Flooding Time of Edge-Markovian Evolving Graphs

We introduce stochastic time-dependency in evolving graphs: starting from an initial graph, at every time step, every edge changes its state (existing or not) according to a two-state Markovian process with probabilities $p$ (edge birth-rate) and $q$ (edge death-rate). If an edge exists at time $t$, then, at time $t+1$, it dies with probability $q$. If instead the edge does not exist at time $t$, then it will come into existence at time $t+1$ with probability $p$. Such an evolving graph model is a wide generalization of time-independent dynamic random graphs [A. E. F. Clementi, A. Monti, F. Pasquale, and R. Silvestri, J. Comput. System Sci., 75 (2009), pp. 213–220] and will be called edge-Markovian evolving graphs. We investigate the speed of information spreading in such evolving graphs. We provide nearly tight bounds (which in fact turn out to be tight for a wide range of probabilities $p$ and $q$) on the completion time of the flooding mechanism aiming to broadcast a piece of information from a source node to all nodes. In particular, we provide i) a tight characterization of the class of edge-Markovian evolving graphs where flooding time is constant and, thus, it does not asymptotically depend on the initial graph; ii) a tight characterization of the class of edge-Markovian evolving graphs where flooding time does not asymptotically depend on the edge death-rate $q$. An interesting consequence of our results is that information spreading can be fast even if the graph, at every time step, is very sparse and disconnected. Furthermore, our bounds imply that the flooding time can be exponentially shorter than the mixing time of the edge-Markovian graph