Flooding time in edge-Markovian dynamic graphs

Abstract

We introduce stochastic time-dependency in evolving graphs: starting from an arbitrary initial edge probability distribution, 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 evolving graph model is a wide generalization of time-independent dynamic random graphs [6] and will be called edge-Markovian dynamic graphs.We investigate the speed of information dissemination in such dynamic 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 dynamic graphs where flooding time is constant and, thus, it does not asymptotically depend on the initial probability distribution. ii) A tight characterization of the class of edge-Markovian dynamic graphs where flooding time does not asymptotically depend on the edge death-rate q.