Homophily-Based Social Group Formation

in a Spin Glass Self-Assembly Framework

in collaboration with

Simon Lindner

Tuan Pham

Rudolf Hanel

Stefan Thurner

Journal Ref.: Phys. Rev. Lett. 130 (2023) 057401

Slides available at: www.slides.com/jankorbel

Graphical abstract

1. soft-matter self-assembly

2. condensed matter spin glasses

3. social group formation

Paper flow

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$

Applications to self-assembly

2. Spin-glass based opinion dynamics

Many opinion dynamics systems follow two basic concepts:

Homophily - people tend to be friends with peers with similar opinions
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

Approximations used in the model

1. Simulated annealing

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

2. Mean-field approximation

- We use the mean-field approximation of the Hamiltonian

$\Rightarrow$ $m^{(k)} = \sum_{i \in group \ of \ size \ k} s_i$ - 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

Results for zero inter-group degree

Theory

MC simulation