Mean field voter model on networks and multi-variate beta distribution

The regional characteristics of elections show a strong spatial correlation and the logarithmic decay with distance suggests that a 2D noisy diffusive equation describes the system. Based on a study of U.S. presidential elections data, we find that the fluctuations of vote shares also exhibit a strong and long-range spatial correlation. It was previously thought that it was difficult to induce strong and long-ranged spatial correlation of the fluctuations without breaking the unimodality of the distribution. However, we demonstrate that a mean field voter model on networks shows such a behavior. In the model, voters in a node are affected by the agents in the node and by agents in the connected nodes. We derive a multivariate Wright-Fisher diffusion equation for the joint probability density of the vote shares. The stationary distribution is a multivariate generalization of the beta distribution. We estimate the equilibrium values and the covariance of the vote shares and obtain the correspondence with a multivariate normal distribution. This approach greatly simplifies the calibration of the parameters in the modeling of elections.