Connectivity in time-graphs

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-carry-forward 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. In this paper, a similar metric is investigated for time-graphs. To this end, a time-graph is represented by a 3-mode reachability tensor which indicates whether a node is reachable from another node at t steps. To evaluate connectivity, we consider the expected hitting time of a random walk, and the time it takes for epidemic routing to infect all vertices. Observations from an extensive set of simulations show that the correlation between the second singular value of the matrix obtained by unfolding the reachability tensor and these indicators is very significant.