Agreement dynamics on directed random graphs

TL;DR

This paper studies how agreement dynamics behave on directed random graphs, revealing how the presence of fixed or flickering sites affects order and disorder depending on the average degree, with different behaviors compared to undirected graphs.

Contribution

It analyzes agreement models on directed random graphs, highlighting the impact of zealots and flickers on system ordering and identifying critical thresholds for disorder.

Findings

Zealots or flickers become prevalent below z~2, leading to disorder.

Directed graphs exhibit a higher threshold for ordering compared to undirected graphs.

Order emerges at z>1 in undirected graphs, but at z~2 in directed graphs.

Abstract

We examine some agreement-dynamics models that are placed on directed random graphs. In such systems a fraction of sites $\exp(-z)$, where $z$ is the average degree, becomes permanently fixed or flickering. In the Voter model, which has no surface tension, such zealots or flickers freely spread their opinions and that makes the system disordered. For models with a surface tension, like the Ising model or the Naming Game model, their role is limited and such systems are ordered at large~$z$. However, when $z$ decreases, the density of zealots or flickers increases, and below a certain threshold ($z\sim 1.9-2.0$) the system becomes disordered. On undirected random graphs agreement dynamics is much different and ordering appears as soon the graph is above the percolation threshold at $z=1$.