Influence and Dynamic Behavior in Random Boolean Networks

Abstract

We present a rigorous mathematical framework for analyzing dynamics of a broad class of Boolean network models. We use this framework to provide the first formal proof of many of the standard critical transition results in Boolean network analysis, and offer analogous characterizations for novel classes of random Boolean networks. We precisely connect the short-run dynamic behavior of a Boolean network to the average influence of the transfer functions. We show that some of the assumptions traditionally made in the more common mean-field analysis of Boolean networks do not hold in general. For example, we offer some evidence that imbalance, or expected internal inhomogeneity, of transfer functions is a crucial feature that tends to drive quiescent behavior far more strongly than previously observed.Introduction. Complex systems can usually be represented as a network of interdependent functional units. Boolean networks were proposed by Kauffman as models of genetic regulatory networks [1, 2] and have received considerable attention across several scientific disciplines. They model a variety of complex phenomena, particularly in theoretical biology and physics [3][4][5][6][7][8].A Boolean network N with n nodes can be described by a directed graph G = (V, E) and a set of transfer functions. We use V and E to denote the sets of nodes and edges respectively, and denote the indegree of node i by K i . Each node i is assigned a K i -ary Boolean function f i : {−1, +1} K i → {−1, +1}, termed transfer function. If the state of node i at time t is x i (t), its state at time t + 1 is described by

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“…S10–12). It would be interesting to further investigate the role of small core motifs in other types of networks [25, 28]. …”

Relating the Chondrocyte Gene Network to Growth Plate Morphology: From Genes to Phenotype

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Characterizing short-term stability for Boolean networks over any distribution of transfer functions

“…Specifically, if a single node has its state flipped, does the effect of this perturbation die out (quiescence), exponentially cascade over time (chaos), or is the system right in between (criticality)? There have been numerous empirical and mathematical observations about the characteristics of critical transition points in classes of Boolean networks [4][5][6][7][8][9][11][12][13][14][15], These results require F to have specific properties: for example, each truth table entry is i.i.d. or that functions are balanced (number of +1 and −1 outcomes is the same) on average.…”

“…Mozeika and Saad [10,14,15] give a powerful generating function framework for analysis of Boolean networks, but do not characterize short-term stability. Seshadhri et al [11] introduced the notion of influence I(F) of transfer function distribution F, an easily computable quantity that determines the short-term behavior for a highly restricted class of balanced families F: on average, functions in F are equally likely to output +1 and −1. We are interested in the sensitivity of a Boolean network state x(t) = {x 1 (t) .…”