Homophily-Based Social Group Formation

in a Spin Glass Self-Assembly Framework

Jan Korbel

Simon Lindner

Tuan Pham Minh

Rudolf Hanel

Stefan Thurner

Phys. Rev. Lett. 130 (2023) 057401 (editors' suggestion) 

Slides available at: www.slides.com/jankorbel

in collaboration with

Graphical abstract

1. soft-matter self-assembly

2. condensed matter spin glasses

3. social group formation

1.

2.

3.

Graphical abstract

1. Thermodynamics of structure-forming systems

Entropy for systems with structures

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

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

\( S = k \cdot \log W\)

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
2.  Social balance - a friend of my friend is my friend, enemy of my friend is my enemy

Group size distribution

Social balance emerges 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

Self-consistency equation

1. Simulated annealing 

 \(A_{ij} \propto k_i k_j\) is proportional to degree \(k_i\)

2. Mean-field approximation

 \(m^{(k)} = \sum_{i \in group \ of \ size \ k} s_i\) -  average opinion vector of a group of size \(k\)

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

Zero inter-group degree

Theory

MC simulation

Application online multiplayer game PARDUS

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