Positive Aging Admits Fast Asynchronous Plurality Consensus

TL;DR

This paper demonstrates that positive aging distributions enable fast asynchronous plurality consensus in distributed systems, significantly improving convergence times compared to synchronous models under certain initial bias conditions.

Contribution

It introduces a new asynchronous consensus algorithm that leverages positive aging distributions to achieve rapid agreement, extending prior synchronous results.

Findings

Consensus is achieved in O(log log_alpha k + log log n) time for most nodes.

Initial bias of at least sqrt(n) log n is sufficient for fast convergence.

Additional density conditions lead to even faster partial consensus.

Abstract

We study distributed plurality consensus among $n$ nodes, each of which initially holds one of $k$ opinions. The goal is to eventually agree on the initially dominant opinion. We consider an asynchronous communication model in which each node is equipped with a random clock. Whenever the clock of a node ticks, it may open communication channels to a constant number of other nodes, chosen uniformly at random or from a list of constantly many addresses acquired in previous steps. The tick rates and the delays for establishing communication channels (channel delays) follow some probability distribution. Once a channel is established, communication between nodes can be performed instantaneously. We consider distributions for the waiting times between ticks and channel delays that have constant mean and the so-called positive aging property. In this setting, asynchronous plurality consensus is fast: if the initial bias between the largest and second largest opinion is at least $\sqrt{n}\log n$, then after $O(\log\log_\alpha k\cdot\log k+\log\log n)$ time all but a $1/ \text{polylog } n$ fraction of nodes have the initial plurality opinion. Here $\alpha$ denotes the initial ratio between the largest and second largest opinion. After additional $O(\log n)$ steps all nodes have the same opinion w.h.p., and this result is tight. If additionally the distributions satisfy a certain density property, which is common in many well-known distributions, we show that consensus is reached in $O(\log \log_\alpha k + \log \log n)$ time for all but $n/\text{polylog } n$ nodes, w.h.p. This implies that for a large range of initial configurations partial consensus can be reached significantly faster in this asynchronous communication model than in the synchronous setting.