Impact of local topological information on random walks on finite graphs

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) = (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(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).