Ergodic Effects in Token Circulation

TL;DR

This paper analyzes a token circulation process on networks, showing that in the recurrent state, token distribution is tightly concentrated around the average, with implications for load balancing and patrolling strategies.

Contribution

The paper establishes ergodic properties of token circulation dynamics, providing tight concentration bounds and linking behavior to Eulerian circuit traversals, a novel approach in this context.

Findings

Time-averaged token distribution concentrates around the mean

Bounds on the maximum time between token appearances on edges

Distributed solution for patrolling with near-optimal idleness

Abstract

We consider a dynamical process in a network which distributes all particles (tokens) located at a node among its neighbors, in a round-robin manner. We show that in the recurrent state of this dynamics (i.e., disregarding a polynomially long initialization phase of the system), the number of particles located on a given edge, averaged over an interval of time, is tightly concentrated around the average particle density in the system. Formally, for a system of $k$ particles in a graph of $m$ edges, during any interval of length $T$, this time-averaged value is $k/m \pm \widetilde{O}(1/T)$, whenever $\gcd(m,k) = \widetilde{O}(1)$ (and so, e.g., whenever $m$ is a prime number). To achieve these bounds, we link the behavior of the studied dynamics to ergodic properties of traversals based on Eulerian circuits on a symmetric directed graph. These results are proved through sum set methods and are likely to be of independent interest. As a corollary, we also obtain bounds on the \emph{idleness} of the studied dynamics, i.e., on the longest possible time between two consecutive appearances of a token on an edge, taken over all edges. Designing trajectories for $k$ tokens in a way which minimizes idleness is fundamental to the study of the patrolling problem in networks. Our results immediately imply a bound of $\widetilde{O}(m/k)$ on the idleness of the studied process, showing that it is a distributed $\widetilde{O}(1)$-competitive solution to the patrolling task, for all of the covered cases. Our work also provides some further insights that may be interesting in load-balancing applications.