Drivers of Social Polarization: Insights from Social Connectivity and Campaign Influence

Jan Korbel

web: jankorbel.eu

slides: slides.com/jankorbel

Part one:

Paper flow

1. Thermodynamics of

structure-forming systems

Entropy for systems with structures

$W(n_i^{(j)}) = \frac{n!}{\prod_{ij} n_i^{(j)}! \textcolor{red}{(j!)^{n_i^{(j)}}}}$

$$\mathcal{S} = - \sum_{ij} \wp_i^{(j)} (\log \wp_i^{(j)} {\color{red}- 1}) {\color{red}- \sum_{ij} \wp_i^{(j)}\log \frac{j!}{n^{j-1}}}$$

$ S = k \cdot \log W$

$ \wp_i^{(j)} = n_i^{(j)}/n$

When we want to calculate the MaxEnt distribution, we maximize the entropy w.r.t. to

normalization $\sum_{ij} j \wp_{i}^{(j)}=1$
average energy $\sum_{ij} \wp_{i}^{(j)} \epsilon_{i}^{(j)} = E$

Equilibrium distribution:

$$\hat{\wp}_i^{(j)} = \frac{n^{j-1}}{j!} \exp(-\alpha j - \beta \epsilon_i^{(j)})$$

Normalization:

$\sum_{ij} j \wp_i^{(j)} = \sum_{ij} \frac{n^{j-1}}{(j-1)!} e^{-{\color{red} \alpha} j - \beta \epsilon_i^{(j)}} = 1$ for ${\color{red} \alpha}$

Free energy:

$ F = U - \beta^{-1} S = - \frac{\alpha}{\beta} {\color{red}- \frac{\mathcal{M}}{\beta}}$

where $\mathcal{M} = \sum_{ij} \wp_{i}^{(j)}$ is the number of molecules

MaxEnt distribution and free energy

Applications to self-assembly

2. Spin-glass based opinion dynamics

Many opinion dynamics systems follow two basic concepts:

1. Homophily - people tend to be friends with peers with similar opinions ("birds of a feather flock together")

2.Social balance - people tend to follow Heider balance relation

("a friend of my friend is my friend, enemy of my friend is my enemy")

2. Spin-glass based opinion dynamics

Group size distribution

Social balance can emerge from homophily

3. Social group formation in the spin-glass self-assembly framework

Hamiltonian of a group $\mathcal{G}$

$H(\mathbf{s}_{i_1},\dots,\mathbf{s}_{i_k}) = \textcolor{red}{\underbrace{- \phi \, \frac{J}{2} \sum_{ij \in \mathcal{G}} A_{ij} \mathbf{s}_i \cdot \mathbf{s}_j}_{intra-group \ social \ stress}} \textcolor{blue}{ + \underbrace{(1-\phi) \frac{J}{2} \sum_{i \in \mathcal{G}, j \notin \mathcal{G}} A_{ij} \mathbf{s}_{i} \cdot \mathbf{s}_j}_{inter-group \ social \ stress}} \\ \qquad \qquad \qquad \qquad - \underbrace{h \sum_{i \in \mathcal{G}} \mathbf{s}_i \cdot \mathbf{w}}_{external \ field}$

Group formation based on opinion= self-assembly of spin glass

Group 1

Group 2

friends

enemies

Approximations used in the model

1. Configuration model

- We do not know the full network, just a degree distribution. $\Rightarrow$ The probability of observing a link between $i$ and $j$ is proportional to the degree of both nodes

2. Mean-field approximation

- We use the mean-field approximation of the Hamiltonian.

These two approximations lead to the set of self-consistency equations:

$$m^{(k)} = k \sum_{q^{(k)} q^{(k,l)}} P(q^{(k)}) P(q^{(k,l)}) \tanh(\beta H^{(k)}(m^{(l)},q^{(k)},q^{(k,l)})) $$

where $q^{(k)}$ is the intra-group degree, $q^{(k,l)}$ is the inter-group degree and $P$ is the degree distribution

Results for zero inter-group degree

Theory

MC simulation

Application online multiplayer game PARDUS

Part two:

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 $h_c$