Threshold-Driven Streaming Graph: Expansion and Rumor Spreading

TL;DR

This paper studies the RAES algorithm in a dynamic streaming graph model with node churn, showing that each snapshot maintains good expansion properties and enabling efficient rumor spreading.

Contribution

It extends the analysis of RAES to dynamic graphs with node churn, demonstrating maintained expansion and fast rumor dissemination.

Findings

Each dynamic graph snapshot has good expansion with high probability.

Rumor spreading protocols complete in logarithmic time on the dynamic graph.

The model captures key features of peer-to-peer networks with churn.

Abstract

A randomized distributed algorithm called RAES was introduced in [Becchetti et al., SODA 2020] to extract a bounded-degree expander from a dense $n$-vertex expander graph $G = (V, E)$. The algorithm relies on a simple threshold-based procedure. A key assumption in [Becchetti et al., SODA 2020] is that the input graph $G$ is static - i.e., both its vertex set $V$ and edge set $E$ remain unchanged throughout the process - while the analysis of RAES in dynamic models is left as a major open question. In this work, we investigate the behavior of RAES under a dynamic graph model induced by a streaming node-churn process (also known as the sliding window model), where, at each discrete round, a new node joins the graph and the oldest node departs. This process yields a bounded-degree dynamic graph $\mathcal{G} =\{ G_t = (V_t, E_t) : t \in \mathbb{N}\}$ that captures essential characteristics of peer-to-peer networks -- specifically, node churn and threshold on the number of connections each node can manage. We prove that every snapshot $G_t$ in the dynamic graph sequence has good expansion properties with high probability. Furthermore, we leverage this property to establish a logarithmic upper bound on the completion time of the well-known PUSH and PULL rumor spreading protocols over the dynamic graph $\mathcal{G}$.