Pair approximation for the noisy threshold $q$-voter model

Abstract

In the standard q-voter model, a given agent can change its opinion only if there is a full consensus of the opposite opinion within a group of influence of size q. A more realistic extension is the threshold q-voter, where a minimal agreement (at least 0 < q0 ≤ q opposite opinions) is sufficient to flip the central agent's opinion, including also the possibility of independent (non conformist) choices. Variants of this model including non-conformist behavior have been previously studied in fully connected networks (mean-field limit). Here we investigate its dynamics in random networks. Particularly, while in the mean-field case it is irrelevant whether repetitions in the influence group are allowed, we show that this is not the case in networks, and we study the impact of both cases, with or without repetition. Furthermore, the results of computer simulations are compared with the predictions of the pair approximation derived for uncorrelated networks of arbitrary degree distributions.

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

“…if ∀ik i = k = q. Finally, ST model with an arbitrary value of r corresponds to the threshold q-voter model on the random regular graph with ∀ik i = k = q [29][30][31][32].…”

“…There are several possibilities to solve the above ambiguity, e.g. we can assume that: (1) a voter prefers to change opinion and therefore will always change it to the opposite one whenever possible [30,32], (2) a voter prefers to keep an old opinion; this assumption overlaps r ≥ 0.5 [29,33] (3) a voter makes a random decision to flip or keep an old state. Each of these scenarios can be used.…”

“…There are two important differences between the Watts threshold model and other models of binary opinions, such as the Galam model [6][7][8], the majority-vote (MV) [9][10][11][12][13][14][15][16][17][18][19][20], the q-voter (qV) [21][22][23][24][25][26][27][28] or the threshold q-voter (TqV) model [29][30][31][32]. The first difference, often considered as the most important, is the heterogeneity -each agent is described by an individual threshold and therefore some agents adopt a new state very easily, whereas others don't [3].…”