Pair approximation for the q-voter models with quenched disorder on networks

TL;DR

This paper develops a pair approximation method to analyze how quenched disorder affects phase transitions in q-voter opinion models on networks, revealing that quenched disorder eliminates discontinuous transitions and alters phase diagrams.

Contribution

The authors introduce a pair approximation formalism for quenched disorder in q-voter models, enabling analysis of phase diagrams on networks beyond mean-field approximations.

Findings

Quenched disorder removes all discontinuous phase transitions.

Ordered phases are broadened by quenched disorder.

Phase diagrams differ between annealed and quenched disorder, especially in finite networks.

Abstract

Using two models of opinion dynamics, the $q$-voter model with independence and the $q$-voter model with anticonformity, we discuss how the change of disorder from annealed to quenched affects phase transitions on networks. To derive phase diagrams on networks, we develop the pair approximation for the quenched versions of the models. This formalism can be also applied to other quenched dynamics of similar kind. The results indicate that such a change of disorder eliminates all discontinuous phase transitions and broadens ordered phases. We show that although the annealed and quenched types of disorder lead to the same result in the $q$-voter model with anticonformity at the mean-field level, they do lead to distinct phase diagrams on networks. These phase diagrams shift towards each other as the average node degree of a network increases, and eventually, they coincide in the mean-field limit. In contrast, for the $q$-voter model with independence, the phase diagrams move towards the same direction regardless of the disorder type, and they do not coincide even in the mean-field limit. To validate our results, we carry out Monte Carlo simulations on random regular graphs and Barab\'{a}si-Albert networks. Although the pair approximation may incorrectly predict the type of phase transitions for the annealed models, we have not observed such errors for their quenched counterparts.