# Detection of an arbitrary number of communities in a block spin Ising model

## TL;DR

The paper introduces a method to detect multiple communities in a block spin Ising model, which can be used to understand complex systems like voting patterns and biological communities.

## Contribution

The novelty lies in solving community detection for any number of groups with varying sizes and interactions, extending beyond previous two-group models.

## Key findings

- The model can identify any number of groups with different sizes and interactions.
- An explicit algorithm is provided to reconstruct the model structure from empirical correlations.
- Applications include real-world voting data and biological communities.

## Abstract

We study the problem of community detection in a general version of the block spin Ising model featuring M groups, a model inspired by the Curie-Weiss model of ferromagnetism in statistical mechanics. We solve the general problem of identifying any number of groups with any possible coupling constants. Up to now, the problem was only solved for the specific situation with two groups of identical size and identical interactions, see [1, 2]. Our results can be applied to the most realistic situations, in which there are many groups of different sizes and different interactions. In addition, we give an explicit algorithm that permits the reconstruction of the structure of the model from a sample of observations based on the comparison of empirical correlations of the spin variables, thus unveiling easy applications of the model to real-world voting data and communities in biology.