# Step-by-Step Community Detection in Volume-Regular Graphs

## TL;DR

This paper extends spectral community detection methods to volume-regular graphs, showing that under certain spectral gap conditions, the community structure can be efficiently recovered without explicit eigenvector computation.

## Contribution

It generalizes previous approaches to a broader class of graphs, establishing a connection between volume regularity and Markov chain lumpability for community detection.

## Key findings

- Community structure can be recovered in logarithmic time.
- The class of volume-regular graphs admits stepwise eigenvectors.
- Spectral gap conditions ensure successful recovery.

## Abstract

Spectral techniques have proved amongst the most effective approaches to graph clustering. However, in general they require explicit computation of the main eigenvectors of a suitable matrix (usually the Laplacian matrix of the graph). Recent work (e.g., Becchetti et al., SODA 2017) suggests that observing the temporal evolution of the power method applied to an initial random vector may, at least in some cases, provide enough information on the space spanned by the first two eigenvectors, so as to allow recovery of a hidden partition without explicit eigenvector computations. While the results of Becchetti et al. apply to perfectly balanced partitions and/or graphs that exhibit very strong forms of regularity, we extend their approach to graphs containing a hidden $k$ partition and characterized by a milder form of volume-regularity. We show that the class of $k$-volume-regular graphs is the largest class of undirected (possibly weighted) graphs whose transition matrix admits $k$ "stepwise" eigenvectors (i.e., vectors that are constant over each set of the hidden partition). To obtain this result, we highlight a connection between volume regularity and lumpability of Markov chains. Moreover, we prove that if the stepwise eigenvectors are those associated to the first $k$ eigenvalues and the gap between the $k$-th and the ($k$+1)-th eigenvalues is sufficiently large, the averaging dynamics of Becchetti et al. recovers the underlying community structure of the graph in logarithmic time, with high probability.