Tight bounds for rumor spreading in graphs of a given conductance

We study the connection between the rate at which a rumor spreads throughout a graph and the \emph{conductance} of the graph---a standard measure of a graph's expansion properties. We show that for any $n$-node graph with conductance $\phi$, the classical PUSH-PULL algorithm distributes a rumor to all nodes of the graph in $O(\phi^{-1} \log n)$ rounds with high probability (w.h.p.). This bound improves a recent result of Chierichetti, Lattanzi, and Panconesi~\cite{Chierichetti2010b}, and it is tight in the sense that there exist graphs where $\Omega(\phi^{-1}\log n)$ rounds of the PUSH-PULL algorithm are required to distribute a rumor w.h.p. We also explore the PUSH and the PULL algorithms, and derive conditions that are both necessary and sufficient for the above upper bound to hold for those algorithms as well. An interesting finding is that every graph contains a node such that the PULL algorithm takes $O(\phi^{-1}\log n)$ rounds w.h.p.\ to distribute a rumor started at that node. In contrast, there are graphs where the PUSH algorithm requires significantly more rounds for any start node