Impact of Local Topological Information on Random Walks on Finite Graphs

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 ^(β)_uv )_u,v∈V be a transition matrix defined by

$$ p_{uv}^{\left( \beta \right)} = \frac{{\exp \left[ { - \beta U\left( {u,v} \right)} \right]}} {{\sum _{w \in N\left( u \right)} \exp \left[ { - \beta U\left( {u,w} \right)} \right]}}foru \in V,v \in N\left( u \right), $$

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 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).