Reversible Random Walks on Dynamic Graphs

TL;DR

This paper provides new bounds on mixing, hitting, and cover times for reversible random walks on dynamic graphs, extending static graph results and applying to various specific walks including the lazy Metropolis walk.

Contribution

It introduces general bounds for random walks with fixed stationary distribution on dynamic graphs, including multiple walks and coalescing times, improving upon previous results.

Findings

Bounds on mixing, hitting, and cover times in terms of static graph parameters.

First bounds for coalescing time and multiple random walks on dynamic graphs.

Tight bounds for lazy Metropolis walk on dynamic graphs: O(n^2) times.

Abstract

Recently, random walks on dynamic graphs have been studied because of their adaptivity to the time-varying structure of real-world networks. In general, there is a tremendous gap between static and dynamic graph settings for the lazy simple random walk: Although $O(n^3)$ cover time was shown for any static graphs of $n$ vertices, there is an edge-changing dynamic graph with an exponential hitting time. On the other hand, previous works indicate that the random walk on a dynamic graph with a time-homogeneous stationary distribution behaves almost identically to that on a static graph. In this paper, we strengthen this insight by obtaining general and improved bounds. Specifically, we consider a random walk according to a sequence $(P_t)_{t\geq 1}$ of irreducible and reversible transition matrices such that all $P_t$ have the same stationary distribution. We bound the mixing, hitting, and cover times in terms of the hitting and relaxation times of the random walk according to the worst fixed $P_t$. Moreover, we obtain the first bounds of the hitting and cover times of multiple random walks and the coalescing time on dynamic graphs. These bounds can be seen as an extension of the well-known bounds of random walks on static graphs. Our results generalize the previous upper bounds for specific random walks on dynamic graphs, e.g., lazy simple random walks and $d_{\max}$-lazy walks, and give improved and tight upper bounds in various cases. As an interesting consequence of our generalization, we obtain tight bounds for the lazy Metropolis walk [Nonaka, Ono, Sadakane, and Yamashita, TCS10] on any dynamic graph: $O(n^2)$ mixing time, $O(n^2)$ hitting time, and $O(n^2\log n)$ cover time. Additionally, our coalescing time bound implies the consensus time bound of the pull voting on a dynamic graph.