Deterministic random walks on finite graphs

Shuji Kijima, Kentaro Koga, Kazuhisa Makino

Research output: Chapter in Book/Report/Conference proceedingConference contribution

10 Citations (Scopus)

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.

Original languageEnglish
Title of host publication9th Meeting on Analytic Algorithmics and Combinatorics 2012, ANALCO 2012
PublisherSociety for Industrial and Applied Mathematics Publications
Pages16-25
Number of pages10
ISBN (Electronic)9781618396235
DOIs
Publication statusPublished - Jan 1 2012
Event9th Meeting on Analytic Algorithmics and Combinatorics, ANALCO 2012 - Kyoto, Japan
Duration: Jan 16 2012 → …

Publication series

Name9th Meeting on Analytic Algorithmics and Combinatorics 2012, ANALCO 2012

Other

Other9th Meeting on Analytic Algorithmics and Combinatorics, ANALCO 2012
CountryJapan
CityKyoto
Period1/16/12 → …

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Materials Chemistry
  • Discrete Mathematics and Combinatorics

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  • Cite this

    Kijima, S., Koga, K., & Makino, K. (2012). Deterministic random walks on finite graphs. In 9th Meeting on Analytic Algorithmics and Combinatorics 2012, ANALCO 2012 (pp. 16-25). (9th Meeting on Analytic Algorithmics and Combinatorics 2012, ANALCO 2012). Society for Industrial and Applied Mathematics Publications. https://doi.org/10.1137/1.9781611973020.3