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.

2.

3.

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 \(h_c\) 

 

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

Acknowledments

Funders

 

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