Opinion Forming in Erdos-Renyi Random Graph and Expanders

TL;DR

This paper analyzes the majority opinion formation process on random graphs and expanders, revealing phase transitions and conditions for rapid consensus, with implications for network immunity and open problems.

Contribution

It provides new insights into the behavior of the majority model on Erdős-Rényi graphs and expanders, including phase transition thresholds and optimal immunity of Ramanujan graphs.

Findings

Phase transition at connectivity threshold $rac{ ext{log} n}{n}$ in Erdős-Rényi graphs.

Majority model rapidly reaches consensus on regular expanders with spectral gap conditions.

Ramanujan graphs are asymptotically optimally immune, confirming an open problem.

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 binomial random graph $\mathcal{G}_{n,p}$ and regular expanders. First we consider the behavior of the majority model in $\mathcal{G}_{n,p}$ 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 $\frac{\log n}{n}$. Furthermore, we discuss the majority model is a `good' and `fast' density classifier on regular expanders. More precisely, we prove if the second-largest absolute eigenvalue of the adjacency matrix of an $n$-node $\Delta$-regular graph is sufficiently smaller than $\Delta$ then the majority model by starting from $(\frac{1}{2}-\delta)n$ blue nodes (for an arbitrarily small constant $\delta>0$) results in fully red configuration in sub-logarithmically many rounds. As a by-product of our results, we show Ramanujan graphs are asymptotically optimally immune, that is for an $n$-node $\Delta$-regular Ramanujan graph if the initial number of blue nodes is $s\leq \beta n$, the number of blue nodes in the next round is at most $\frac{cs}{\Delta}$ for some constants $c,\beta>0$. This settles an open problem by Peleg.