Towards Constant Time Multi-Call Rumor Spreading on Small-Set Expanders

TL;DR

This paper introduces a multi-call rumor spreading protocol on small-set expanders, demonstrating it can significantly reduce spreading time by increasing communication, especially on graphs with strong local connectivity and small diameter.

Contribution

It generalizes the classic PUSH&PULL rumor spreading to multiple contacts per round and analyzes its efficiency on small-set expanders, providing both upper and lower bounds.

Findings

Rumor spreading completes in Θ(log_k n) rounds on complete graphs.

On small-set expanders with expansion φ > 1, the protocol takes O(log_φ n · log_k n) rounds.

Lower bound of Ω(log_φ n + log_k n) rounds established.

Abstract

We study a multi-call variant of the classic PUSH&PULL rumor spreading process where nodes can contact $k$ of their neighbors instead of a single one during both PUSH and PULL operations. We show that rumor spreading can be made faster at the cost of an increased amount of communication between the nodes. As a motivating example, consider the process on a complete graph of $n$ nodes: while the standard PUSH&PULL protocol takes $\Theta(\log n)$ rounds, we prove that our $k$-PUSH&PULL variant completes in $\Theta(\log_{k} n)$ rounds, with high probability. We generalize this result in an expansion-sensitive way, as has been done for the classic PUSH&PULL protocol for different notions of expansion, e.g., conductance and vertex expansion. We consider small-set vertex expanders, graphs in which every sufficiently small subset of nodes has a large neighborhood, ensuring strong local connectivity. In particular, when the expansion parameter satisfies $\phi > 1$, these graphs have a diameter of $o(\log n)$, as opposed to other standard notions of expansion. Since the graph's diameter is a lower bound on the number of rounds required for rumor spreading, this makes small-set expanders particularly well-suited for fast information dissemination. We prove that $k$-PUSH&PULL takes $O(\log_{\phi} n \cdot \log_{k} n)$ rounds in these expanders, with high probability. We complement this with a simple lower bound of $\Omega(\log_{\phi} n+ \log_{k} n)$ rounds.