Networks maximizing the consensus time of voter models

TL;DR

This paper identifies specific network structures that maximize the average consensus time in voter models across different update rules, revealing that certain graphs scale as O(N^3) with the number of nodes.

Contribution

It analytically and numerically demonstrates which networks maximize consensus time for various update rules in voter models, providing exact scaling behaviors.

Findings

Lollipop, barbell, and double star graphs maximize consensus time for different update rules.

Maximum mean consensus time scales as O(N^3).

Different network structures optimize consensus delay depending on the update rule.

Abstract

We explore the networks that yield the largest mean consensus time of voter models under different update rules. By analytical and numerical means, we show that the so-called lollipop graph, barbell graph, and double star graph maximise the mean consensus time under the update rules called the link dynamics, voter model, and invasion process, respectively. For each update rule, the largest mean consensus time scales as O(N^3), where N is the number of nodes in the network.