Asymptotics for Pull on the Complete Graph

Abstract

We study the randomized rumor spreading algorithm \emph{pull} on complete graphs with $n$ vertices. Starting with one informed vertex and proceeding in rounds, each vertex yet uninformed connects to a neighbor chosen uniformly at random and receives the information, if the vertex it connected to is informed. The goal is to study the number of rounds needed to spread the information to everybody, also known as the \emph{runtime}. In our main result we provide a description, as $n$ gets large, for the distribution of the runtime that involves a martingale limit. %We provide a description of the distribution of the runtime in terms of a limit of a sequence of martingales. This allows us to establish that in general there is no limiting distribution and that convergence occurs only on suitably chosen subsequences $(n_i)_{i \in \mathbb{N}}$ of $\mathbb{N}$, namely when the fractional part of $(\log_2 n_i +\log_2\ln n_i)_{i \in \mathbb{N}}$ converges.