The generating functional method is employed to investigate the synchronous dynamics of Boolean networks, providing an exact result for the system dynamics via a set of macroscopic order parameters. The topology of the networks studied and its constituent Boolean functions represent the system's quenched disorder and are sampled from a given distribution. The framework accommodates a variety of topologies and Boolean function distributions and can be used to study both the noisy and noiseless regimes; it enables one to calculate correlation functions at different times that are inaccessible via commonly used approximations. It is also used to determine conditions for the annealed approximation to be valid, explore phases of the system under different levels of noise and obtain results for models with strong memory effects, where existing approximations break down. Links between Boolean networks and general Boolean formulas are identified and results common to both system types are highlighted. © 2012 Copyright Taylor and Francis Group, LLC.
Bibliographical noteThis is an electronic version of an article published in Mozeika, A & Saad, D 2012, 'Phase transitions and memory effects in the dynamics of Boolean networks', Philosophical Magazine Part B, vol 92, no. 1-3, pp. 210-229. Philosophical Magazine Part B is available online at http://www.tandfonline.com/openurl?genre=article&issn=1478-6435&volume=92&issue=1-3&spage=210
- Boolean networks
- disordered systems
- generating functional analysis
- dynamical systems