Majority Model on Random Regular Graphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

Consider a graph \(G=(V,E)\) and an initial random coloring where each vertex \(v \in V\) is blue with probability \(P_b\) and red otherwise, independently from all other vertices. In each round, all vertices simultaneously switch their color to the most frequent color in their neighborhood and in case of a tie, a vertex keeps its current color. The main goal of the present paper is to analyze the behavior of this basic and natural process on the random d-regular graph \(\mathbb {G}_{n,d}\). It is shown that for \(\epsilon >0\), \(P_b \le 1/2-\epsilon \) results in final complete occupancy by red in \(\mathcal {O}(\log _d\log n)\) rounds with high probability, provided that \(d\ge c/\epsilon ^2\) for a sufficiently large constant c. We argue that the bound \(\mathcal {O}(\log _d\log n)\) is asymptomatically tight. Furthermore, we show that with high probability, \(\mathbb {G}_{n,d}\) is immune; i.e., the smallest dynamic monopoly is of linear size. A dynamic monopoly is a subset of vertices that can “take over” in the sense that a commonly chosen initial color eventually spreads throughout the whole graph, irrespective of the colors of other vertices. This answers an open question of Peleg [22].

Keywords

Majority model Random regular graph Bootstrap percolation Density classification Threshold behavior Dynamic monopoly 

Notes

Acknowledgments

The authors would like to thank Mohsen Ghaffari for several stimulating conversations and Jozsef Balogh and Nick Wormald for referring to some relevant prior results.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceETH ZurichZürichSwitzerland

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