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

TL;DR

This work studies a multi-call extension of the PUSH&PULL rumor-spreading model, where each informed and each uninformed node contacts k neighbors per round. On complete graphs, the process completes in Θ(log_k n) rounds with high probability, establishing a precise speedup over the classical Θ(log n) bound. The authors then extend the analysis to small-set vertex expanders with φ>1, showing that rumor spreading takes O((log_φ n + φ^{-1}(α − 1/(2+2φ))^{-1}) log_k n) rounds with high probability, complemented by a lower bound Ω(log_φ n + log_k n). The technical core combines a boundary-decomposition approach by node degree, careful handling of overlapping parallel calls, and concentration via bounded-differences, building on prior work for φ≤1. These results identify regime conditions under which sub-logarithmic rumor spreading is feasible and highlight the role of expansion and diameter in accelerating information dissemination, with practical implications for bandwidth-conscious dissemination on sparse, highly expanding networks.

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 $Θ(\log n)$ rounds, we prove that our $k$-PUSH&PULL variant completes in $Θ(\log_{k} n)$ 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 $φ> 1$, 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 $O(\log_φ n \cdot \log_{k} n)$ rounds in these expanders, with high probability. We complement this with a simple lower bound of $Ω(\log_φ n+ \log_{k} n)$ rounds.

Theorems & Definitions (38)

• definition 1.1
• theorem 1.2
• theorem 1.3
• lemma 1.4
• lemma 1.5
• lemma 2.1
• lemma 2.2
• proof
• lemma 2.3
• proof