### Abstract

Random walk is a powerful tool for searching a network, especially for a very large network such as the Internet. The cover time is an important measure of a random walk on a finite graph, and has been studied well. For the purpose of searching a network, it is quite natural to think that multiple crawlers might cover a network faster than a single crawler. Alon et al. (2011) showed that a multiple random walk by k crawlers covers a network k times faster than a single random walk in certain graphs such as complete graphs, random graphs, etc., while the speeding up ratio is limited only to log k times in other graphs such as cycles and paths. This paper investigates a multiplex random walk by k tokens, in which k tokens randomly walks on a graph independently according to an individual transition probability. For the cover time of a multiplex random walk, we present new inequalities similar to celebrated Matthews' inequalities for a single random walk, that provide upper and lower bounds of the cover time by its hitting times. We also show that the bounds are tight in certain graphs, namely complete graphs, bipartite complete graphs, and random graphs.

Original language | English |
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Title of host publication | Proceedings of the 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops, IPDPSW 2012 |

Pages | 872-877 |

Number of pages | 6 |

DOIs | |

Publication status | Published - Oct 18 2012 |

Event | 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops, IPDPSW 2012 - Shanghai, China Duration: May 21 2012 → May 25 2012 |

### Publication series

Name | Proceedings of the 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops, IPDPSW 2012 |
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### Other

Other | 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops, IPDPSW 2012 |
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Country | China |

City | Shanghai |

Period | 5/21/12 → 5/25/12 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software

### Cite this

*Proceedings of the 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops, IPDPSW 2012*(pp. 872-877). [6270730] (Proceedings of the 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops, IPDPSW 2012). https://doi.org/10.1109/IPDPSW.2012.107

**An extension of Matthews' bound to multiplex random walks.** / Hosaka, Yusuke; Yamauchi, Yukiko; Kijima, Shuji; Ono, Hirotaka; Yamashita, Masafumi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops, IPDPSW 2012.*, 6270730, Proceedings of the 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops, IPDPSW 2012, pp. 872-877, 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops, IPDPSW 2012, Shanghai, China, 5/21/12. https://doi.org/10.1109/IPDPSW.2012.107

}

TY - GEN

T1 - An extension of Matthews' bound to multiplex random walks

AU - Hosaka, Yusuke

AU - Yamauchi, Yukiko

AU - Kijima, Shuji

AU - Ono, Hirotaka

AU - Yamashita, Masafumi

PY - 2012/10/18

Y1 - 2012/10/18

N2 - Random walk is a powerful tool for searching a network, especially for a very large network such as the Internet. The cover time is an important measure of a random walk on a finite graph, and has been studied well. For the purpose of searching a network, it is quite natural to think that multiple crawlers might cover a network faster than a single crawler. Alon et al. (2011) showed that a multiple random walk by k crawlers covers a network k times faster than a single random walk in certain graphs such as complete graphs, random graphs, etc., while the speeding up ratio is limited only to log k times in other graphs such as cycles and paths. This paper investigates a multiplex random walk by k tokens, in which k tokens randomly walks on a graph independently according to an individual transition probability. For the cover time of a multiplex random walk, we present new inequalities similar to celebrated Matthews' inequalities for a single random walk, that provide upper and lower bounds of the cover time by its hitting times. We also show that the bounds are tight in certain graphs, namely complete graphs, bipartite complete graphs, and random graphs.

AB - Random walk is a powerful tool for searching a network, especially for a very large network such as the Internet. The cover time is an important measure of a random walk on a finite graph, and has been studied well. For the purpose of searching a network, it is quite natural to think that multiple crawlers might cover a network faster than a single crawler. Alon et al. (2011) showed that a multiple random walk by k crawlers covers a network k times faster than a single random walk in certain graphs such as complete graphs, random graphs, etc., while the speeding up ratio is limited only to log k times in other graphs such as cycles and paths. This paper investigates a multiplex random walk by k tokens, in which k tokens randomly walks on a graph independently according to an individual transition probability. For the cover time of a multiplex random walk, we present new inequalities similar to celebrated Matthews' inequalities for a single random walk, that provide upper and lower bounds of the cover time by its hitting times. We also show that the bounds are tight in certain graphs, namely complete graphs, bipartite complete graphs, and random graphs.

UR - http://www.scopus.com/inward/record.url?scp=84867430779&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867430779&partnerID=8YFLogxK

U2 - 10.1109/IPDPSW.2012.107

DO - 10.1109/IPDPSW.2012.107

M3 - Conference contribution

AN - SCOPUS:84867430779

SN - 9780769546766

T3 - Proceedings of the 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops, IPDPSW 2012

SP - 872

EP - 877

BT - Proceedings of the 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops, IPDPSW 2012

ER -