Consensus and diversity in multistate noisy voter models

We study a variant of the voter model with multiple opinions; individuals can imitate each other and also change their opinion randomly in mutation events. We focus on the case of a population with all-to-all interaction. A noise-driven transition between regimes with multimodal and unimodal stationary distributions is observed. In the former, the population is mostly in consensus states; in the latter, opinions are mixed. We derive an effective death-birth process, describing the dynamics from the perspective of one of the opinions and use it to analytically compute marginals of the stationary distribution. These calculations are exact for models with homogeneous imitation and mutation rates and an approximation if rates are heterogeneous. Our approach can be used to characterize the noise-driven transition and to obtain mean switching times between consensus states.

Figure 1

Sample trajectories and stationary distribution. Panels (a), (b), (d), and (e) show single realizations of the model dynamics; the distributions in panels (c) and (f) are from an average over many realizations. Panels (a) and (d) are for $m=2$, (b) and (e) for $m=5$, and (c) and (f) for $m=3$. Panels (a)–(c) are for a population size of $N=50$; the system is frequently in states of full consensus. In panels (d)–(f) $N=500$, and diversity of opinions is observed; states of consensus are rarely visited. The imitation and mutation rates are uniform across species; we use $r=1$ and $\varepsilon ={10}^{-2}/(m-1)$.

Figure 2

Illustration of the concepts of arrival and switching time. We show the time line of a model with $m\ge 4$ opinion states. Times at which the system reaches a consensus state are marked above the time axis by circled numbers. Times during which the system resides at a consensus state are indicated as filled bars on the time axis. Between these times the population is in mixed states. Arrivals at a new consensus state, as defined in the text, are marked by stars below the time axis. The switching time $\tau$ is the mean time between subsequent arrivals at new consensus states.

Figure 3

Stationary distribution of the model with homogeneous rates across species. Panel (a) is for $m=2$; panel (b) shows the marginal distribution for single species for the model with $m=5$. The different curves are for different population sizes in the range $N=50$ (top) to $N=900$ (bottom). Markers are from simulations; lines show the analytical predicitions from the theory described in the text. Remaining model parameters are $r=1$ and $\epsilon ={10}^{-2}/(m-1)$.

Figure 4

Phase diagram for the model with homogeneous rates. The critical system size is plotted as a function of the number of strategies. The upper continuous line (blue) is $N_L$ as calculated from Eq. (15), and the lower continuous line (purple) shows $N_R$ as obtained from Eq. (16). The remaining lines are from a diffusion approximation to the model, as discussed in Appendix pp1. Markers show results from simulations. Mutation rates are $\varepsilon ={10}^{-2}$ in panel (a) and $\varepsilon ={10}^{-2}/(m-1)$ in panel (b). We set $r=1$ in both panels.

Figure 5

Switching times in the model with homogeneous imitation and mutation rates. The figure shows the switching time $\tau$ between consensus states for different choices of the number of species $m$. Lines are from Eq. (17); markers show simulation results. In all cases $N=100$ and $r=1$.

Figure 6

Marginals of the stationary distribution for the heterogeneous multistate noisy voter model. We show the distributions ${\mathcal{P}}_i\left(x_i\right)$ for the different individual species in a model with $m=5$ for different population sizes, as indicated in the figure. Markers represent simulation results; lines are evaluations of Eq. (7), using the rates in Eq. (18) with the approximation in Eq. (20). The choice of mutation and imitation rates is as described in the text (see Sec. 5b), using $r=1,(m-1)\varepsilon ={10}^{-2}$, and $\delta =0.05$.

Figure 7

Phase diagram of the model with $m=5$ species and heterogeneous rates. Upper and lower dashed lines show $N_L^{\max }$ and $N_R^{\max }$, respectively; upper and lower dotted lines are $N_L^{\min }$ and $N_R^{\min }$. Solid lines are $N_L^{\mathrm{hom}}$ and $N_R^{\mathrm{hom}}$ (see text for definitions). Markers are from simulations (with $▾,▴,▿,▵$ showing $N_L^{\max },N_L^{\min },N_R^{\max },N_R^{\min }$, respectively). Mutation and imitation rates for each species are chosen as described in the text (see Sec. 5b).

Figure 8

Switching times in the noisy voter model with heterogeneous rates. Panel (a) shows the quantities $t_i^{0\to N}$ for the different species $i=1,\cdots ,5$ (bottom to top) in the model with $m=5$. Panel (b) shows the resulting switching time $\tau$ for models with $m=2,3,4,5$ species from top to bottom. Lines are from the analytical approximation [Eq. (13) with the rates as approximated in Eq. (20)]; markers are from simulations. In all cases, $N=100,\delta =0.05$; imitation and mutation rates are distributed as described in Sec. 5b, using $r=1$ and $\varepsilon$ as indicated on the horizontal axis.