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

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.