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arXiv · 2205.08203

On the Hierarchy of Distributed Majority Protocols

Petra Berenbrink·Amin Coja-Oghlan·Oliver Gebhard·Max Hahn-Klimroth·Dominik Kaaser·Malin Rau
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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].

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Petra Berenbrink, Amin Coja-Oghlan, Oliver Gebhard, Max Hahn-Klimroth, Dominik Kaaser, Malin Rau. 2022-05-17. On the Hierarchy of Distributed Majority Protocols. https://arxiv.org/abs/2205.08203

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