Phase Transition of the k-Majority Dynamics in Biased Communication Models

TL;DR

This paper studies how biased external influences can cause a sudden shift in the majority opinion in a network under the k-Majority dynamics, revealing a sharp phase transition depending on the bias strength.

Contribution

It introduces a model analyzing the robustness of k-Majority dynamics against two types of bias, identifying a phase transition in the time to subversion based on bias probability.

Findings

Existence of a critical bias threshold p* for subversion.

Sharp phase transition in subversion time at p*.

For k<3, no phase transition, quick disruption for any bias >0.

Abstract

Consider a graph where each of the $n$ nodes is either in state $\mathcal{R}$ or $\mathcal{B}$. Herein, we analyze the \emph{synchronous $k$-Majority dynamics}, where in each discrete-time round nodes simultaneously sample $k$ neighbors uniformly at random with replacement and adopt the majority state among those of the nodes in the sample (breaking ties uniformly at random). Differently from previous work, we study the robustness of the $k$-Majority in \emph{maintaining a $\mathcal{R}$ majority}, when the dynamics is subject to two forms of \emph{bias} toward state $\mathcal{B}$. The bias models an external agent that attempts to subvert the initial majority by altering the communication between nodes, with a probability of success $p$ in each round: in the first form of bias, the agent tries to alter the communication links by transmitting state $\mathcal{B}$; in the second form of bias, the agent tries to corrupt nodes directly by making them update to $\mathcal{B}$. Our main result shows a \emph{sharp phase transition} in both forms of bias. By considering initial configurations in which every node has probability $q \in (\frac{1}{2},1]$ of being in state $\mathcal{R}$, we prove that for every $k\geq3$ there exists a critical value $p_{k,q}^*$ such that, with high probability, the external agent is able to subvert the initial majority either in $n^{\omega(1)}$ rounds, if $p<p_{k,q}^*$, or in $O(1)$ rounds, if $p>p_{k,q}^*$. When $k<3$, instead, no phase transition phenomenon is observed and the disruption happens in $O(1)$ rounds for $p>0$.