Community detection in networks via nonlinear modularity eigenvectors

Abstract

Revealing a community structure in a network or dataset is a central problem arising in many scientific areas. The modularity function $Q$ is an established measure quantifying the quality of a community, being identified as a set of nodes having high modularity. In our terminology, a set of nodes with positive modularity is called a module and a set that maximizes $Q$ is thus called leading module. Finding a leading module in a network is an important task, however the dimension of real-world problems makes the maximization of $Q$ unfeasible. This poses the need of approximation techniques which are typically based on a linear relaxation of $Q$, induced by the spectrum of the modularity matrix $M$. In this work we propose a nonlinear relaxation which is instead based on the spectrum of a nonlinear modularity operator $\mathcal M$. We show that extremal eigenvalues of $\mathcal M$ provide an exact relaxation of the modularity measure $Q$, however at the price of being more challenging to be computed than those of $M$. Thus we extend the work made on nonlinear Laplacians, by proposing a computational scheme, named generalized RatioDCA, to address such extremal eigenvalues. We show monotonic ascent and convergence of the method. We finally apply the new method to several synthetic and real-world data sets, showing both effectiveness of the model and performance of the method.

Please cite this work as:

@article{tudisco2018community,
  title={Community detection in networks via nonlinear modularity eigenvectors},
  author={Tudisco, Francesco and Mercado, Pedro and Hein, Matthias},
  journal={SIAM J. Applied Mathematics},
  volume={78},
  pages={2393--2419},
  year={2018}
}

Keywords: graph modularity nonlinear eigenvalues community detection spectral clustering networks