Threshold-Driven Streaming Graph: Expansion and Rumor Spreading

2025-07-31

AI Summary

This work investigates the graph-structural maintenance and rumor-spreading performance of the RAES distributed algorithm in dynamic streaming graphs—where nodes continuously join and depart (high churn). Addressing the limitations of conventional static-graph assumptions, we extend RAES—originally designed for static settings—to high-churn environments by integrating threshold-based connectivity and sliding-window mechanisms. Leveraging stochastic analysis and expander graph theory, we rigorously prove that, at any time, the graph snapshot is an expander with high probability. Consequently, we derive an $O(log n)$ upper bound on the rumor-spreading time under both PUSH and PULL protocols. Our core contribution is the first theoretical framework for dynamic networks that simultaneously ensures structural robustness (via expansion guarantees) and efficient distributed communication (via tight convergence bounds), thereby breaking the static-graph constraint and enabling provably robust and scalable distributed protocols in highly dynamic settings.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Content-based information diffusion

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 $mathcal{G}$.

Problem

• Analyzing RAES algorithm in dynamic graph models
• Studying expansion properties under node-churn streaming process
• Bounding rumor spreading time in dynamic expander graphs

Innovation

• Randomized distributed algorithm RAES for expanders
• Dynamic graph model with node churn
• Logarithmic rumor spreading time bound