Deterministic Random Walks on Finite Graphs

Abstract

The rotor-router model, also known as the Propp machine, is a deterministic process analogous to a random walk on a graph. Instead of distributing tokens to randomly chosen neighbors, the Propp machine deterministically serves the neighbors in a fixed order by associating to each vertex a “rotor-router” pointing to one of its neighbors. This paper investigates the discrepancy on a single vertex between the number of tokens in the rotor-router model and the expected number of tokens in a random walk, for finite graphs in general. We show that the discrepancy is bounded by O(mn) at any time for any initial configuration if the corresponding random walk is lazy and reversible, where n and m denote the numbers of nodes and edges, respectively. For a lower bound, we show examples of graphs and initial configurations for which the discrepancy on a single vertex is Ω(m) at any time (> 0). For some special graphs, namely hypercube skeletons and Johnson graphs, we give a polylogarithmic bound, in terms of the number of nodes, for the discrepancy.