Effect of social behaviors in the opinion dynamics $q$-voter model

The mean-field approximation postulates that the system is homogeneous and isotropic. In other words, all fluctuation in the system is neglected, implying that local and global concentration can be considered the same [39, 40]. In addition, the system does not depend on the dimensions of space. At this point, we can describe the system’s state using one parameter of the system, such as the fraction opinion $k$ in this model. Regarding graph structure, the graph’s complexity is reduced in the mean-field regime, allowing us to analyze the system analytically more easily. Even though the complexity is reduced, we can observe statistical physics features such as continuous and discontinuous phase transitions, scaling, universality, relaxation time, and exit probability, which are parameters that have been studied frequently in recent years [12, 13, 41, 42, 43, 28].

Model with independent voters- An independent agent acts independently to change their opinion in the population; the agents do not follow the neighboring agents ($q$-sized agent). We define a probability $p$, a parameter that controls the independent behavior of the agents, and another parameter $s$, which represents skepticism of the agents. In this model, with probability $p$, agents act independently of other agents whenever agents influence the voter-agent, while with probability $s$, they act skeptically to the agents’ opinions. With probability $1/2$ the voter-agent change its opinion $\pm S_{i}(t)=\mp S_{i}(t+1)$. Therefore, the total probability that makes the agent-voter change its opinion is $ps/2$. To describe the dynamics of the model, we follow the procedure in Ref. [23], in defining the probability of fraction opinion increases $\varphi_{+}$ and decreases $\varphi_{+}$ and decreases $\varphi_{-}$ for $N_{\uparrow}/N\leq 1/2$ in finite system as:

and for $N_{\uparrow}/N>1/2$:

Equations (4)-(7) describe the increasing (decreasing) fraction opinion density by $+1/N(-1/N)$ or remain constant during the dynamics process with probability $(1-\varphi_{+}-\varphi_{-})$ for a specific range of fraction opinion density $N_{\uparrow}/N$. To test the accuracy of the analytical result with the M-C result, we can expand Eqs. (4)-(7) for any $N$. However, the mean-field results are more suitable for large $N\gg 1$. Therefore, we are more interested in examining Eqs. (4)-(7) as:

where $k=N_{\uparrow}/N$. It should be emphasized that because we are analyzing the model with the mean-field approach, it is more suitable when considering the system for a large population $N$ such in Eqs. (8)-(9) instead of Eqs.(4)-(7).

Model with anticonformist voters- In this model, with probability $p$, the agent-voter act as anticonformist, while probability $s$ acts skeptically to the opinion’ $q$-sized agent. When the $q$-sized agent shares the same opinion, the voter adopts the opposite opinion to the opinion’ $q$-sized agent $\pm,S_{i}(t)=\mp S_{i}(t+1)$. Similar to the model with independent voters, the probability of the fraction opinion up increases $\varphi_{+}$ and $\varphi_{-}$ decreases (in the finite system for $N_{\uparrow}/N\leq 1/2$), respectively are:

and for $N_{\uparrow}/N>1/2$:

Again for comparison to the numerical simulation, defined on the complete graph, we consider Eqs. (10)-(13) for a large population size $N$. Therefore, Eqs. (10)-(13) reduced to more simple forms as:

One can see in Eqs. (8)-(9) and Eq. (14)-(15), the model reduces to the original $q$-voter model for $p=0$ [31]. For the stationary condition of the fraction opinion, $k$ will be discussed in the next subsection.

The recursive formula of the fraction opinion $k$ at time $t$ can be driven from the discrete-time master equation, which can be written as:

where $\varphi(k^{\prime}\to k)$ is a transition probability of fraction opinion from state $k^{\prime}$ to $k$ at a single time $t$. By combining the equation of the probability of finding fraction opinion $k$ at time $t$, $k(t)=\sum_{k}k\,\mathrm{P}(k,t)$ to Eq. (16), the recursive formula of the fraction opinion $k$ at a time step $t$ can be written as:

The functions $\varphi_{+}(k^{\prime})$ and $\varphi_{-}(k^{\prime})$ are the probability density of fraction opinion density increases and decreases which defined in Eqs. (8)-(9) and Eq. (14)-(15). Note that the time step $\Delta t=1/N$ since the total time for $N$ step is $1$. Therefore we can rewrite Eq. (17) in continue version by taking limit $N\to\infty$, or $\Delta t\to 0$ as:

Equation Eq. (17) is also called the rate equation of fraction opinion $k$, which describes the [44, 45].

The macroscopic behavior of the system can be analyzed easily by considering the equilibrium condition of Eq. (18) $\mathrm{d}k(t)/\mathrm{d}t$ = 0, implies $\varphi_{+}(k)=\varphi_{-}(k)$. Therefore, from Eqs. (8)-(9) and (18) the probability of independence $p$ at a stationary state is:

Eq. (19) describes the relation of the probability $p$ and skepticism $s$ for stationary state for specific ranges of $k>1/2$ and $k\leq 1/2$ and to be the same for $s=1$. The transition of skepticism $s$ decreases non-smoothly at low $p$ and a large $q$-sized agent and decreases smoothly for small $q$ and all values of $q$-sized agent, with $k$ increases as $s$ and $p$ increase  [23].

We can also observe the order-disorder phase transition of the model from the equilibrium condition of Eqs. (8)-(9). Thus, the independence probability $p$ at the equilibrium state can be written as:

Note that Eqs. (19) and (20) are slightly different, where Eq. (19) describes the order parameter for a specific range of $k$, while Eq. (20) describes the order parameter for all value of $k=\{0,1\}$. The probability $p_{c}$ in Eq. (20) is the critical independence that makes the system undergoes an order-disorder phase transition. In Ref. [29], authors have shown that for $s=1$, the model undergoes a discontinuous phase transition for $q>5$ and continuous phase transition for $q\leq 5$. In this model, we also show that model undergoes a discontinuous phase transition for $q>5$ and a continuous phase transition for $q\leq 5$ for all values of $s\neq 0$. Although responsible for the occurrence of the phase transition, the parameter $s$ has no impact on changing the phase-transition kind; namely cannot change from continuous to discontinuous phase transitions, and vice versa, but only shifts the critical point of the system. Panel (a) in Fig. 1 shows the plot of Eq. (20) for several values of $q$-sized agent and $s=1/2$. One can see the model undergoes a continuous phase transition for $q\leq 5$ and discontinuous for $q>5$. Based on phenomena, in the social context, one can say that at $p<p_{c}$, the system is in the consensus (there is an agreement), the complete consensus occurred when there is no independent agent $p=0$, and the consensus decreases as $p$ increases until at the critical point $p_{c}$, and at $p\geq p_{c}$ the system is in a status-quo or stalemate situation. As we can see that the skepticism $s$ also controls the critical point $p_{c}$ since for $s=1$, the model is reduced to the model introduced in the previous study [29].

Similar to the model with independent voters, we are interested in analyzing the effect of the probability $p$ on the transition of the skepticism level $s$. Therefore by considering the steady state, $\varphi_{+}(k)=\varphi_{-}(k)$, the anticonformity probability at equilibrium state $p$ for specific range of $k$ can be written as:

Based on Eq. (21), the stationary concentration $k$ is affected by the probability $p$, skepticism level $s$, $q$-sized agent, and the initial concentration $k_{0}$. Similar to the for $k\leq 1/2$, the transition of skepticism $s$ is smooth both for $q=3$ and $q=9$ for all range $p$ (see Fig. 2), which indicates the existence of the continuous phase transition in this model. The effect of probability $p$ to the skepticism $s$ differs from the model with independent voters, showing the non-smoothness (discontinuous) transition of $s$ for $q=9$ at low $p$ [23].

In the same way as the model with independent voters, by considering the stationary condition of Eq. (18), the anticonformity probability $p$ at the equilibrium state, which makes the system undergoes an order-disorder phase transition when $k=1/2$ can be written as:

Plot of Eq. (22) for several values of $q$-sized agents and $s=1/2$ is exhibited in Fig. 1, panel (b). One can see that the model undergoes a continuous phase transition for all values $q$-sized agents. As shown in Eq. (22), besides depending on the $q$-sized agent, the critical point $p_{c}$ also depends on the skepticism $s$, i.e., it decreases as $s$ increases. For a specific case $s=1$, the model is reduced to the one introduced in Ref. [29].

In general, Eqs. (20) and (22) can be written as $m\sim[(p-p_{c})/p_{c}]^{\beta}$ (after re-scaling $k\to(m+1)/2$), where $\beta=1/2$ for all values of $q\neq 1$. For example, for the model with independent voters, for $q=2$, we have $m(p,s)=\pm[((1+2s)p-1)/(1-p)]^{1/2}$, where the critical point is $p_{c}=1/(1+2s)$, that is when $m=0$. The $\beta=1/2$ is typical of the critical exponent of the mean-field Ising model for the magnetization that makes the best data collapse near the critical point $p_{c}$ [46]. We also find the same $\beta$ for the model with anticonformist voters based on Eq. (22). Other critical exponents corresponding to susceptibility $\chi$ and Binder cumulant $U$ will be obtained using M-C simulation in section the next section.

The time evolution of the fraction opinion $k$ can be analyzed from Eq. (17) or Eq. (18) for a large population version. One can see that, obtaining the exact solution of Eq. (18) for $k$ in time step $t$, namely $\int\mathrm{d}k/[\varphi_{+}(k)-\varphi_{-}(k)]=\int\mathrm{d}t$, for any values of $q$-sized agent and $s$ will much need work. However, we can solve (17) numerically using for example, Runge-Kutta 4th order (R-K) [47] . The comparison of the M-C simulation (data point) to the R-K method (dashed line) for the model with independent voters for typical values of $q$ and $p<p_{c}$ is exhibited in Fig. 3. One can see that the M-C result agrees very well with the R-K method. Based on Fig. 3, for $q=3$ and $q=5$ the fraction opinion $k$ evolves to two stable states, while for $q=7$ and $q=9$ the fraction opinion $k$ evolves to three stable states, indicating the model undergoes a continuous phase transition for $q=3,5$ and discontinuous phase transition for $q=7,5$ as discussed on previous section [see Fig. 1 panel (a)]. Fig. 4 shows the time evolution of the fraction opinion $k$ for the model with anticonformist voter. In this model, the fraction opinion $k$ evolves to two stable states for all $q$, indicating the model undergoes a continuous phase transition as discussed in the previous section [see Fig. (1) panel (b)]. The case of this stability can be analyzed from the effective potential and the probability of the density function of the system, which is discussed in the section.

Statistically, we can formulate the scaling parameter of the model that makes data collapse. We can follow the procedure introduced in Ref. [48] to find the scaling parameter for several macroscopic parameters of the model, such as the order parameter $m$, susceptibility $\chi$, and Binder cumulant $U$. Firstly, we prepare the system’s initial state in a disordered state, i.e., the fraction opinion $k=1/2$. We randomly select a group of a $q$-sized agent and another agent (voter) interacting with each other based on the model. In this case, there are three rules followed by the voter. Rule (1): With probability $p$, the voter acts as a nonconformist (anticonformist or independent), and with probability $s$, the voters act skeptical. Rule (2): With probability $1-p$, the voter acts as a conformist. For the model with anticonformist voters, the voters will change their opinion whenever they have the same opinion as the $q$-sized agent (homogeneous state). For the model with independent voters, with probability $1/2$, the voter changes their opinion. Thus the total probability of the voters changing their opinion is $ps$ for the model with anticonformist voters and $ps/2$ for the model with independent voters. Rule (3) with probability $p\,(1-s)$ and $p\,(1-s/2)$, the voters do not change their opinion, respectively. Therefore, rule (3) does not change the system’s state. Then, based on these rules, the scaling parameter depends on the probability of nonconformist $p$ and skepticism $s$.

During the dynamic process, the voters follow one of the three rules above and make the sequence states where the state depends on the initial condition and the probability rules (1) and (2) with ratio $r=\text{probability rule (2)}/\text{probability rule (1)}$, where probability rule (1) and $(2)$ generally depends on the model. We can say that if we have two systems with the same ratio $r$, they are statistically the same. To compare the two systems, we need to obtain the average time during the dynamic process as a function of probability rules (1) and (2). The average time $\tau$ of two systems is inversely proportional to the total probability that makes the voters change their opinion [48]. If two systems have the same ratio probability $r$ and state at an instant time $t/\tau$, then the macroscopic parameters of the systems will be the same, namely $m_{1}=m_{2},\chi_{1}=\chi_{2},$ and $U_{1}=U_{2}$. Therefore, by plotting

The data will be collapsed. Theoretically, these scaling parameters work for any system which statistically follows the rules (1), (2), and (3). This paper will test the model on the complete graph and the 2D square lattice, which will be discussed in the next section.

The Landau theory of free energy can be used to analyze an equilibrium phase transition of a thermodynamics system [49, 50]. In his hypothesis, Landau argued that the free energy could be expanded in power series near the critical point in terms of the order parameter. Landau potential also can be applied to analyze a nonequilibrium system such as the Langevin equation for two absorbing states using mean-field approximation [51, 52]. Generally, the Landau potential is not only described by thermodynamic parameters such as temperature, pressure, and volume but can also depend on the order parameters of the system as in Eq. (3). Here, we use the Landau potential to analyze the order-disorder phase transition of the model. Based on the Landau theory, a potential $V$ can be written as $V(m)=\sum_{i=0}^{n}V_{i}m^{i}$, where $m$ is defined as in Eq. (3), and $V_{i}$ in general as a function of the macroscopic parameters of the system such as the probability $p$, and the skepticism $s$. The potential $V(m)$ is symmetric under inversion $m\to-m$; therefore, the odd terms will vanish.

To analyze the order-disorder phase transition of the model, we only need to consider at least two even terms of the potential $V(m)$ as:

where $m$ is defined in Eq. (3), and $V_{1},V_{2}$ depend of the model. The maximum-minimum conditions that describe the order-disorder phase transitions can be analyzed from the extremum conditions of Eq. (24). In addition, the continuous and discontinuous phase conditions can be analyzed by selecting the appropriate values of $V_{1}$ and $V_{2}$. For $V_{1}=0$, both continuous and discontinuous phase transition conditions occur. At the critical point, the continuous and discontinuous phase occurs for $V_{2}\geq 0$ and $V_{2}<0$, respectively.

To obtain the effective potential of the system, we will use the probability of the fraction opinion $k$ increases and decreases as shown in Eqs. (8) and (9) for the model with independent voters and Eqs. (14) and (15) for the model with anticonformist voters to define an effective force of the model. In general, the effective potential of the system can be obtained as $V(k)=-\int f(k)\,\mathrm{d}k$, where $f(k)=\varphi_{+}(k)-\varphi_{-}(k)$ is called an effective force that drives the spins to flip [53].

The general solution of the effective potential $V$ for the model with independent voters is:

The plot of Eq. (III.4) for typical values of $q$ and probability $p$ is exhibited in Fig. 5. As seen for $q=3$ and $q=5$, the model undergoes a continuous phase transition characterized by the bistable potential for $p<p_{c}(q)$ (dashed line), and the potential is monostable for $p>p_{c}$. However, three stable states for $q=7$ and $q=9$ indicate that the model undergoes a discontinuous phase transition. These potentials confirm the time evolution of the fraction opinion $k$ in Fig. 3; that is, the fraction opinion $k$ evolves to two stable states for $q=3$ and $q=5$ and three stable states for $q=7$ and $q=9$.

For the model with independent voters, the general solution of potential $V$ is:

The plot of Eq. (III.4) can be seen in Fig. 6. For all values of $q$, the potential $V$ has two stable states for $p<p_{c}$ and monostable for $p>p_{c}$, indicating the model undergoes a continuous phase transition. This potential also confirms the time evolution of the fraction opinion $k$ in the previous section (see Fig. 4).

The effective potential in Eqs. (III.4) and (III.4) maximum at $k=1/2$, that is $\partial^{2}V(k,q,p,s)/\partial k^{2}|_{k=1/2}<0$, with two bistable states, minimum at $k=1/2$, that is $\partial^{2}V(k,q,p,s)/\partial k^{2}|_{k=1/2}>0$, with one stable state, and the transition maximum-minimum at $k=1/2$, that is $\partial^{2}V(k,q,p,s)/\partial k^{2}|_{k=1/2}=0$. Therefore we can obtain the critical point of the system as follows:

for the model with independent voters, and

for the mode with anticonformist voters.

The potential $V$ in Eqs. (III.4) and (III.4) are functions of the fraction opinion $k$. However, we can easily convert $V$ as a function of the order parameter $m$ by substituting $k=(m+1)/2$ into equations (III.4) and (III.4) to get a similar form as the Landau potential in Eq. (24). Therefore, for the model with the anticonformist agents, we obtain

We have the same result as Eq. (27) for the critical point $p_{c}$ by setting $V_{1}(q,p,s)=0$. The parameter $V_{2}(q,s)$ is always positive for $q\leq 5$ (typically a continuous phase transition) and negative for $q>5$ (typically a discontinuous phase transition) for all values of $s\neq 0$.

As seen in Eq. (31), we also find the same for as $p_{c}$ in Eq. (28). The parameter $V_{2}(q,s)$ is always positive for all values of $s\neq 0$ and $q\neq 1$, indicating the typical of the continuous phase transition for all $s\neq 0$ and $q\neq 1$.

In the previous section, we analyzed the model’s order-disorder phase transition by considering the equilibrium condition of $k$ and the effective potential $V(k)$. Here, we analyze the order-disorder phase transition of the model by analyzing the stationary of the probability density function, which can be obtained from the Fokker-Planck (F-P) equation below [54]:

where $\xi_{1}=\left(\varphi_{+}-\varphi_{-}\right)$ and $\xi_{2}=\left(\varphi_{+}+\varphi_{-}\right)/N$ are the diffusion-like and drift-like coefficients which depend on the model. As we can see, Eq. (33) is just a one-dimensional of the F-P equation since the model is defined on the complete graph. The diffusion-like $\xi_{1}$ and drift-like $\xi_{2}$ coefficients for the model with independent voters, respectively, are:

and for the model with anticonformist voters, the diffusion-like $\xi_{1}$ and drift-like $\xi_{2}$ coefficients, respectively, are:

We only need the stationary condition of Eq. (33) to analyze the order-disorder phase transition. By a little mathematical modification, the general solution of Eq. (33) for the stationary condition can be written as:

where $C$ is a constant that satisfied the normalized condition of $P(k)$. One can see that obtaining the exact solution of Eq. (38) for any values of the $q$-sized agent, probability $p$, and $s$ are difficult to carry out. However, we can solve numerically Eq. (38) for typical values of $q,p$, and $s$. The plot of Eq. (38) for the model with independent voters is exhibited in Fig. 7. One can see the typical of the discontinuous phase transition, namely $P(k)_{st}$ has three maxima corresponding to three stable states on the effective potential in Eq. (III.4). For low $p$, $P(k)_{st}$ has two maxima, and the maxima appearing in $k=1/2$ is getting higher as $p$ increases. For high $p$, the probability density function $P(k)_{st}$ has only one maximum at $k$, indicating the system is stable at $k=1/2$.

The plot of Eq. (38) for the model with anticonformist voters is exhibited in Fig. 8. In this model, the probability density function $P(k)_{st}$ has two maxima at low $p<p_{c}$ corresponding to two stable states in the effective potential in Eq. (III.4). The ‘peaks’ of $P(k)_{st}$ are getting lower as $p$ increases and only one maximum for $p>p_{c}$ at $k=1/2$. This $P(k)_{st}$ corresponds to the occurrence of a continuous phase transition.

Exit probability is of the parameters that compute in social dynamics. Exit probability can be defined as the probability that the system is in a homogeneous state with one of two homogeneous states. In this paper, we consider the model to reach a homogeneous state with $m=1$, namely the probability of all agents having the same state up in the end. The differential equation of the exit probability can be derived from the F-P equation as [55]:

with boundary conditions $E(0)=0$ and $E(1)=1$, since the probability of the system in opinion up is zero when $k=0$ and one when $k=1$.

Theoretically, we can solve Eq. (39) exactly but obtaining $E(k)$ in any values of $q,p,$ and $s$ can need much work. However, in this paper, we solve Eq. (39) numerically using finite-different method [47]. As an illustration of the exact solution of Eq. (39), we can solve it easily for a simple case, for example, $q=2$ and without anticonformist voter ($p=0$), we have $E(k)=1/2[1+\mathrm{erf}(\sqrt{N/2}\,(2\,k-1))/\mathrm{erf}(\sqrt{N/2})]$. One can check that the exit probability $E(k)$ will coincide in $k=1/2$ for all values of $N$, indicating the existence of the separator point $k_{c}$, which separates two ordered states $m=1(k=1)$ and $m=-1(k=0)$, with the same probability as shown in panel (a) of Figs. 9. It is easy to understand that for the case without the nonconformist voter ($p=0$), all agents reach a consensus state depending on the state of the initial number of fractions opinion $k$. If the fraction opinion $k>0\,(k<0)$, then the system will reach a full consensus state with all agents having the same state or opinion up (down) [28].

The plot of Eq. (39) for typical values of $q,p\leq 0$ and $s=1/2$ for the model with independent voters is shown in Figs. 9 [panel (b)]. We also obtain that the exit probability of the model coincides at $k=1/2$, indicating the separation of two consensus states with the same probability at $k_{c}=1/2$. One can see that the effect of the probability $p$ does not affect the ‘symmetry form’ of the exit probability $E(k)$ to the fraction opinion $k$ because the probability $p$ ‘acts’ homogeneous in reaching the order-disorder phase transition of the model. Therefore the probability of the system reaching two consensus states are the same. The same result is obtained for the model with anticonformist voters (not shown). It can also be seen in Fig. 1 that the fraction opinion $k$ is symmetrical to either the independence or anticonformity probability $p$.

We can also obtain the scaling parameters between the exit probability $E(k)$ versus fraction opinion $k$ using the standard finite-size scaling $E(k)=N^{-\delta_{1}}\phi((k-k_{c})\,N^{\delta_{2}})$, and $k-k_{c}=\delta_{1}N^{-\delta_{2}}$ [56]. As seen in the main panels of Fig. 9, the data are collapsed for $\delta_{1}=0$ and $\delta_{2}=1/2$. These $\delta_{1}=0$ and $\delta_{2}=1/2$ are universal; we obtain the same value for all values of N. These scaling parameters are the same as the majority and the $q$-voter models with mass media effect, defined on the complete graph [28, 27].