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We study a dynamical system motivated by our earlier work on the statisticalphysics of social balance on graphs that can be viewed as a generalization ofannihilating walks along two directions: first, the interaction topology is ahypergraph; second, the ``number of particles`` at a vertex of the hypergraphis an element of a finite field ${\bf Z}_{p}$ of integers modulo $p$, $p\geq3$. Equivalently, particles move on a hypergraph, with a moving particle at avertex being replaced by one indistinguishable copy at each neighbor in a givenhyperedge; particles at a vertex collectively annihilate when their numberreaches $p$. The system we study can also be regarded as a naturalgeneralization of certain lights-out games to finite fields and hypergraphtopologies. Our result shows that under a liberal sufficient condition on thenature of the interaction hypergraph there exists a polynomial time algorithm(based on linear algebra over ${\bf Z}_{p}$) for deciding reachability andrecurrence of this dynamical system. Interestingly, we provide a counterexamplethat shows that this connection does not extend to all graphs.

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