### Abstract

For any undirected connected graph G = (V, E) with order n, let P ^{(β)} = (p_{uv}^{(β)})_{u,v∈V} be a transition matrix defined by p_{uv}^{(β)} = exp[-βU(u,v)]/∑_{w∈N(u)}exp[-βU(u,w)] for u ∈ V, v ∈ N(u), where β is a real number, N(u) is the set of vertices adjacent to a vertex u, deg(u) = |N(u)|, and U(·, ·) is a potential function defined as U(u, v) = U(v) = log deg(v) for v ∈ N(u), u ∈ V. In this paper, we show that for any undirected graph with order n, the cover time and the mean hitting time for P^{(1/2)} are bounded by O(n^{2} log n) and O(n^{2}), respectively. Since the mean hitting time of a path graph of order n, for any transition matrix, is ω(n^{2}), P^{(1/2)} is best possible with respect to the mean hitting time.

Original language | English |
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Title of host publication | Proceedings of the International Conference on VLSI, VLSI 03 |

Editors | H.R. Arbania, L.T. Yang |

Pages | 203-207 |

Number of pages | 5 |

Publication status | Published - Dec 1 2003 |

Event | Proceedings of the International Conference on VLSI, VLSI'03 - Las Vegas, NV, United States Duration: Jun 23 2003 → Jun 26 2003 |

### Publication series

Name | Proceedings of the International Conference on VLSI |
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### Other

Other | Proceedings of the International Conference on VLSI, VLSI'03 |
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Country | United States |

City | Las Vegas, NV |

Period | 6/23/03 → 6/26/03 |

### All Science Journal Classification (ASJC) codes

- Hardware and Architecture
- Electrical and Electronic Engineering

### Cite this

*Proceedings of the International Conference on VLSI, VLSI 03*(pp. 203-207). (Proceedings of the International Conference on VLSI).

**Reducing the Hitting and the Cover Times of Random Walks on Finite Graphs by Local Topological Information.** / Ikeda, Satoshi; Kubo, Izumi; Yamashita, Masafumi.

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

*Proceedings of the International Conference on VLSI, VLSI 03.*Proceedings of the International Conference on VLSI, pp. 203-207, Proceedings of the International Conference on VLSI, VLSI'03, Las Vegas, NV, United States, 6/23/03.

}

TY - GEN

T1 - Reducing the Hitting and the Cover Times of Random Walks on Finite Graphs by Local Topological Information

AU - Ikeda, Satoshi

AU - Kubo, Izumi

AU - Yamashita, Masafumi

PY - 2003/12/1

Y1 - 2003/12/1

N2 - For any undirected connected graph G = (V, E) with order n, let P (β) = (puv(β))u,v∈V be a transition matrix defined by puv(β) = exp[-βU(u,v)]/∑w∈N(u)exp[-βU(u,w)] for u ∈ V, v ∈ N(u), where β is a real number, N(u) is the set of vertices adjacent to a vertex u, deg(u) = |N(u)|, and U(·, ·) is a potential function defined as U(u, v) = U(v) = log deg(v) for v ∈ N(u), u ∈ V. In this paper, we show that for any undirected graph with order n, the cover time and the mean hitting time for P(1/2) are bounded by O(n2 log n) and O(n2), respectively. Since the mean hitting time of a path graph of order n, for any transition matrix, is ω(n2), P(1/2) is best possible with respect to the mean hitting time.

AB - For any undirected connected graph G = (V, E) with order n, let P (β) = (puv(β))u,v∈V be a transition matrix defined by puv(β) = exp[-βU(u,v)]/∑w∈N(u)exp[-βU(u,w)] for u ∈ V, v ∈ N(u), where β is a real number, N(u) is the set of vertices adjacent to a vertex u, deg(u) = |N(u)|, and U(·, ·) is a potential function defined as U(u, v) = U(v) = log deg(v) for v ∈ N(u), u ∈ V. In this paper, we show that for any undirected graph with order n, the cover time and the mean hitting time for P(1/2) are bounded by O(n2 log n) and O(n2), respectively. Since the mean hitting time of a path graph of order n, for any transition matrix, is ω(n2), P(1/2) is best possible with respect to the mean hitting time.

UR - http://www.scopus.com/inward/record.url?scp=1642378864&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1642378864&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:1642378864

SN - 1932415106

T3 - Proceedings of the International Conference on VLSI

SP - 203

EP - 207

BT - Proceedings of the International Conference on VLSI, VLSI 03

A2 - Arbania, H.R.

A2 - Yang, L.T.

ER -