Voter model on networks and the multivariate beta distribution

In elections, the vote shares or turnout rates show a strong spatial correlation. The logarithmic decay with distance suggests that a two-dimensional (2D) noisy diffusive equation describes the system. Based on the study of U.S. presidential elections data, it was determined that the fluctuations of vote shares also exhibit a strong and long-range spatial correlation. Previously, it was considered difficult to induce strong and long-range spatial correlation of the vote shares without breaking the empirically observed narrow distribution. We demonstrate that a voter model on networks shows such a behavior. In the model, there are many voters in a node who are affected by the agents in the node and by the agents in the linked nodes. A multivariate Wright-Fisher diffusion equation for the joint probability density of the vote shares is derived. The stationary distribution is a multivariate generalization of the beta distribution. In addition, we also estimate the equilibrium values and the covariance matrix of the vote shares and obtain a correspondence with a multivariate normal distribution. This approach largely simplifies the calibration of the parameters in the modeling of elections.

Figure 1

(a) Plot of the distribution of $\mathrm{\Delta }{v}_d(i,t)=v_d(i,t)-v_d\left(i\right)$. The dotted line shows the normal distribution with the same mean and the variance. (b) Plot of the variance of $v_d(i,t)$ and its decomposition to the variance of $v_d\left(i\right)$ and $\mathrm{\Delta }{v}_d(i,t)$ vs. the inverse of the average value of $N\left(i\right)$. The solid line shows the sum of the two variances. The broken line shows $1/\left(4N\right)$.

Figure 2

Semilogarithmic plots of the (a) covariance and (b) correlation of $v_d(i,t),v_d\left(i\right)$, and $\mathrm{\Delta }{v}_d(i,t)$ vs. $r$. The solid line in the left figure shows the sum of the covariance of $v_d\left(i\right)$ and $\mathrm{\Delta }{v}_d(i,t)$. According to the decomposition of the covariance, the contribution of $\mathrm{\Delta }{v}_d(i,t)$ is larger than that of $v_d\left(i\right)$. The correlation coefficient of $\mathrm{\Delta }{v}_d(i,t)$ is also larger than that of $v_d\left(i\right)$.

Figure 3

Illustration of the voter model. The big circles represent node $i$ and node $j\in J\left(i\right)$ that is linked with node $i$. There are $N_i$ and $N_j$ voters in node $i$ and node $j$, respectively. Among them $n_i\left(n_j\right)$ voters (up arrow) and $N_i-n_i(N_j-n_j)$ voters (down arrow) are for and against an option in node $i\left(j\right)$. Initially, a susceptible voter is chosen at random from node $i$. Next, an infectious voter is chosen from node $i$ or node $j\in J\left(i\right)$. Node $i$ has $N_i-1$ infectious voters after a voter is chosen from the node. In addition, there are ${\theta }_1=a_i+b_i$ voters who do not change their decisions. Among them, $a_i$ and $b_i$ voters are for and against the option, respectively, ${\mu }_i=a_i/{\theta }_1$ is the intrinsic tendency of node $i$ to vote for the option and ${\theta }_1$ controls the strength of the tendency. Node $j$ has ${\theta }_2$ infectious voters.

Figure 4

(a) Plots of $E\left(v_1\right)$ vs. ${\theta }_2$ for case II. Solid line plots of Eq. (3) vs. ${\theta }_2$. (b) Plots of $V\left(v_1\right)$ vs. ${\theta }_2$ for case I (gray circle), and case II (hollow black circle). The theoretical results for Eq. (4) are plotted using solid curves in black and gray for case I and case II, respectively. (c) Plots of $\rho$ vs. ${\theta }_c$ for case I (gray circle), case II (hollow black circle) and the theoretical results (solid curves) of Eq. (5).

Figure 5

(a) Plots of $V\left(v_i\right)$ vs. ${\theta }_2$ for a 1D lattice with $L={10}^2$. (b) Plots of $\rho$ vs. $l$ for a 1D lattice with $L={10}^2$. (c) Plots of $V\left[v_{(i,j)}\right]$ vs. ${\theta }_2$ for a 2D lattice with $L=40$. (d) Semilogarithmic plot of $\rho$ vs. $l$ for a 2D with $L=40$. We adopt $\theta =10$ and ${\mu }_i={\mu }_{(i,j)}={\mu }_{\mathrm{avg}}=1/2$. In (a) and (c), solid line plots Eq. (6) vs. ${\theta }_2$ for a 1D and Eq. (8) vs. ${\theta }_2$ for a 2D. We adopt ${\theta }_2={10}^2$ and ${10}^3$ for a 1D lattice in (b) and ${\theta }_2={10}^2,{10}^3$ and ${\theta }_2={10}^4$ for a 2D lattice in (d). In (b) and (d), we plot the theoretical results from Eq. (7) for 1D and those from Eq. (9) for 2D by solid and broken curves.

Figure 6

Semilogarithmic plots of the correlation of $v_i$ (model) vs. $r$. We adopt ${\theta }_1=\{0.044,0.034,0.03\}$ and ${\theta }_2=\{73,50,40\}$ for $z=\{3,4,5\}$, respectively. The Gaussian approximation is used to estimate the correlation of the models. The symbols ($\circ$) is used to plot the correlation of $\mathrm{\Delta }{v}_d$ vs. $r$.