On the Asymptotic $u_0$-Expected Flooding Time of Stationary Edge-Markovian Graphs

Abstract

Consider that $u_0$ nodes are aware of some piece of data $d_0$. This note derives the expected time required for the data $d_0$ to be disseminated through-out a network of $n$ nodes, when communication between nodes evolves according to a graphical Markov model $\overline{ \mathcal{G}}_{n,\hat{p}}$ with probability parameter $\hat{p}$. In this model, an edge between two nodes exists at discrete time $k \in \mathbb{N}^+$ with probability $\hat{p}$ if this edge existed at $k-1$, and with probability $(1-\hat{p})$ if this edge did not exist at $k-1$. Each edge is interpreted as a bidirectional communication link over which data between neighbors is shared. The initial communication graph is assumed to be an Erdos-Renyi random graph with parameters $(n,\hat{p})$, hence we consider a \emph{stationary} Markov model $\overline{\mathcal{G}}_{n,\hat{p}}$. The asymptotic "$u_0$-expected flooding time" of $\overline{\mathcal{G}}_{n,\hat{p}}$ is defined as the expected number of iterations required to transmit the data $d_0$ from $u_0$ nodes to $n$ nodes, in the limit as $n$ approaches infinity. Although most previous results on the asymptotic flooding time in graphical Markov models are either \emph{almost sure} or \emph{with high probability}, the bounds obtained here are \emph{in expectation}. However, our bounds are tighter and can be more complete than previous results.