Tight Bounds for Rumor Spreading with Vertex Expansion

TL;DR

This paper proves that the PUSH-PULL rumor spreading protocol completes in nearly optimal time on graphs with a given vertex expansion, establishing a tight bound that matches the lower bound and answering an open question.

Contribution

It provides a tight bound on rumor spreading time in terms of vertex expansion, resolving an open problem and offering new insights into initial node selection for fast spreading.

Findings

O(log^2(n)/α) rounds suffice with high probability

Bound matches the known lower bound, confirming optimality

New methods for selecting initial nodes to accelerate spreading

Abstract

We establish a bound for the classic PUSH-PULL rumor spreading protocol on arbitrary graphs, in terms of the vertex expansion of the graph. We show that O(log^2(n)/\alpha) rounds suffice with high probability to spread a rumor from a single node to all n nodes, in any graph with vertex expansion at least \alpha. This bound matches the known lower bound, and settles the question on the relationship between rumor spreading and vertex expansion asked by Chierichetti, Lattanzi, and Panconesi (SODA 2010). Further, some of the arguments used in the proof may be of independent interest, as they give new insights, for example, on how to choose a small set of nodes in which to plant the rumor initially, to guarantee fast rumor spreading.