Is the Voter Model a Model for Voters?

The voter model has been studied extensively as a paradigmatic opinion dynamics model. However, its ability to model real opinion dynamics has not been addressed. We introduce a noisy voter model (accounting for social influence) with recurrent mobility of agents (as a proxy for social context), where the spatial and population diversity are taken as inputs to the model. We show that the dynamics can be described as a noisy diffusive process that contains the proper anisotropic coupling topology given by population and mobility heterogeneity. The model captures statistical features of U.S. presidential elections as the stationary vote-share fluctuations across counties and the long-range spatial correlations that decay logarithmically with the distance. Furthermore, it recovers the behavior of these properties when the geographical space is coarse grained at different scales—from the county level through congressional districts, and up to states. Finally, we analyze the role of the mobility range and the randomness in decision making, which are consistent with the empirical observations.

Figure 1

U.S. electoral results 1980–2012. (a) County vote-share probability density functions. (b) Spatial vote-share correlations as a function of distance. The dashed lines indicate logarithmic decay.

Figure 2

(a) Vote-share standard deviation versus noise intensity $D$. The dashed black line marks the dispersion of the empirical data (${\sigma }_e=0.11$). Boxes surrounding the main plot display results obtained with the level of noise marked as squares and include the distribution of vote shares shifted to have zero mean, along with their spatial correlations. Darker (black) curves are initial conditions. Results are shown for increasing time steps from darker to lighter: 10 and 20 MC steps (top right panel); 100 and 200 MC steps (bottom right panel); 40 and 140 MC steps (bottom left panel). (b) The average dispersion in the Democrat vote share is plotted versus the number of elections. Best agreement is obtained for 2.5 MC steps/year.

Figure 3

Parameters of the simulation are $\alpha =1/2$, $D=0.03$. (a) Time traces of the vote shares for Democrats in different counties: one with high population, Los Angeles, California [darkest (black) symbols and curves, $9.5\times {10}^6$ inhabitants]; one with a medium population, Blane, Idaho [light gray (orange), $19\times {10}^3$ inhabitants]; and one with low population, Loving, Texas [medium gray (green) line, 67 inhabitants]. Symbols represent data and dashed lines represent the results of a single realization of the model with initial conditions taken from year 2000 data; solid lines represent the average of 100 realizations of the model and dotted lines their standard deviation. (b), (c), and (d) Democratic vote-share probability density functions, showing the cumulative probability density functions (PDF) [except for (d) that shows the cumulative PDF] as predicted by the model for counties, congressional districts, and states, respectively. Initial condition at $t=0$ (black circles): vote shares obtained from the 2000 elections. (e) Vote-share spatial correlations as a function of the distance. (f) and (g) Distribution of ratio between model predictions and data observations for the Democrat vote shares at county level (f) and for congressional districts (g). The colored areas mark 80% confidence intervals.

Figure 4

(a) Calibrated noise intensity $D^{*}$ on networks thresholded at different distances. (b) Vote-share spatial correlations on different networks.