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

Remah Dahdoul

Jan Korbel

Stefan Thurner

web: jankorbel.eu

slides: slides.com/jankorbel

accepted to Physical Review Letters

Motivation

Recent elections show increasing political polarization.

However, it is unclear how campaign spending interacts with social influence among voters.

Key question:

How does campaign spending influence this polarization?

We use a simple physics-inspired model combining two key factors: voter homophily and campaign intensity.

Illustration of the model

Ising model with random bimodal field

We model the individual opinion as a binary spin $$s_i \in \{-1,+1\}$$
The homophily is given by the standard Ising model $$ - J \sum_{ij} A_{ij} s_i s_j \quad \textrm{where} \ A_{ij} \textrm{\ is \ adj. \ matrix} $$
The external campaign is given by the random external field $$- \sum_i h_i s_i \quad \textrm{where} \ h_i \in \{+h^+,-h^-\}$$
Here $h^+$ denotes the intensity of the campaign to promote opinion $s = +1$ and $h^-$ is the intensity of the campaign to promote opinion $s = -1$

Mean-field approximation

The Hamiltonian describes a Random-Field Ising model $$H(s) = - J\sum_{ij} A_{ij} s_i s_j - \sum_i h_i s_i$$ with a bimodal (and binary) field
By using mean-field approximation and configuration model approximation we obtain $$H^{MF}(s) = - \sum_i (\tilde{J} m + h_i) \sigma_i$$ where $\tilde{J} = J/\langle k \rangle$ and $h_i$ is a binary random variable with the probability distribution $(p(h^+),p(h^-))$ where $p \equiv p({h^+})$
The random component models affiliation to one or the other campaign

Self-consistency equation

By averaging of the random field, the opinion distribution is $$p(s) = \frac{p e^{- \beta(\tilde{J} m + {h^+}) s) } + (1-p) e^{-\beta (\tilde{J} m - h^-)s}}{Z}$$
The average magnetization can be determined as $$m = \langle s \rangle = p \langle s \rangle_{h=h^+}+ (1-p) \langle s \rangle_{h=h^-}$$ from which we get the self-consistency equation

$$m= p \tanh(\beta(\tilde{J} m + h^+)) + (1-p) \tanh(\beta \tilde{J} m - h^-)$$

Critical point and phase transition

For $T \geq 1$, we observe no phase transition
For $T<1$ we observe a phase transition where the transition is observed at critical points$$h_c^+ = T \, \textrm{arctanh}\left(\sqrt{(1-T) \frac{1-p}{p}}\right)$$$$h_c^- = T \, \textrm{arctanh}\left(\sqrt{(1-T) \frac{p}{1-p}}\right)$$
This is a generalization of the well-known critical curve for symmetric case $ (p=1/2) $
Thus, for $T<1$ the system has a hysteresis region

Polarization

To measure whether the society is in consensus or polarized, we define a quantity

$$\pi = \frac{1}{2} \left( \langle s \rangle_{h=h^+} - \langle s \rangle_{h=h^-} \right)$$

This quantity, which is called campaign polarization, measures the difference between the opinion of groups affected by the opposite campaigns.

Phase diagrams

Magnetization

Polarization

Application to US House elections 1980-2020

We apply our model to the US House elections
We include only bipartisan races (all races with more than 2 candidates with non-negligible outcome are excluded)
Altogether, we analyze 6357 races
We use our model as a winner classifier
- If m>0, the Republican wins
- If m<0 , the Democrat wins
The hysteresis region is interpreted as incumbency region (i.e., incumbent wins even if they spend less)
We use the hysteresis region to calibrate the model parameters $T$ and $h_c$

Classifiers

incumbency region

optimal temperature

Application to US House elections 1980-2020

full region

results for close races

spending above $h_c$

Application to US House elections 1980-2020

1. Polarization phase transition

Campaign spending can trigger a phase transition from socially driven voting to strongly polarized elections.

2. Hysteresis and incumbency

The model exhibits hysteresis, which can be interpreted as an incumbency advantage.

3. Estimation of social temperature

Thermodynamic quantities like temperature can be estimated from the shape of the hysteresis region.

4. Empirical relevance

Analysis of US House elections (1980–2020) suggests the existence of a hysteresis region and a critical spending threshold.
Above a critical spending level (~$1.8M), campaign spending dominates social interactions, and polarization increases.

Summary

Thank you for your attention

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