Slow links, fast links, and the cost of gossip

Consider the classical problem of information dissemination: one (or more) nodes in a network have some information that they want to distribute to the remainder of the network. In this paper, we study the cost of information dissemination in networks where edges have latencies, i.e., sending a message from one node to another takes some amount of time. We first generalize the idea of conductance to weighted graphs by defining $\phi_*$ to be the "critical conductance" and $\ell_*$ to be the "critical latency". % One goal of this paper is to argue that $\phi_*$ % characterizes the connectivity of a weighted graph with latencies in much the same way that conductance characterizes the connectivity of unweighted graphs. % We give near tight lower and upper bounds on the problem of information dissemination, up to polylogarithmic factors. Specifically, we show that in a graph with (weighted) diameter $D$ (with latencies as weights) and maximum degree $\Delta$, any information dissemination algorithm requires at least $\Omega(\min(D+\Delta, \ell_*/\phi_*))$ time % in the worst case. We show several variants of the lower bound (e.g., for graphs with small diameter, graphs with small max-degree, etc.) by reduction to a simple combinatorial game.