Deterministic Random Walks on Finite Graphs

The rotor-router model, also known as the Propp machine, is a deterministic process analogous to a random walk on a graph [1]. Instead of distributing tokens to randomly chosen neighbors, the rotor-router model deterministically serves the neighbors in a fixed order by associating to each vertex a “rotor-router” pointing to one of its neighbors. The rotor-router model sometimes appears under the name of deterministic random walk, meaning a “derandomized, hence deterministic, version of a random walk.” In this talk, we investigate the discrepancy at 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 multigraphs. In case that the random walk is ergodic, reversible and lazy, we show that the discrepancy is O(nm), where n denotes the number of vertices, and m denotes the total number of multiple edges [3]. For irreducible transition matrix P in general, we show that the discrepancy is O(α∗n2m/(1− λ∗)), where λ∗ denotes the second largest eigenvalue of P , and α∗ is a parameter defined by P [2]. We also propose a new deterministic process, which we call functional-router model, in a similar fashion to the rotor-router model [4]. While the rotor-router is an analogy with random walks consisting of only rational transition probabilities using parallel edges, the functional-router can imitate random walks containing irrational transition probabilities. In fact, the functional-router can also emulate the rotor-router, thus the functional-router model is a generalization of the rotor-router model.