Reachability and recurrence in a modular generalization of annihilating random walks (and lights-out games) on hypergraphs

TL;DR

This paper investigates a generalized dynamical system on hypergraphs involving particles in finite fields, providing a polynomial-time algorithm for reachability and recurrence, with notable exceptions demonstrated through counterexamples.

Contribution

It introduces a new hypergraph-based model of annihilating walks over finite fields and develops an efficient linear algebra method for key dynamical questions.

Findings

Polynomial-time algorithm for reachability and recurrence under certain conditions

Counterexample showing limitations of the algorithm's applicability

Generalization of lights-out games to hypergraphs and finite fields

Abstract

We study a dynamical system motivated by our earlier work on the statistical physics of social balance on graphs that can be viewed as a generalization of annihilating walks along two directions: first, the interaction topology is a hypergraph; second, the ``number of particles`` at a vertex of the hypergraph is an element of a finite field ${\bf Z}_{p}$ of integers modulo $p$, $p\geq 3$. Equivalently, particles move on a hypergraph, with a moving particle at a vertex being replaced by one indistinguishable copy at each neighbor in a given hyperedge; particles at a vertex collectively annihilate when their number reaches $p$. The system we study can also be regarded as a natural generalization of certain lights-out games to finite fields and hypergraph topologies. Our result shows that under a liberal sufficient condition on the nature of the interaction hypergraph there exists a polynomial time algorithm (based on linear algebra over ${\bf Z}_{p}$) for deciding reachability and recurrence of this dynamical system. Interestingly, we provide a counterexample that shows that this connection does not extend to all graphs.