A voter model on networks and multivariate beta distribution

TL;DR

This paper introduces a network-based voter model that captures long-range spatial correlations in vote shares, deriving a multivariate beta distribution and simplifying election data analysis.

Contribution

It presents a novel voter model on networks that reproduces observed spatial correlations and derives a multivariate beta distribution for vote shares.

Findings

The model reproduces long-range spatial correlations in vote shares.

Derived a multivariate beta distribution for joint vote share probabilities.

Established a link between the model and a multivariate normal distribution for calibration.

Abstract

In elections, the vote shares or turnout rates show a strong spatial correlation. The logarithmic decay with distance suggests that a 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.