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DOI: 10.4230/LIPIcs.ISAAC.2018.4
URN: urn:nbn:de:0030-drops-99529
URL: http://drops.dagstuhl.de/opus/volltexte/2018/9952/
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N. Zehmakan, Ahad

Opinion Forming in Erdös-Rényi Random Graph and Expanders

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LIPIcs-ISAAC-2018-4.pdf (0.4 MB)


Abstract

Assume for a graph G=(V,E) and an initial configuration, where each node is blue or red, in each discrete-time round all nodes simultaneously update their color to the most frequent color in their neighborhood and a node keeps its color in case of a tie. We study the behavior of this basic process, which is called majority model, on the Erdös-Rényi random graph G_{n,p} and regular expanders. First we consider the behavior of the majority model on G_{n,p} with an initial random configuration, where each node is blue independently with probability p_b and red otherwise. It is shown that in this setting the process goes through a phase transition at the connectivity threshold, namely (log n)/n. Furthermore, we say a graph G is lambda-expander if the second-largest absolute eigenvalue of its adjacency matrix is lambda. We prove that for a Delta-regular lambda-expander graph if lambda/Delta is sufficiently small, then the majority model by starting from (1/2-delta)n blue nodes (for an arbitrarily small constant delta>0) results in fully red configuration in sub-logarithmically many rounds. Roughly speaking, this means the majority model is an "efficient" and "fast" density classifier on regular expanders. As a by-product of our results, we show regular Ramanujan graphs are asymptotically optimally immune, that is for an n-node Delta-regular Ramanujan graph if the initial number of blue nodes is s <= beta n, the number of blue nodes in the next round is at most cs/Delta for some constants c,beta>0. This settles an open problem by Peleg [Peleg, 2014].

BibTeX - Entry

@InProceedings{nzehmakan:LIPIcs:2018:9952,
  author =	{Ahad N. Zehmakan},
  title =	{{Opinion Forming in Erd{\"o}s-R{\'e}nyi Random Graph and Expanders}},
  booktitle =	{29th International Symposium on Algorithms and Computation  (ISAAC 2018)},
  pages =	{4:1--4:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Wen-Lian Hsu and Der-Tsai Lee and Chung-Shou Liao},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9952},
  URN =		{urn:nbn:de:0030-drops-99529},
  doi =		{10.4230/LIPIcs.ISAAC.2018.4},
  annote =	{Keywords: majority model, random graph, expander graphs, dynamic monopoly, bootstrap percolation}
}

Keywords: majority model, random graph, expander graphs, dynamic monopoly, bootstrap percolation
Seminar: 29th International Symposium on Algorithms and Computation (ISAAC 2018)
Issue Date: 2018
Date of publication: 27.11.2018


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