Boolean Networks

Random Boolean networks were originally introduced as simple conceptual models for genetic regulatory systems. The contrast between their simple design and their complex emergent behaviour has motivated researchers in diverse fields including the biological, neurological, computational and evolutionary sciences to use these or related models as test beds for ideas about self-organization. Complimentary to their application as models, this dichotomy between their straightforward formulation and their complex dynamics make them worthy of study in their own right.

The dynamics of Boolean networks can be represented by a directed state space network by linking each dynamical state, represented as a node, to its temporal successor. Like all finite discrete deterministic systems the dynamics must eventually settle into a periodic attractor cycle. Berdahl et al. (2008, 2009) clarify how different weighting schemes and sampling methods affect the estimates for attractor length distributions in random Boolean networks. We find that the unbiased distribution of attractor lengths decays as a power-law for all K > 1, thus power-law behaviour in this distribution is not an indicator of criticality. However, we observe a power-law in the distribution of the sizes of “avalanches” in critical random Boolean networks only.