Empirical validation of the polarization transition in a double-random field model of elections

Abstract

We model bipartisan elections where voters are exposed to two forces: local homophilic interactions and external influence from two political campaigns. The model is mathematically equivalent to the random field Ising model with a bimodal field. When both parties exceed a critical campaign spending, the system undergoes a phase transition to a highly polarized state where homophilic influence becomes negligible, and election outcomes mirror the proportion of voters aligned with each campaign, independent of total spending. The model predicts a hysteresis region, where the election results are not determined by campaign spending but by incumbency. Calibrating the model with historical data from US House elections between 1980 and 2020, we find the critical campaign spending to be $\sim 1.8$ million USD. Campaigns exceeding critical expenditures increased in 2018 and 2020, suggesting a boost in political polarization.

Figures (9)

• Figure 1: Phase diagrams of the election model. Phase diagram for magnetization, $m$ (a-d), and polarization, $\pi$, (e-h) in the $(h^+,h^-)$ plane for temperature $T=1$ (a,c,e,g), and for $T=0.75$ (b,d,f,h), and a prior probability, $p=0.5$ (a,b,e,f) and $p=0.6$ (c,d,g,h). The black dashed line shows $m=0$. The purple point marks the maximal point of the hysteresis, as derived in the main text. For $T=1$, the expected behavior is that the magnetization is directly affected by the relative strength of the two fields, affected by $p$. For $T=0.75$, we observe a more interesting behavior of the phase diagram. For the case of low field strength, we observe hysteresis (striped region). In both cases, the campaign polarization, $\pi$, starts to increase rapidly when both field strengths are above a critical value, $h_c$ (red).
• Figure 2: Estimation of model parameters for US House of Representatives. We compare campaign spending and election results for 6357 of 9135 races between 1980–2020, focusing on close races ($p=0.5 \pm 0.05$). (a) For $T \geq 1$, the classification model (see Supplemental Material) predicts the higher-spending candidate wins. (b) For $T<1$, it predicts an incumbency region (yellow) where incumbents win despite lower spending. (c) Optimal parameters $T$ and $h_c$ are estimated by maximizing classification accuracy across all 6357 races, yielding $T^=0.922$ and $h_c=\$1.83$M. Cases where incumbents win with lower spending are highlighted, with additional black borders for points in the hysteresis region. The spending diagram is truncated to show the incumbency region (yellow). (d) Accuracy across $T$ is shown, with the maximum marked by a red star. The inset shows a McNemar contingency table comparing the optimal model ($T=T^*$) to the null model ($T=1$). The McNemar test gives $p<0.0001$, indicating significantly better performance of the optimal model.
• Figure 3: Illustration of the model of voters influenced by homophily and election campaign. Every individual has a binary opinion, expressing their voting preference. Everyone is following one of the political campaigns, while also being influenced by their local social environment (friends) in homophilic interactions with the neighbors in the social network.
• Figure 4: Emergence of campaign polarization in the US House of Representatives elections. We compare campaign spending and election results for races from 1980–2020, focusing on close contests with $p = 0.5 \pm 0.05$. (a) Phase diagram as in Fig. \ref{['fig:3']}, now covering the full range of spending, including the polarized region; the gray area marks races with close spending. (b) Election outcomes near $h^{DEM} \approx h^{REP}$, where $|h^{DEM}-h^{REP}|<\$100,000$. The $x$-axis shows average spending $\tfrac{1}{2}(h^{DEM}+h^{REP})$. Below $h_c$, outcomes are mostly decisive, with only 39% close races ($|m|<0.1$) and stronger incumbency effects. Above $h_c$, over 70% of outcomes are near 50:50, consistent with RFIM predictions. (c) Percentage of close races where both campaigns exceed $h_c$, i.e., within the polarized region $\pi$. This share rises sharply in 2018 and 2020.
• Figure 5: Estimation of model parameters for US House of Representatives for Republican-leaning races ($p=0.6 \pm 0.05$). Similarly to \ref{['fig:3']} in the main text, we use the classification model for $p=0.6$ to estimate the parameters of the model. (a) Classification model for $T=1$ without hysteresis. (b) Classification model for $T<1$ with incumbent region. (c) The plot of the election results with the campaign spending, with the incumbent region predicted by the optimal model. (d) The accuracy of the classification model in a range of temperatures; the star denotes the model with the best accuracy, corresponding to $T^\star=0.845$.