# An algebraic analysis of the graph modularity

Dario Fasino,
Francesco Tudisco,

**SIAM J. Matrix Analysis and Applications**,
35 :
997--1018
(2014)

### Abstract

One of the most relevant tasks in network analysis is the detection of
community structures, or clustering. Most popular techniques for community
detection are based on the maximization of a quality function called
modularity, which in turn is based upon particular quadratic forms associated
to a real symmetric modularity matrix $M$, defined in terms of the adjacency
matrix and a rank one null model matrix. That matrix could be posed inside the
set of relevant matrices involved in graph theory, alongside adjacency,
incidence and Laplacian matrices. This is the reason we propose a graph
analysis based on the algebraic and spectral properties of such matrix. In
particular, we propose a nodal domain theorem for the eigenvectors of $M$; we
point out several relations occurring between graph’s communities and
nonnegative eigenvalues of $M$; and we derive a Cheeger-type inequality for the
graph optimal modularity.

### Please cite this work as:

```
@article{fasino2014algebraic,
title={An algebraic analysis of the graph modularity},
author={Fasino, Dario and Tudisco, Francesco},
journal={SIAM Journal on Matrix Analysis and Applications},
volume={35},
number={3},
pages={997--1018},
year={2014},
publisher={SIAM}
}
```

**Links:**
doi
arxiv

Keywords:
Graph partitioning
Community detection
Nodal domains
graph modularity
Spectral clustering
networks
cheeger inequality