# The Geometry of Community Detection via the MMSE Matrix

## TL;DR

This paper introduces a geometric framework for community detection in networks with variable community sizes, using an effective signal-to-noise ratio matrix to characterize detection limits and improve understanding of real-world network behaviors.

## Contribution

It extends existing models by incorporating community variability and develops a matrix-based geometric approach to analyze detection limits, generalizing previous scalar SNR concepts.

## Key findings

- Effective SNR matrix characterizes community detectability.
- Explicit formulas for mutual information and MSE bounds.
- Numerical simulations validate theoretical predictions.

## Abstract

The information-theoretic limits of community detection have been studied extensively for network models with high levels of symmetry or homogeneity. The contribution of this paper is to study a broader class of network models that allow for variability in the sizes and behaviors of the different communities, and thus better reflect the behaviors observed in real-world networks. Our results show that the ability to detect communities can be described succinctly in terms of a matrix of effective signal-to-noise ratios that provides a geometrical representation of the relationships between the different communities. This characterization follows from a matrix version of the I-MMSE relationship and generalizes the concept of an effective scalar signal-to-noise ratio introduced in previous work. We provide explicit formulas for the asymptotic per-node mutual information and upper bounds on the minimum mean-squared error. The theoretical results are supported by numerical simulations.