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

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 ${\mathbf{Z}}_p$ 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 ${\mathbf{Z}}_p$) 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.