Asymptotics for Pull on the Complete Graph

TL;DR

This paper analyzes the asymptotic distribution of the number of rounds needed for the pull rumor spreading algorithm on complete graphs, revealing complex convergence behavior involving martingale limits as the graph size grows.

Contribution

It provides a detailed asymptotic description of the runtime distribution, including conditions for convergence and the role of martingale limits, which was previously unexplored.

Findings

Distribution described via martingale limits

No universal limiting distribution, convergence depends on subsequences

Convergence occurs when fractional parts of specific logarithmic sequences converge

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.