### Abstract

It is just amazing that both of the mean hitting time and the cover time of a random walk on a finite graph, in which the vertex visited next is selected from the adjacent vertices at random with the same probability, are bounded by O(n^{3}) for any undirected graph with order n, despite of the lack of global topological information. Thus a natural guess is that a better transition matrix is designable if more topological information is available. For any undirected connected graph G = (V, E), let P^{(β)} = (p_{uv}^{(β)})_{u,v∈V} be a transition matrix defined by (Equation Presented) 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) - log (max {deg(u), deg(v)}) for it u ∈ V, v ∈ N(u). In this paper, we show that for any undirected graph with order n, the cover time and the mean hitting time with respect to P^{(1)} are bounded by O(n^{2}log n) and O(n^{2}), respectively. We further show that P^{(1)} is best possible with respect to the mean hitting time, in the sense that the mean hitting time of a path graph of order n, with respect to any transition matrix, is Ω(n^{2}).

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

Pages (from-to) | 1054-1067 |

Number of pages | 14 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 2719 |

Publication status | Published - Dec 1 2003 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*,

*2719*, 1054-1067.

**Impact of local topological information on random walks on finite graphs.** / Ikeda, Satoshi; Kubo, Izumi; Okumoto, Norihiro; Yamashita, Masafumi.

Research output: Contribution to journal › Article

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, vol. 2719, pp. 1054-1067.

}

TY - JOUR

T1 - Impact of local topological information on random walks on finite graphs

AU - Ikeda, Satoshi

AU - Kubo, Izumi

AU - Okumoto, Norihiro

AU - Yamashita, Masafumi

PY - 2003/12/1

Y1 - 2003/12/1

N2 - It is just amazing that both of the mean hitting time and the cover time of a random walk on a finite graph, in which the vertex visited next is selected from the adjacent vertices at random with the same probability, are bounded by O(n3) for any undirected graph with order n, despite of the lack of global topological information. Thus a natural guess is that a better transition matrix is designable if more topological information is available. For any undirected connected graph G = (V, E), let P(β) = (puv(β))u,v∈V be a transition matrix defined by (Equation Presented) 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) - log (max {deg(u), deg(v)}) for it u ∈ V, v ∈ N(u). In this paper, we show that for any undirected graph with order n, the cover time and the mean hitting time with respect to P(1) are bounded by O(n2log n) and O(n2), respectively. We further show that P(1) is best possible with respect to the mean hitting time, in the sense that the mean hitting time of a path graph of order n, with respect to any transition matrix, is Ω(n2).

AB - It is just amazing that both of the mean hitting time and the cover time of a random walk on a finite graph, in which the vertex visited next is selected from the adjacent vertices at random with the same probability, are bounded by O(n3) for any undirected graph with order n, despite of the lack of global topological information. Thus a natural guess is that a better transition matrix is designable if more topological information is available. For any undirected connected graph G = (V, E), let P(β) = (puv(β))u,v∈V be a transition matrix defined by (Equation Presented) 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) - log (max {deg(u), deg(v)}) for it u ∈ V, v ∈ N(u). In this paper, we show that for any undirected graph with order n, the cover time and the mean hitting time with respect to P(1) are bounded by O(n2log n) and O(n2), respectively. We further show that P(1) is best possible with respect to the mean hitting time, in the sense that the mean hitting time of a path graph of order n, with respect to any transition matrix, is Ω(n2).

UR - http://www.scopus.com/inward/record.url?scp=35248892347&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35248892347&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:35248892347

VL - 2719

SP - 1054

EP - 1067

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -