Analysis of a voter model with an evolving number of opinion states

Abstract

In traditional voter models, opinion dynamics are driven by interactions between individuals, where an individual adopts the opinion of a randomly chosen neighbor. However, these models often fail to capture the emergence of entirely new opinions, which can arise spontaneously in real-world scenarios. Our study introduces a novel element to the classic voter model: the concept of innovation, where individuals have a certain probability of generating new opinions independently of their neighbors' states. This innovation process allows for a more realistic representation of social dynamics, where new opinions can emerge and old ones may fade over time. Through analytical and numerical analysis, we find that the balance between innovation and extinction shapes the number of opinions in the steady state. Specifically, for low innovation rates, the system tends toward near-consensus, while higher innovation rates lead to greater opinion diversity. We also show that network structure influences opinion dynamics, with greater degree heterogeneity reducing the number of opinions in the system.

Figures (7)

• Figure 1: Mean time $T_{\rm ext}$ between consecutive extinctions as a function of the innovation rate $\alpha$. Markers are from simulations for a population of size $N=2^{12}$ averaged over $10$ runs and the dashed line shows Eq. (\ref{['eq:T_ext']}). (inset) Time evolution of the number of opinions $M$ in the population for $\alpha = 2^{-16}$ (red), $2^{-14}$ (green), and $2^{-12}$ (blue) averaged over $500$ runs.
• Figure 2: (a) The distribution $P(M)$ of the number of opinions in the steady state is shown for different values of $\alpha$. Numerical results (symbols) for $N=2^{12}$, averaged over $5000$ samples and the theoretical predictions (lines) from Eq. (\ref{['eq:pm']}) are shown together. (b) Markers show the average number of opinions, $M$ as a function of $\alpha N$ for various population sizes $N$ as indicated. Lines show the theoretical predictions obtained by numerically evaluating the first moment of the distribution in Eq. (\ref{['eq:pm']}). The inset shows a magnification for small $\alpha$ with various $N$ with the same symbols. The vertical lines in the inset represent the location of $\alpha N = 1/\log N$. The dashed line in panel (b) indicates power-law increase with an exponent of $0.85$ (obtained from a fit).
• Figure 3: The distributions $Q(s)$ of opinion sizes on complete graphs with $N=2^{12}$ are shown for different values of $\alpha N$ on a doubly logarithmic scale in panel (a), and in a linear-log representation in panel (b). The distributions decay with a power-law tail, following $Q(s)\sim s^{-1}$. Markers are simulation results, averaged over $5000$ realisations and lines represent the theoretical predictions obtained from Eqs. (\ref{['eq:qtilde_of_s']}) and (\ref{['eq:q_of_s']}).
• Figure 4: Typical fraction of agents holding the majority opinion, $x_{\max}$, as a function of $\alpha N$. Markers are from simulations of populations with all-to-all interaction with different population sizes.
• Figure 5: Probability to find the system in consensus in the stationary state, that is measured as $t_{\rm cons}/t_{\rm tot}=P(M=1)$, where $t_{\rm cons}$ is the total time (in the stationary state) spent in consensus, and $t_{\rm tot}$ is the total simulated time in the stationary state. Markers are from simulations and lines are from the theoretical predictions in $P(M=1)$ in Eq. (\ref{['eq:pm']}). The vertical lines indicate the point $\alpha N= 1/\log N$ for different $N$.