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

2025-08-25

AI Summary

This paper investigates rumor spreading efficiency on small-set vertex expanders under a multi-call (k-PUSH&PULL) mechanism, addressing the slow convergence of classical single-neighbor protocols. Method: We propose a model where each node actively or passively communicates with k distinct neighbors per round, and for the first time integrate this multi-call paradigm with small-set vertex expansion. Using probabilistic analysis coupled with expansion theory, we derive tight bounds on the number of rounds required for complete dissemination. Contribution/Results: We establish a high-probability upper bound of $O(log_phi n cdot log_k n)$ rounds and a matching lower bound of $Omega(log_phi n + log_k n)$, where $phi$ denotes the small-set vertex expansion parameter. Our results demonstrate that strong local connectivity, synergized with multi-call communication, enables near-constant-time spreading—significantly improving upon the classical $O(log n)$ bound—and provide a new theoretical foundation and parameterized design principle for efficient distributed information diffusion.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Distributed Search
• Multiagent Systems: Mechanism Design

Application Category

• Web Mining and Content Analysis: Content-based information diffusion
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on 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.

Problem

• Speeding up rumor spreading with multi-call PUSH&PULL operations
• Analyzing communication rounds on small-set vertex expander graphs
• Establishing tight bounds for k-call rumor dissemination efficiency

Innovation

• Multi-call PUSH&PULL variant with k neighbors
• Constant-time rumor spreading on small-set expanders
• Expansion-sensitive analysis with O(log_φ n · log_k n) rounds