2 Search Results for "N. Zehmakan, Ahad"


Document
Two Phase Transitions in Two-Way Bootstrap Percolation

Authors: Ahad N. Zehmakan

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)


Abstract
Consider a graph G and an initial random configuration, where each node is black with probability p and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least r black neighbors and white otherwise. We prove that this basic process exhibits a threshold behavior with two phase transitions when the underlying graph is a d-dimensional torus and identify the threshold values.

Cite as

Ahad N. Zehmakan. Two Phase Transitions in Two-Way Bootstrap Percolation. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{zehmakan:LIPIcs.ISAAC.2019.5,
  author =	{Zehmakan, Ahad N.},
  title =	{{Two Phase Transitions in Two-Way Bootstrap Percolation}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{5:1--5:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Lu, Pinyan and Zhang, Guochuan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.5},
  URN =		{urn:nbn:de:0030-drops-115017},
  doi =		{10.4230/LIPIcs.ISAAC.2019.5},
  annote =	{Keywords: bootstrap percolation, cellular automata, phase transition, d-dimensional torus, r-threshold model, biased majority}
}
Document
Opinion Forming in Erdös-Rényi Random Graph and Expanders

Authors: Ahad N. Zehmakan

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)


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].

Cite as

Ahad N. Zehmakan. Opinion Forming in Erdös-Rényi Random Graph and Expanders. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 4:1-4:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{n.zehmakan:LIPIcs.ISAAC.2018.4,
  author =	{N. Zehmakan, Ahad},
  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 =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.4},
  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}
}
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