Boolean Gossip Networks

TL;DR

This paper introduces a Boolean gossip model for probabilistic Boolean networks, analyzing convergence, communication classes, and the influence of network structure on dynamics, with implications for understanding complex network behaviors.

Contribution

It provides a comprehensive analysis of Boolean gossip networks, including convergence properties, communication class enumeration, and conditions for absorbing dynamics, linking network topology to Boolean dynamics.

Findings

Node states converge to a binary distribution in large networks.

Number of communication classes depends on network topology.

Most Boolean interaction rules lead to topology-independent absorption, given connectivity.

Abstract

This paper proposes and investigates a Boolean gossip model as a simplified but non-trivial probabilistic Boolean network. With positive node interactions, in view of standard theories from Markov chains, we prove that the node states asymptotically converge to an agreement at a binary random variable, whose distribution is characterized for large-scale networks by mean-field approximation. Using combinatorial analysis, we also successfully count the number of communication classes of the positive Boolean network explicitly in terms of the topology of the underlying interaction graph, where remarkably minor variation in local structures can drastically change the number of network communication classes. With general Boolean interaction rules, emergence of absorbing network Boolean dynamics is shown to be determined by the network structure with necessary and sufficient conditions established regarding when the Boolean gossip process defines absorbing Markov chains. Particularly, it is shown that for the majority of the Boolean interaction rules, except for nine out of the total $2^{16}-1$ possible nonempty sets of binary Boolean functions, whether the induced chain is absorbing has nothing to do with the topology of the underlying interaction graph, as long as connectivity is assumed. These results illustrate possibilities of {relating dynamical} properties of Boolean networks to graphical properties of the underlying interactions.