# Loop-erased partitioning of a graph: mean-field analysis

## TL;DR

This paper analyzes a graph partitioning method based on loop-erased random walks, revealing phase transitions in community detection and providing insights into the macroscopic structure of complex networks.

## Contribution

It introduces a novel loop-erased random walk-based partitioning approach and characterizes its phase transition behavior in community detection.

## Key findings

- Derived an interaction potential for vertex pairs based on non-membership probability.
- Computed the potential and its scaling limits on complete and non-homogeneous graphs.
- Identified a phase transition in community detectability depending on parameters.

## Abstract

We consider a random partition of the vertex set of an arbitrary graph that can be sampled using loop-erased random walks stopped at a random independent exponential time of parameter $q>0$, that we see as a tuning parameter.The related random blocks tend to cluster nodes visited by the random walk on time scale $1/q$. We explore the emerging macroscopic structure by analyzing 2-point correlations. To this aim, it is defined an interaction potential between pair of vertices, as the probability that they do not belong to the same block of the random partition. This interaction potential can be seen as an affinity measure for ``densely connected nodes'' and capture well-separated regions in network models presenting non-homogeneous landscapes. In this spirit, we compute this potential and its scaling limits on a complete graph and on a non-homogeneous weighted version with community structures. For the latter geometry we show a phase-transition for ``community detectability'' as a function of the tuning parameter and the edge weights.