Influence and Dynamic Behavior in Random Boolean Networks

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.

💡 Research Summary

The paper develops a rigorous mathematical framework that links the short‑run sensitivity of a Boolean network to a single scalar quantity: the average influence I of its transfer functions. A Boolean network N consists of n nodes arranged on a directed random graph G. Each node i receives K_i inputs (drawn independently from a prescribed indegree distribution D) and updates its state via a K_i‑ary Boolean function f_i sampled independently from a distribution T. The authors consider two families of T: (1) full independence, where every entry of the truth table is i.i.d., and (2) average balance, where the expected fraction of +1 outputs equals ½.

Influence is defined in the classic sense of Kahn, Kalai, and Linial: for a Boolean function f, Inf_i(f) is the probability that flipping the i‑th input changes the output. The total influence of a d‑input function is the sum of Inf_i over all inputs, and the expected total influence under the d‑input distribution T_d is denoted I(d). The overall network influence is then I = K_max ∑_{d=1}^{K_max} p(d) I(d), where p(d) is the probability that a node has indegree d and K_max is the maximal indegree.

The central result (Theorem 1) shows that, for a uniformly random initial state and a single‑bit perturbation, the expected Hamming distance after t steps satisfies
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