Generalized Independence in the q-Voter Model: How Do Parameters Influence the Phase Transition?

Abstract

We study the q-voter model with flexibility, which allows for describing a broad spectrum of independence from zealots, inflexibility, or stubbornness through noisy voters to self-anticonformity. Analyzing the model within the pair approximation allows us to derive the analytical formula for the critical point, below which an ordered (agreement) phase is stable. We determine the role of flexibility, which can be understood as an amount of variability associated with an independent behavior, as well as the role of the average network degree in shaping the character of the phase transition. We check the existence of the scaling relation, which previously was derived for the Sznajd model. We show that the scaling is universal, in a sense that it does not depend neither on the size of the group of influence nor on the average network degree. Analyzing the model in terms of the rescaled parameter, we determine the critical point, the jump of the order parameter, as well as the width of the hysteresis as a function of the average network degree ⟨ k ⟩ and the size of the group of influence q.

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

“…At the pair approximation level, the only network parameter that affects the model dynamics under the quenched approach is the average node degree. The same applies to the q-voter models under the annealed approach [30,[32][33][34][35]. Interestingly, if repetition of voters in the influence group is allowed, other moments of the degree distribution may appear in the solution [31,35].…”

“…The pair approximation is a general technique used to study various dynamics on static [28][29][30][31][32][33][34][35][36][37][38][39][40] as well as coevolutionary networks [41][42][43][44][45]. This method has already been applied to the q-voter models with nonconformity under the annealed approach.…”

“…The question which naturally arises is whether the above results remain valid on networks. Although both the models have been studied extensively under the annealed approach on networks, by the use of the pair approximation and Monte Carlo simulations [30][31][32][33][34][35][36], such a comprehensive analysis is missing for their quenched counterparts.…”

Phase transitions in biology: from bird flocks to population dynamics

“…from firstorder to critical) has been studied in a model of social contagion with a heterogeneous threshold for adoption of new behaviour [75]. Mathematical models of this type of problem have been investigated in the voter model in probability theory, in which phase transitions have also been observed [76].…”

In search for the social hysteresis -- the symmetrical threshold model with independence on Watts-Strogatz graphs

“…We study the model on Watts-Strogatz (WS) graph [39] because it allows to tune the structure from (1) the complete graph, for which the mean-field approximation gives exact result, through (2) random graphs for which the pair approximation should work properly, to (3) small-world networks which resembles the basic features of the real social networks. Because it has been shown recently that the size of the hysteresis may depend on the graph's properties, we focus on this issue and check to what extend results found within the MV model and the qV model are universal [14,15,19,20,28,40].…”