On the Dynamics of Social Balance on General Networks (with an application to XOR-SAT)

Abstract

We study nondeterministic and probabilistic versions of a discrete dynamical system (due to T. Antal, P. L. Krapivsky, and S. Redner [3]) 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 experimental observations in [25].

We also point out an application to the analysis of the random walk algorithm on certain instances of the 3-XOR-SAT problem.