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**Published in:** LIPIcs, Volume 286, 27th International Conference on Principles of Distributed Systems (OPODIS 2023)

We consider a type of pull voting suitable for discrete numeric opinions which can be compared on a linear scale, for example, 1 ("disagree strongly"), 2 ("disagree"), …, 5 ("agree strongly"). On observing the opinion of a random neighbour, a vertex changes its opinion incrementally towards the value of the neighbour’s opinion, if different. For opinions drawn from a set {1,2,…,k}, the opinion of the vertex would change by +1 if the opinion of the neighbour is larger, or by -1, if it is smaller.
It is not clear how to predict the outcome of this process, but we observe that the total weight of the system, that is, the sum of the individual opinions of all vertices, is a martingale. This allows us analyse the outcome of the process on some classes of dense expanders such as complete graphs K_n and random graphs G_{n,p} for suitably large p. If the average of the original opinions satisfies i ≤ c ≤ i+1 for some integer i, then the asymptotic probability that opinion i wins is i+1-c, and the probability that opinion i+1 wins is c-i. With high probability, the winning opinion cannot be other than i or i+1.
To contrast this, we show that for a path and opinions 0,1,2 arranged initially in non-decreasing order along the path, the outcome is very different. Any of the opinions can win with constant probability, provided that each of the two extreme opinions 0 and 2 is initially supported by a constant fraction of vertices.

Colin Cooper, Tomasz Radzik, and Takeharu Shiraga. Discrete Incremental Voting. In 27th International Conference on Principles of Distributed Systems (OPODIS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 286, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cooper_et_al:LIPIcs.OPODIS.2023.10, author = {Cooper, Colin and Radzik, Tomasz and Shiraga, Takeharu}, title = {{Discrete Incremental Voting}}, booktitle = {27th International Conference on Principles of Distributed Systems (OPODIS 2023)}, pages = {10:1--10:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-308-9}, ISSN = {1868-8969}, year = {2024}, volume = {286}, editor = {Bessani, Alysson and D\'{e}fago, Xavier and Nakamura, Junya and Wada, Koichi and Yamauchi, Yukiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2023.10}, URN = {urn:nbn:de:0030-drops-195005}, doi = {10.4230/LIPIcs.OPODIS.2023.10}, annote = {Keywords: Random distributed processes, Pull voting} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Consider a distributed graph where each vertex holds one of two distinct opinions. In this paper, we are interested in synchronous voting processes where each vertex updates its opinion according to a predefined common local updating rule. For example, each vertex adopts the majority opinion among 1) itself and two randomly picked neighbors in best-of-two or 2) three randomly picked neighbors in best-of-three. Previous works intensively studied specific rules including best-of-two and best-of-three individually.
In this paper, we generalize and extend previous works of best-of-two and best-of-three on expander graphs by proposing a new model, quasi-majority functional voting. This new model contains best-of-two and best-of-three as special cases. We show that, on expander graphs with sufficiently large initial bias, any quasi-majority functional voting reaches consensus within O(log n) steps with high probability. Moreover, we show that, for any initial opinion configuration, any quasi-majority functional voting on expander graphs with higher expansion (e.g., Erdős-Rényi graph G(n,p) with p = Ω(1/√n)) reaches consensus within O(log n) with high probability. Furthermore, we show that the consensus time is O(log n/log k) of best-of-(2k+1) for k = o(n/log n).

Nobutaka Shimizu and Takeharu Shiraga. Quasi-Majority Functional Voting on Expander Graphs. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 97:1-97:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{shimizu_et_al:LIPIcs.ICALP.2020.97, author = {Shimizu, Nobutaka and Shiraga, Takeharu}, title = {{Quasi-Majority Functional Voting on Expander Graphs}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {97:1--97:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.97}, URN = {urn:nbn:de:0030-drops-125042}, doi = {10.4230/LIPIcs.ICALP.2020.97}, annote = {Keywords: Distributed voting, consensus problem, expander graph, Markov chain} }

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**Published in:** LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

This paper is concerned with voting processes on graphs where each vertex holds one of two different opinions. In particular, we study the Best-of-two and the Best-of-three. Here at each synchronous and discrete time step, each vertex updates its opinion to match the majority among the opinions of two random neighbors and itself (the Best-of-two) or the opinions of three random neighbors (the Best-of-three). Previous studies have explored these processes on complete graphs and expander graphs, but we understand significantly less about their properties on graphs with more complicated structures.
In this paper, we study the Best-of-two and the Best-of-three on the stochastic block model G(2n,p,q), which is a random graph consisting of two distinct Erdős-Rényi graphs G(n,p) joined by random edges with density q <= p. We obtain two main results. First, if p=omega(log n/n) and r=q/p is a constant, we show that there is a phase transition in r with threshold r^* (specifically, r^*=sqrt{5}-2 for the Best-of-two, and r^*=1/7 for the Best-of-three). If r>r^*, the process reaches consensus within O(log log n+log n/log (np)) steps for any initial opinion configuration with a bias of Omega(n). By contrast, if r<r^*, then there exists an initial opinion configuration with a bias of Omega(n) from which the process requires at least 2^{Omega(n)} steps to reach consensus. Second, if p is a constant and r>r^*, we show that, for any initial opinion configuration, the process reaches consensus within O(log n) steps. To the best of our knowledge, this is the first result concerning multiple-choice voting for arbitrary initial opinion configurations on non-complete graphs.

Nobutaka Shimizu and Takeharu Shiraga. Phase Transitions of Best-of-Two and Best-of-Three on Stochastic Block Models. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{shimizu_et_al:LIPIcs.DISC.2019.32, author = {Shimizu, Nobutaka and Shiraga, Takeharu}, title = {{Phase Transitions of Best-of-Two and Best-of-Three on Stochastic Block Models}}, booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)}, pages = {32:1--32:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-126-9}, ISSN = {1868-8969}, year = {2019}, volume = {146}, editor = {Suomela, Jukka}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.32}, URN = {urn:nbn:de:0030-drops-113397}, doi = {10.4230/LIPIcs.DISC.2019.32}, annote = {Keywords: Distributed Voting, Consensus Problem, Random Graph} }

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**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

In a voting process on a graph vertices revise their opinions in a distributed way based on the opinions of nearby vertices. The voting completes when the vertices reach consensus, that is, they all have the same opinion. The classic example is synchronous pull voting where at each step, each vertex adopts the opinion of a random neighbour. This very simple process, however, can be slow and the final opinion is not necessarily the one with the initial largest support. It was shown earlier that if there are initially only two opposing opinions, then both these drawbacks can be overcome by a synchronous two-sample voting, in which at each step each vertex considers its own opinion and the opinions of two random neighbours.
If there are initially three or more opinions, a problem arises when there is no clear majority. One class of opinions may be largest (the plurality opinion), although its total size is less than that of two other opinions put together. We analyse the performance of the two-sample voting on d-regular graphs for this case. We show that, if the difference between the initial sizes A_1 and A_2 of the largest and second largest opinions is at least C n max{sqrt((log n)/A_1), lambda}, then the largest opinion wins in O((n log n)/A_1) steps with high probability. Here C is a suitable constant and lambda is the absolute second eigenvalue of transition matrix P=Adj(G)/d of a simple random walk on the graph G. Our bound generalizes the results of Becchetti et al. [SPAA 2014] for the related three-sample voting process on complete graphs. Our bound implies that if lambda = o(1), then the two-sample voting can consistently converge to the largest opinion, even if A_1 - A_2 = o(n). If lambda is constant, we show that the case A_1 - A_2 = o(n) can be dealt with by sampling using short random walks. Finally, we give a simple and efficient push voting algorithm for the case when there are a number of large opinions and any of them is acceptable as the final winning opinion.

Colin Cooper, Tomasz Radzik, Nicolás Rivera, and Takeharu Shiraga. Fast Plurality Consensus in Regular Expanders. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{cooper_et_al:LIPIcs.DISC.2017.13, author = {Cooper, Colin and Radzik, Tomasz and Rivera, Nicol\'{a}s and Shiraga, Takeharu}, title = {{Fast Plurality Consensus in Regular Expanders}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {13:1--13:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.13}, URN = {urn:nbn:de:0030-drops-79778}, doi = {10.4230/LIPIcs.DISC.2017.13}, annote = {Keywords: Plurality consensus, Regular expanders} }

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