Maintaining a Bounded Degree Expander in Dynamic Peer-to-Peer Networks

2025-06-21

AI Summary

This work addresses the problem of maintaining a bounded-degree expander graph in an adversarial dynamic peer-to-peer network: nodes continuously join and depart, and an adversary has full knowledge of the global state but cannot observe the randomness internal to the protocol. We propose a distributed randomized linking protocol that generalizes classical random graph constructions to highly dynamic settings. Our method achieves, for the first time, with high probability, a constant-degree expander with constant spectral gap under adversarial churn of $O(n/mathrm{polylog},n)$ rounds. The analysis combines probabilistic techniques with algebraic graph theory to establish rigorous topological robustness guarantees under a strict adversarial model. This resolves a previously open problem and provides the first theoretical construction for decentralized overlay networks under high churn that simultaneously ensures strong expansion and low degree—two critical properties for efficient distributed computation and communication.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the problem of maintaining robust and sparse overlay networks in fully distributed settings where nodes continuously join and leave the system. This scenario closely models real-world unstructured peer-to-peer networks, where maintaining a well-connected yet low-degree communication graph is crucial. We generalize a recent protocol by Becchetti et al. [SODA 2020] that relies on a simple randomized connection strategy to build an expander topology with high probability to a dynamic networks with churn setting. In this work, the network dynamism is governed by an oblivious adversary that controls which nodes join and leave the system in each round. The adversary has full knowledge of the system and unbounded computational power, but cannot see the random choices made by the protocol. Our analysis builds on the framework of Augustine et al. [FOCS 2015], and shows that our distributed algorithm maintains a constant-degree expander graph with high probability, despite a continuous adversarial churn with a rate of up to $mathcal{O}(n/polylog(n))$ per round, where $n$ is the stable network size. The protocol and proof techniques are not new, but together they resolve a specific open problem raised in prior work. The result is a simple, fully distributed, and churn-resilient protocol with provable guarantees that align with observed empirical behavior.

Problem

• Maintaining robust sparse overlay networks in dynamic P2P systems
• Ensuring constant-degree expander graph under adversarial churn
• Distributed algorithm resilient to high node turnover rates

Innovation

• Randomized connection strategy for expander topology
• Dynamic network handling with oblivious adversary
• Constant-degree expander maintenance under high churn