Dynamics of Boolean Networks: An Exact Solution

Abstract

The dynamics of Boolean networks (BN) with quenched disorder and thermal noise is studied via the generating functional method. A general formulation, suitable for BN with any distribution of Boolean functions, is developed. It provides exact solutions and insight into the evolution of order parameters and properties of the stationary states, which are inaccessible via existing methodology.We identify cases where the commonly used annealed approximation is valid and others where it breaks down. Broader links between BN and general Boolean formulas are highlighted.

Transfer matrix analysis of one-dimensional majority cellular automata with thermal noise

“…This result contrasts with the result of [20] for non-uniform (random) topologies, where one bit of information can be remembered in the presence of noise. However, numerical experiments show that the 1d-MAJ3-CA has a quite long memory about its initial state.…”

“…And what are the effects of the self-interaction (in (1) the state s i (t + 1) is directly dependent on s i (t)) on this property of the system? In systems with non-uniform topologies, the presence of self-interaction yields strong memory effects [20]. A direct approach to this problem is to use the dynamical transfer matrix method [21].…”

“…Then intuitively, both models are in the ordered state. This intuition turns out to be correct for models with non-uniform topologies [19,20], but whether this is true in a more general case is not clear.…”

Opportunity, challenge and countermeasure for studying relationship among multiple ecosystem services

“…BN 是由 Kauffman [43] 于 1969 年首次提出的, 是一种简单的机理-调控网络 模型. 至今, BN 及其相关理论仍不断发展 [44,45] , 并且 在复杂的生物学调控网络的模拟和分析中扮演着非 常重要的角色 [41,42] . Theoretical understanding behind the relationships among multiple ecosystem services has been the hotspot of sustainable development.…”

Phase transitions and memory effects in the dynamics of Boolean networks

“…The aim of the current paper is to develop further a framework for exact analysis of N-k models based on the generating functional analysis (GFA) framework [15,16], which has been employed successfully in the study of various Ising spin models [17]. The newly developed framework is then employed to determine the conditions under which the annealed approximation is valid, to investigate the possible phases of BN depending on the noise level and to demonstrate its efficacy for analyzing systems with memory, where the annealed approximation is known to break down [14].…”