Fast Consensus for Voting on General Expander Graphs

Cooper, Colin; Elsässer, Robert; Radzik, Tomasz; Rivera, Nicolás; Shiraga, Takeharu

doi:10.1007/978-3-662-48653-5_17

Cited by 28 publications

(53 citation statements)

References 17 publications

(31 reference statements)

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“…All previous work on non-complete graphs has involved some special bias setting (e.g. an initial bias [13,14,15], or a random initial opinion configuration [3,18,28]). In this paper, we present the following first worst-case analysis of non-complete graphs.…”

Section: Results Ii: Worst-case Analysismentioning

confidence: 99%

Phase transitions of Best‐of‐two and Best‐of‐three on stochastic block models

Shiraga

2021

Random Struct Algorithms

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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 = ω(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 * = √ 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 Ω(n). By contrast, if r < r * , then there exists an initial opinion configuration with a bias of Ω(n) from which the process requires at least 2 Ω(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.

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Section: Results Ii: Worst-case Analysismentioning

confidence: 99%

Phase transitions of Best‐of‐two and Best‐of‐three on stochastic block models

Shiraga

2021

Random Struct Algorithms

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“…The simplest non-trivial example is then arguably the 2-Choices dynamics, in which agents choose two random neighbors and switch to their opinion only if they coincide [19] (see Definition 1) instead of copying the color a priori as in the Voter Model. Still, the analysis of the 2-Choices dynamics has been limited to networks with good expansion properties, and the theoretical guarantees provided so far are essentially independent from the positioning of ini-tial opinions [20]. Only recently the 2-Choices dynamics has also been analyzed on classes of graphs presenting a clustered structure [27,57], as discussed in more detail in Sect.…”

Section: Introductionmentioning

confidence: 99%

Phase transition of the 2-Choices dynamics on core–periphery networks

et al. 2021

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The 2-Choices dynamics is a process that models voting behavior on networks and works as follows: Each agent initially holds either opinion blue or red; then, in each round, each agent looks at two random neighbors and, if the two have the same opinion, the agent adopts it. We study its behavior on a class of networks with core–periphery structure. Assume that a densely-connected subset of agents, the core, holds a different opinion from the rest of the network, the periphery. We prove that, depending on the strength of the cut between core and periphery, a phase-transition phenomenon occurs: Either the core’s opinion rapidly spreads across the network, or a metastability phase takes place in which both opinions coexist for superpolynomial time. The interest of our result, which we also validate with extensive experiments on real networks, is twofold. First, it sheds light on the influence of the core on the rest of the network as a function of its connectivity toward the latter. Second, it is one of the first analytical results which shows a heterogeneous behavior of a simple dynamics as a function of structural parameters of the network.

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“…In order to reach double logarithmic speed the graph requires a tractable local structure around each vertex, and we need to be able to keep track of the configuration of opinions around each vertex at each time step. In this regard, the techniques used in [4] and [5] are not necessarily useful to tackle the problem in our case, even though they work on a large class of graphs. This is because in their work, the authors track the number of red (and blue) opinions instead of the actual configuration of the opinions of the vertices.…”

Section: Resultsmentioning

confidence: 99%

“…The authors showed that if the imbalance between the number of red and blue opinions is greater than Kn 1/d + d/n initially, where K is a large constant, then w.h.p the process reaches consensus towards majority in O(log n) time-steps. In [5], the result was extended and refined to general graphs with large expansion. Denote by R 0 and B 0 the initial sets of vertices with red and blue opinions respectively.…”

Section: Introductionmentioning

confidence: 99%

Best-of-Three Voting on Dense Graphs

Kang

Rivera

2019

The 31st ACM Symposium on Parallelism in Algorithms and Architectures

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Given a graph G of n vertices, where each vertex is initially attached an opinion of either red or blue. We investigate a random process known as the Bestof-three voting. In this process, at each time step, every vertex chooses three neighbours at random and adopts the majority colour. We study this process for a class of graphs with minimum degree d = n α , where α = Ω (log log n) −1 . We prove that if initially each vertex is red with probability greater than 1/2 + δ, and blue otherwise, where δ ≥ (log d) −C for some C > 0, then with high probability this dynamic reaches a final state where all vertices are red within O (log log n) + O log δ −1 steps .

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Fast Consensus for Voting on General Expander Graphs

Cited by 28 publications

References 17 publications

Phase transitions of Best‐of‐two and Best‐of‐three on stochastic block models

Phase transitions of Best‐of‐two and Best‐of‐three on stochastic block models

Phase transition of the 2-Choices dynamics on core–periphery networks

Best-of-Three Voting on Dense Graphs

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