Effect of zealotry in high-dimensional opinion dynamics models

Most of the work on opinion dynamics models focuses on the case of two or three opinion types. We consider the case of an arbitrary number of opinions in the mean field case of the naming game model in which it is assumed the population is infinite and all individuals are neighbors. A particular challenge of the naming game model is that the number of variables, which corresponds to the number of possible sets of opinions, grows exponentially with the number of possible opinions. We present a method for generating mean field dynamical equations for the general case of $k$ opinions. We calculate the steady states in two important special cases in arbitrarily high dimension: the case in which there exist zealots of only one type, and the case in which there are an equal number of zealots for each opinion. We show that in these special cases a phase transition occurs at critical values $p_c$ of the parameter $p$ describing the fraction of zealots. In the former case, the critical value determines the threshold value beyond which it is not possible for the opinion with no zealots to be held by more nodes than the opinion with zealots, and this point remains fixed regardless of dimension. In the latter case, the critical point $p_c$ is the threshold value beyond which a stalemate between all $k$ opinions is guaranteed, and we show that it decays precisely as a lognormal curve in $k$.

Figure 1

The fraction of followers for each of three opinions: $A,B$, and $C$, with system size $n=1000000$. Initially, $A$ has 60% of the followers, $B$ and $C$ both have 20%, and there are no initial undecideds. The solid line indicates expected value obtained from Eqs. (4, 5, 6) and the dots indicate direct simulation. Note that direct simulation yields a less accurate approximation of steady states due to minor fluctuations in the stochastic process. The $x$ axis is scaled by $nln\left(n\right)$ and not by $n$ as might seem more natural, due to the mixing time of the process, which appears to be of order $nln\left(n\right)$ in both the subcritical and supercritical regimes.

Figure 2

The naming game with zealots of only one type for $k=4$ and $p=0.09$, with initial conditions that strongly favor an opinion without zealots. In steady state, an opinion without zealots dominates due to the initial conditions which favor that opinion. Note that all but two opinions are eliminated, consistent with the assumption made in Sec. 4 in order to calculate critical points for all $k$.

Figure 3

The naming game with zealots of only one type for $k=4$ and $p=0.11$, with initial conditions that strongly favor an opinion without zealots. In steady state, the opinion with zealots dominates despite initial conditions which are unfavorable for that opinion, and all others opinions are eliminated. This is consistent with the assumption made in Sec. 4 in order to calculate critical points for all $k$.

Figure 4

The naming game with equal zealots of all opinions, random initial conditions, $k=5$, and $p=0.05$. In steady state, it appears that the decideds for a single opinion dominate while the remaining decideds tie at a lower value, consistent with the assumption made in Sec. 4 in order to calculate critical points for all $k$.

Figure 5

Critical value $p_c$ of $p$ above which a stalemate is guaranteed, plotted against a fitted lognormal curve up to $k=20$. The residual sum of squares is $2.05\times {10}^{-6}$.

Figure 6

Critical points plotted against a fitted lognormal curve up to $k=100$. Note that the lognormal curve plotted here has the same parameters and is normalized as the lognormal curve in Fig. 5. The residual sum of squares is $2.34\times {10}^{-6}$.

Figure 7

Simplified versus original process with $k=2,p=q=0.05$, and initial condition $x=0.3$ compared to the original process with $p=q=0.05$ and initial conditions $x=0.3$ and $z=f\left(x\right)$.

Figure 8

Simplified versus original process with $k=2,p=q=0.05$, and initial condition $x=0.3$ compared to the original process with $p=q=0.05$ and initial conditions $x=0.3$ and $z=0.0$.

Figure 9

The original process is run with $x=0.3,y=0.7$, and $z=0.0$. The simplified process is run starting at time $0.15nln\left(n\right)$ with initial conditions equal to the values of the original process at that time.

Figure 10

The original process is run with $x=0.3,y=0.7$, and $z=0.0$. The simplified process is run starting at time $0.3nln\left(n\right)$ with initial conditions equal to the values of the original process at that time.