When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.20
URN: urn:nbn:de:0030-drops-126239
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12623/
 Go to the corresponding LIPIcs Volume Portal

### Reaching a Consensus on Random Networks: The Power of Few

 pdf-format:

### Abstract

A community of n individuals splits into two camps, Red and Blue. The individuals are connected by a social network, which influences their colors. Everyday, each person changes his/her color according to the majority of his/her neighbors. Red (Blue) wins if everyone in the community becomes Red (Blue) at some point.
We study this process when the underlying network is the random Erdos-Renyi graph G(n, p). With a balanced initial state (n/2 persons in each camp), it is clear that each color wins with the same probability.
Our study reveals that for any constants p and ε, there is a constant c such that if one camp has n/2 + c individuals at the initial state, then it wins with probability at least 1 - ε. The surprising fact here is that c does not depend on n, the population of the community. When p = 1/2 and ε = .1, one can set c = 6, meaning one camp has n/2 + 6 members initially. In other words, it takes only 6 extra people to win an election with overwhelming odds. We also generalize the result to p = p_n = o(1) in a separate paper.

### BibTeX - Entry

```@InProceedings{tran_et_al:LIPIcs:2020:12623,
author =	{Linh Tran and Van Vu},
title =	{{Reaching a Consensus on Random Networks: The Power of Few}},
booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
pages =	{20:1--20:15},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-164-1},
ISSN =	{1868-8969},
year =	{2020},
volume =	{176},
editor =	{Jaros{\l}aw Byrka and Raghu Meka},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},