Cover time in edge-uniform stochastically-evolving graphs

Abstract

We define a general model of stochastically evolving graphs, namely the Edge-Uniform Stochastically-Evolving Graphs. In this model, each possible edge of an underlying general static graph evolves independently being either alive or dead at each discrete time step of evolution following a (Markovian) stochastic rule. The stochastic rule is identical for each possible edge and may depend on the past k ≥ 0 observations of the edge’s state. We examine two kinds of random walks for a single agent taking place in such a dynamic graph: (i) The Random Walk with a Delay (RWD), where at each step the agent chooses (uniformly at random) an incident possible edge (i.e. an incident edge in the underlying static graph) and then it waits till the edge becomes alive to traverse it. (ii) The more natural Random Walk on what is Available (RWA) where the agent only looks at alive incident edges at each time step and traverses one of them uniformly at random. Our study is on bounding the cover time, i.e. the expected time until each node is visited at least once by the agent. For RWD, we provide the first upper bounds for the cases k = 0, 1 by correlating RWD with a simple random walk on a static graph. Moreover, we present a modified electrical network theory capturing the k = 0 case and a mixing-time argument toward an upper bound for the case k = 1. For RWA, we derive the first upper bounds for the cases k = 0, 1, too, by reducing RWA to an RWD-equivalent walk with a modified delay. Finally, for the case k = 1, we prove that when the underlying graph is complete, then the cover time is O(nlog n) (i.e. it matches the cover time on the static complete graph) under only a mild condition on the edge-existence probabilities determined by the stochastic rule.