$Q$-voter model with nonconformity in freely forming groups: Does the size distribution matter?

We study a $q$-voter model with stochastic driving on a complete graph with $q$ being a random variable described by probability density function $P\left(q\right)$, instead of a constant value. We investigate two types of $P\left(q\right)$: (1) artificial with the fixed expected value $\langle q\rangle$, but a changing variance and (2) empirical of freely forming groups in informal places. We investigate also two types of stochasticity that can be interpreted as different kinds of nonconformity (anticonformity or independence) to answer the question about differences observed at the macroscopic level between these two types of nonconformity in real social systems. Moreover, we ask the question if the behavior of a system depends on the average value of the group size $q$ or rather on probability distribution function $P\left(q\right)$.

Figure 1

Schematic illustration of the model's assumptions. Left panel: the size $q$ of the influence group is not longer a parameter, as in the original $q$-voter model, but is given by a certain density probability distribution $P\left(q\right)$. This means that the $q$ panel (green up spinsons in this illustration), which influences a voter (red down spinson), can change size in each update. Right panel: social network is described here by the complete graph (clique) and therefore each agent is connected with any other agent. Yet, in a single update an agent is influenced only by a group of $q$ spinsons (here $q=3$).

Figure 2

Example of a single update for the $q$ voter with anticonformity (model A). In this example $q=3$ and a chosen target spinson (in the square) has initially positive opinion.

Figure 3

Example of a single update for the $q$ voter with independence (model I). In this example $q=3$ and a target spinson (in the square) has initially positive opinion. In this paper we will use $f=1/2$, analogous as in [4].

Figure 4

Dependencies between the average public opinion $\langle m\rangle$ in the steady state and the level of nonconformity $p$ for the model with conformity+anticonformity (model A). In this case the size of the $q$ panel is constant [$\epsilon =0$ in Eq. (1)] and $q$ increases from left to right. Results have been obtained from Eq. (11). It is seen that phase transition is continuous for all values of $q$ and critical temperature increases with $q$. The same results have been obtained within less general model with constant $q$ in [4].

Figure 5

Dependencies between the average public opinion $\langle m\rangle$ in the steady state and the level of nonconformity $p$ for the model with conformity+independence (model I). In this case the size of the $q$ panel is constant [$\epsilon =0$ in Eq. (1)] and $q$ decreases from left to right. Results have been obtained from Eq. (12). Dotted lines have been used to mark instability of the solution, whereas solid lines denote stable solution. This shows that for model I the phase transition is continuous only $q\in [2,5]$, whereas for $q\ge 6$ the phase transition is discontinuous. Moreover, the critical value of $p$ decreases with $q$, contrary to model A. The same results have been obtained within a less general model with constant $q$ in [4].

Figure 6

Dependencies between the average public opinion $\langle m\rangle$ in the steady state and the level of nonconformity $p$ for the probability density function of $q$ is given by Eq. (1) with $\langle q\rangle =6,\delta =1$, and $\epsilon =0.1$. This means that $\vec{q}=(5,6,7)$ with probabilities $\vec{w}=(0.1,0.8,0.1)$. Results for the annealed approach are presented in the upper panels, whereas in the bottom panels results for the quenched approach are shown. Besides, results for the model with anticonformity (model A) are presented in the left column and for the model with independence (model I) in the right column: (a) model A with the annealed approach; (b) model I with the annealed approach; (c) model A with the quenched approach; (d) model I with the quenched approach. The simulations were performed for the system consisting of ${10}^4$ spinsons. The results were averaged over ${10}^2$ realizations after ${10}^3$ Monte Carlo steps.

Figure 7

Dependencies between the average public opinion $\langle m\rangle$ in the steady state and the level of nonconformity $p$ for the models with anticonformity (model A; upper panels) and independence (model I; bottom panels) for $P\left(q\right)$ given by Eq. (1) with $\langle q\rangle =6$ and two values of $\delta$: left panels (a),(c) $\delta =3$; right panels (b),(d) $\delta =1$. In each panel results for several values of $\epsilon$ are shown: $\epsilon$ changes with a step $\mathrm{\Delta }\epsilon =0.1$ from zero to 1 (from right to left for model A and from left to right for model I).

Figure 8

Phase diagram for model I (conformity+independence) with $P\left(q\right)$ given by Eq. (1) for $\langle q\rangle =6$ and $\epsilon ,\delta$ being two model's parameters. In the black region the system undergoes continuous phase transition (there is no jump in order parameter, i.e., $\mathrm{\Delta }m=0$) and in the brighter area there is discontinuous phase transition.

Figure 9

Empirical probability density functions of $P\left(q\right)$ (upper panels) for fans gathered to watch a college football game [34] (a) and pedestrians on the streets in Eugene (Oregon) [33] (b). Corresponding dependencies between the average public opinion $\langle m\rangle$ in the steady state and the level of nonconformity are presented in bottom panels.