On the dynamics of Social Balance on general networks (with an application to XOR-SAT)

TL;DR

This paper analyzes the convergence dynamics of social balance models on various networks, revealing connections to hypergraph walks and providing bounds on convergence time, with applications to XOR-SAT problems.

Contribution

It introduces a novel connection between triad dynamics and hypergraph annihilating walks, characterizes recurrent states, and bounds convergence times on specific graph classes.

Findings

Complete characterization of recurrent states in certain graphs.

Linear convergence time on triadic cycle graphs.

Cubic and Cheeger constant-based bounds on convergence time.

Abstract

We study nondeterministic and probabilistic versions of a discrete dynamical system (due to T. Antal, P. L. Krapivsky, and S. Redner) inspired by Heider's social balance theory. We investigate the convergence time of this dynamics on several classes of graphs. Our contributions include: 1. We point out the connection between the triad dynamics and a generalization of annihilating walks to hypergraphs. In particular, this connection allows us to completely characterize the recurrent states in graphs where each edge belongs to at most two triangles. 2. We also solve the case of hypergraphs that do not contain edges consisting of one or two vertices. 3. We show that on the so-called "triadic cycle" graph, the convergence time is linear. 4. We obtain a cubic upper bound on the convergence time on 2-regular triadic simplexes G. This bound can be further improved to a quantity that depends on the Cheeger constant of G. In particular this provides some rigorous counterparts to previous experimental observations. We also point out an application to the analysis of the random walk algorithm on certain instances of the 3-XOR-SAT problem.