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 ${\bf Z}{p}$ 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.

💡 Research Summary

The paper investigates a discrete-time asynchronous dynamical system that generalizes annihilating random walks in two directions: the interaction topology is a hypergraph rather than a simple graph, and the number of particles at each vertex is taken modulo a prime p ≥ 3 (i.e., an element of the finite field ℤₚ). In each step a vertex v that currently holds at least one particle is chosen together with a hyperedge e containing v. The particle at v is removed (its count is decreased by one) and a new particle is added to every other vertex of e. Whenever a vertex’s count reaches p, the p particles annihilate simultaneously, which corresponds to resetting the count to zero in ℤₚ.

Two decision problems are defined: Reachability – given two configurations w₁ and w₂, decide whether w₂ can be obtained from w₁ by a finite sequence of allowed moves; and Recurrence – decide whether w₂ is reachable from every configuration that is itself reachable from w₁. Naïvely exploring the state space would require exponential time (PSPACE for reachability, EXPSPACE for recurrence), because the state space has size p^|V|.

The authors’ main contribution is to translate these reachability questions into a system of linear equations over ℤₚ. For each ordered pair (e, v) with v ∈ e they introduce a variable X_{e,v} that records, modulo p, how many times the move (v, e) is applied. The effect of all moves on a vertex v can be expressed as

 ∑{e∋v} X{e,v} − ∑{e∋v}∑{u∈e{v}} X_{e,u} = w₂