Random walks in time-graphs

Abstract

Dynamic networks are characterized by topologies that vary with time and are represented by time-graphs. The notion of connectivity in time-graphs is fundamentally different than that in static graphs. End-to-end connectivity is achieved opportunistically by store-forward-carry paradigm if the network is so sparse that source-destination pairs are usually not connected by complete paths. In static graphs, it is well known that the network connectivity is tied to the spectral gap of the underlying adjacency matrix of the topology: if the gap is large, the network is well connected and a random walk on this graph has a small hitting time. In this paper, we investigate a similar metric for time-graphs, which indicates how quickly opportunistic methods deliver packets to destinations, speed of convergence in estimating an entity and quickness in the online optimization of protocol parameters, etc. To this end, a time-graph is represented by a 3-mode reachability tensor which yields whether a vertex is reachable from another node within t steps. Our observations from an extensive set of simulations show that the correlation between the expected hitting time of a random walk in the time-graph (following a non-homogenous Markov Chain) and the second singular value of the matrix obtained by unfolding the reachability tensor is significantly large, above 90%.