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 \textit{module} and a set that maximizes Q is thus
called \textit{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 of M. Thus we extend the
work made on nonlinear Laplacians, by proposing a computational scheme, named
\textit{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