Community detection in networks via nonlinear modularity eigenvectors

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 M. We show that extremal eigenvalues of M provide an exact relaxation of the modularity measure Q, however at the price of being more challenging to be computed than those ofM . 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.