Simple dynamics for plurality consensus
Simple dynamics for plurality consensus
复制标题
多元化共识的简单动态
作者：
L. Becchetti;A. Clementi;Emanuele Natale;F. Pasquale;R. Silvestri;L. Trevisan
我们研究了一个复杂的过程，其中通信网络的每种匿名代理都会支持从集合中选择的颜色，然后在每个回合中，每个代理都可以根据目前的颜色来修改他的颜色。他的邻居的随机样本都假定最初的颜色配置表现出对固定多数颜色的足够大偏见，即支持多数的节点超过节点的数量通过S其他节点支持任何其他颜色。该过程的3多数动力学）是：每个代理商查看三个随机邻居的颜色，然后应用多数规则（均匀打破领带）。 o（min {k，（n/logn）1/3} logn）文档class [12pt] {minimal} usepackage {amsmath} usepackage {wasySym}升级} setLength {oddSideMargin} { - 69pt} egin {document} $$ MATHCAL {o}（min {k，（n/log n）^{1/3}}，log n）$ end End {概率提供了s⩾cmin{2k，（n/logn）1/3} nlogndocumentClass [12pt] {minimal} usepackage {amsmath} usepackage {wasysym} useymym} usepackage} usepackage {amsfonts} } setLength {oddSidemargin} { - 69pt} egin {document} $$ s geqslant c sqrt {min {2k，（n/log n）^{1/3}}，n log n log n} $$证明我们上方的上限很紧，只要k⩽（n/logn）1/4DocumentClass [12pt] {minimal} USEPACKAGE {AMSMATH} USEPACKAGE {wasySym} usepackage {amsfonts} usepackage} } { - 69pt} egin {document} $$ k leqslant（n/log n）^{1/4} $$ end {document {document}，这意味着多数的时间间隔和中位数过程（请参阅DOERR）等。 ACM关于算法和体系结构中的平行性（SPAA'11），第149-158页。回答这个问题：尤其是，我们表明，聚集膜大小的样本只能通过一个多粒子因子加快过程。
We study a plurality-consensus process in which each of n anonymous agents of a communication network initially supports a color chosen from the set [k]. Then, in every round, each agent can revise his color according to the colors currently held by a random sample of his neighbors. It is assumed that the initial color configuration exhibits a sufficiently large biass towards a fixed plurality color, that is, the number of nodes supporting the plurality color exceeds the number of nodes supporting any other color by s additional nodes. The goal is having the process to converge to the stable configuration in which all nodes support the initial plurality. We consider a basic model in which the network is a clique and the update rule (called here the 3-majority dynamics) of the process is the following: each agent looks at the colors of three random neighbors and then applies the majority rule (breaking ties uniformly). We prove that the process converges in time O(min{k,(n/logn)1/3}logn)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathcal {O}( min { k, (n/log n)^{1/3} } , log n )$$end{document} with high probability, provided that s⩾cmin{2k,(n/logn)1/3}nlogndocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$s geqslant c sqrt{ min { 2k, (n/log n)^{1/3} }, n log n}$$end{document}. We then prove that our upper bound above is tight as long as k⩽(n/logn)1/4documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$k leqslant (n/log n)^{1/4}$$end{document}. This fact implies an exponential time-gap between the plurality-consensus process and the median process (see Doerr et al. in Proceedings of the 23rd annual ACM symposium on parallelism in algorithms and architectures (SPAA’11), pp 149–158. ACM, 2011). A natural question is whether looking at more (than three) random neighbors can significantly speed up the process. We provide a negative answer to this question: in particular, we show that samples of polylogarithmic size can speed up the process by a polylogarithmic factor only.
以两种选择的力量稳定共识
DOI：
10.1145/1989493.1989516
发表时间：
2011
期刊：
--
影响因子：
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作者：
Doerr B
通讯作者：
Doerr B