Abstract

Voting protocols, such as the push and the pull protocol, model the behavior of people during an election. These processes have been studied in distributed computing in peer-to-peer networks, and to describe how viruses or rumors spread in a community. We determine the asymptotic behavior of the runtime of discordant linear protocols on the cycle graph and the probability for each consensus to win.

Highlights

  • Models of voting in finite graphs have been studied intensively for decades, see e.g., [6, 16, 12, 1, 15, 7]

  • Opinion is replaced by a piece of information that each computer has at a given time, and they share the data in a randomized way

  • We show that in case of the cycle graph the probability of each opinion to win with the discordant push, pull or oblivious protocol is asymptotically proportionate to the number of vertices with that opinion in the initial state, provided that the initial state be tame

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Summary

Introduction

Models of voting in finite graphs have been studied intensively for decades, see e.g., [6, 16, 12, 1, 15, 7]. Usually many idle rounds go by before the opinion of some vertex is altered This example demonstrates the advantage of discordant (oblivious, push, pull) voting protocols, defined in [3]. The topic of the present paper is the expected time to reach consensus with the discordant push, pull and oblivious processes on the electronic journal of combinatorics 27(1) (2020), #P1.58 the n-cycle. We show that in case of the cycle graph the probability of each opinion to win with the discordant push, pull or oblivious protocol is asymptotically proportionate to the number of vertices with that opinion in the initial state, provided that the initial state be tame. This paper is a demonstration of how the iterative application of that elementary lemma can yield asymptotically sharp results to basic questions about evolutionary processes, where the transition matrix is typically large but sparse and easy to describe

General tools
Further terminology
Expected time to absorption on the cycle
Estimations of the expected runtime on the n-cycle
Winning probabilities on the cycle
Further results and future work
Full Text
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