On the Hierarchy of Distributed Majority Protocols

TL;DR

This paper investigates the speed hierarchy of distributed majority consensus protocols under different communication models, proving that protocols sampling more opinions converge faster, and resolves an open question about their stochastic ordering.

Contribution

It establishes a hierarchy among $j$-Majority protocols, showing $(j+1)$-Majority converges faster than $j$-Majority, using coupling techniques and addressing an open problem.

Findings

$(j+1)$-Majority converges faster than $j$-Majority

Hierarchy holds under both gossip and population models

Addresses an open question by Berenbrink et al. (2017)

Abstract

We study the Consensus problem among $n$ agents, defined as follows. Initially, each agent holds one of two possible opinions. The goal is to reach a consensus configuration in which every agent shares the same opinion. To this end, agents randomly sample other agents and update their opinion according to a simple update function depending on the sampled opinions. We consider two communication models: the gossip model and a variant of the population model. In the gossip model, agents are activated in parallel, synchronous rounds. In the population model, one agent is activated after the other in a sequence of discrete time steps. For both models we analyze the following natural family of majority processes called $j$-Majority: when activated, every agent samples $j$ other agents uniformly at random (with replacement) and adopts the majority opinion among the sample (breaking ties uniformly at random). As our main result we show a hierarchy among majority protocols: $(j+1)$-Majority (for $j > 1$) converges stochastically faster than $j$-Majority for any initial opinion configuration. In our analysis we use Strassen's Theorem to prove the existence of a coupling. This gives an affirmative answer for the case of two opinions to an open question asked by Berenbrink et al. [2017].