### 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 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 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 at a single vertex is Ω(m) at any time (> 0). For some special graphs, namely hypercube skeletons and Johnson graphs, we give a polylogarithmic upper bound, in terms of the number of nodes, for the discrepancy.

Original language | English |
---|---|

Pages (from-to) | 739-761 |

Number of pages | 23 |

Journal | Random Structures and Algorithms |

Volume | 46 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jul 1 2015 |

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### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics

### Cite this

*Random Structures and Algorithms*,

*46*(4), 739-761. https://doi.org/10.1002/rsa.20533

**Deterministic random walks on finite graphs.** / Kijima, Shuji; Koga, Kentaro; Makino, Kazuhisa.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 46, no. 4, pp. 739-761. https://doi.org/10.1002/rsa.20533

}

TY - JOUR

T1 - Deterministic random walks on finite graphs

AU - Kijima, Shuji

AU - Koga, Kentaro

AU - Makino, Kazuhisa

PY - 2015/7/1

Y1 - 2015/7/1

N2 - 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 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 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 at a single vertex is Ω(m) at any time (> 0). For some special graphs, namely hypercube skeletons and Johnson graphs, we give a polylogarithmic upper bound, in terms of the number of nodes, for the discrepancy.

AB - 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 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 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 at a single vertex is Ω(m) at any time (> 0). For some special graphs, namely hypercube skeletons and Johnson graphs, we give a polylogarithmic upper bound, in terms of the number of nodes, for the discrepancy.

UR - http://www.scopus.com/inward/record.url?scp=84929291712&partnerID=8YFLogxK

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U2 - 10.1002/rsa.20533

DO - 10.1002/rsa.20533

M3 - Article

AN - SCOPUS:84929291712

VL - 46

SP - 739

EP - 761

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 4

ER -