Loosely-Stabilizing Phase Clocks and the Adaptive Majority Problem

TL;DR

This paper introduces a leaderless, loosely-stabilizing phase clock for population protocols that quickly recovers from arbitrary states and enables a protocol to reliably track the majority opinion among agents with a large bias, within polylogarithmic interactions.

Contribution

It presents a novel loosely-stabilizing phase clock with minimal states that ensures rapid recovery and sustained synchronization, facilitating reliable majority detection in population protocols.

Findings

The phase clock requires O(log n) states and runs forever with high probability.

The clock recovers from arbitrary configurations within O(n log n) interactions.

The majority protocol converges correctly within O(n log n) interactions when the majority support is at least logarithmic in n.

Abstract

We present a loosely-stabilizing phase clock for population protocols. In the population model we are given a system of $n$ identical agents which interact in a sequence of randomly chosen pairs. Our phase clock is leaderless and it requires $O(\log n)$ states. It runs forever and is, at any point of time, in a synchronous state w.h.p. When started in an arbitrary configuration, it recovers rapidly and enters a synchronous configuration within $O(n\log n)$ interactions w.h.p. Once the clock is synchronized, it stays in a synchronous configuration for at least poly $n$ parallel time w.h.p. We use our clock to design a loosely-stabilizing protocol that solves the comparison problem introduced by Alistarh et al., 2021. In this problem, a subset of agents has at any time either $A$ or $B$ as input. The goal is to keep track which of the two opinions is (momentarily) the majority. We show that if the majority has a support of at least $\Omega(\log n)$ agents and a sufficiently large bias is present, then the protocol converges to a correct output within $O(n\log n)$ interactions and stays in a correct configuration for poly $n$ interactions, w.h.p.