Resilience of 3-Majority Dynamics to Non-Uniform Schedulers

Abstract

In recent years there has been great interest in networks of passive, computationally-weak nodes, whose interactions are controlled by the outside environment; examples include population protocols, chemical reactions networks (CRNs), DNA computing, and more. Such networks are usually studied under one of two extreme regimes: the schedule of interactions is either assumed to be adversarial, or it is assumed to be chosen uniformly at random. In this paper we study an intermediate regime, where the interaction at each step is chosen from some not-necessarily-uniform distribution: we introduce the definition of a (p,ε)-scheduler, where the distribution that the scheduler chooses at every round can be arbitrary, but it must have 𝓁_p-distance at most ε from the uniform distribution. We ask how far from uniform we can get before the dynamics of the model break down.
For simplicity, we focus on the 3-majority dynamics, a type of chemical reaction network where the nodes of the network interact in triplets. Each node initially has an opinion of either 𝖷 or 𝖸, and when a triplet of nodes interact, all three nodes change their opinion to the majority of their three opinions. It is known that under a uniformly random scheduler, if we have an initial gap of Ω(√{n log n}) in favor of one value, then w.h.p. all nodes converge to the majority value within O(n log n) steps.
For the 3-majority dynamics, we prove that among all non-uniform schedulers with a given 𝓁_1- or 𝓁_∞-distance to the uniform scheduler, the worst case is a scheduler that creates a partition in the network, disconnecting some nodes from the rest: under any (p,ε)-close scheduler, if the scheduler’s distance from uniform only suffices to disconnect a set of size at most S nodes and we start from a configuration with a gap of Ω(S+√{n log n}) in favor of one value, then we are guaranteed that all but O(S) nodes will convert to the majority value. We also show that creating a partition is not necessary to cause the system to converge to the wrong value, or to fail to converge at all. We believe that our work can serve as a first step towards understanding the resilience of chemical reaction networks and population protocols under non-uniform schedulers.