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arXiv · 1511.05435

Reaching consensus on a connected graph

John Haslegrave·Mate Puljiz
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Abstract

We study a simple random process in which vertices of a connected graph reach consensus through pairwise interactions. We compute outcome probabilities, which do not depend on the graph structure, and consider the expected time until a consensus is reached. In some cases we are able to show that this is minimised by $K_n$. We prove an upper bound for the case $p=0$ and give a family of graphs which asymptotically achieve this bound. In order to obtain the mean of the waiting time we also study a gambler's ruin process with delays. We give the mean absorption time and prove that it monotonically increases with $p\in[0,1/2]$ for symmetric delays.

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John Haslegrave, Mate Puljiz. 2015-11-17. Reaching consensus on a connected graph. https://doi.org/10.1017/jpr.2016.94

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