Coppersmith, S., and Kadanoff, L. P. (2003). Boolean dynamics with random couplings. In Kaplan, E., Marsden, J. E., and Sreenivasan, K. R., editors, Perspectives and Problems in Nonlinear Science. A Celebratory Volume in Honor of Lawrence Sirovich. Springer Applied Mathematical Sciences Series.
• Kauffman, S. A. (1969). Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology, 22:437-467.
• Kauffman, S. A. (1993). Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press. Technical monograph. ISBN 0-19-507951-5
• Gershenson, C. (2002). *Classification of random Boolean networks. In Standish, R. K., Bedau, M. A., and Abbass, H. A., editors, Artificial Life VIII:Proceedings of the Eight International Conference on Artificial Life, pages 1-8. MIT Press.
• Harvey, I. and Bossomaier, T. (1997). Time out of joint: Attractors in asynchronous random Boolean networks. In Husbands, P. and Harvey, I., editors, Proceedings of the Fourth European Conference on Artificial Life (ECAL97), pages 67-75. MIT Press.
• Wuensche, A. (1998). *Discrete dynamical networks and their attractor basins. In Standish, R., Henry, B., Watt, S., Marks, R., Stocker, R., Green, D., Keen, S., and Bossomaier, T., editors, Complex Systems'98, University of New South Wales, Sydney, Australia.