  In this article, we study spectral methods for community detection based on $
\alpha$-parametrized normalized modularity matrix hereafter called $ {\bf
L}_\alpha $ in heterogeneous graph models. We show, in a regime where community
detection is not asymptotically trivial, that $ {\bf L}_\alpha $ can be well
approximated by a more tractable random matrix which falls in the family of
spiked random matrices. The analysis of this equivalent spiked random matrix
allows us to improve spectral methods for community detection and assess their
performances in the regime under study. In particular, we prove the existence
of an optimal value $ \alpha_{\rm opt} $ of the parameter $ \alpha $ for which
the detection of communities is best ensured and we provide an on-line
estimation of $ \alpha_{\rm opt} $ only based on the knowledge of the graph
adjacency matrix. Unlike classical spectral methods for community detection
where clustering is performed on the eigenvectors associated with extreme
eigenvalues, we show through our theoretical analysis that a regularization
should instead be performed on those eigenvectors prior to clustering in
heterogeneous graphs. Finally, through a deeper study of the regularized
eigenvectors used for clustering, we assess the performances of our new
algorithm for community detection. Numerical simulations in the course of the
article show that our methods outperform state-of-the-art spectral methods on
dense heterogeneous graphs.
