The multi-agent rotor-router on the ring: a deterministic alternative to parallel random walks

The rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, an agent is initially placed at one of the nodes of the graph. Each node maintains a cyclic ordering of its outgoing arcs, and during successive visits of the agent, propagates it along arcs chosen according to this ordering in round-robin fashion. In this work we consider the setting in which multiple, indistinguishable agents are deployed in parallel in the nodes of the graph, and move around the graph in synchronous rounds, interacting with a single rotor-router system. We propose new techniques which allow us to perform a theoretical analysis of the multi-agent rotor-router model, and to compare it to the scenario of parallel independent random walks in a graph. Our main results concern the n-node ring, and suggest a strong similarity between the performance characteristics of this deterministic model and random walks.

We show that on the ring the rotor-router with k agents admits a cover time of between Θ(n²/k²) in the best case and Θ(n²/ log k) in the worst case, depending on the initial locations of the agents, and that both these bounds are tight. The corresponding expected value of cover time for k random walks, depending on the initial locations of the walkers, is proven to belong to a similar range, namely between Θ(n²/(k²/ log² k)) and Θ(n²/ log k).

Finally, we study the limit behavior of the rotor-router system. We show that, once the rotor-router system has stabilized, all the nodes of the ring are always visited by some agent every Θ(n/k) steps, regardless of how the system was initialized. This asymptotic bound corresponds to the expected time between successive visits to a node in the case of k random walks. All our results hold up to a polynomially large number of agents (1≤k < n^1/11).