Detection of an Arbitrary Number of Communities in a Block Spin Ising Model

Gabor Toth; Jos\'e Luis P\'erez; Miguel Ballesteros; Rams\'es H. Mena

Detection of an Arbitrary Number of Communities in a Block Spin Ising Model

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arxiv 2311.18112 v1 pith:IFGPEX4T submitted 2023-11-29 math.PR math.STstat.TH

Detection of an Arbitrary Number of Communities in a Block Spin Ising Model

Miguel Ballesteros , Rams\'es H. Mena , Jos\'e Luis P\'erez , Gabor Toth This is my paper

classification math.PR math.STstat.TH

keywords modelgroupsproblemspinblockcommunitiesdetectiondifferent

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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. 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.

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