  We are interested in multilayer graph clustering, which aims at dividing the
graph nodes into categories or communities. To do so, we propose to learn a
clustering-friendly embedding of the graph nodes by solving an optimization
problem that involves a fidelity term to the layers of a given multilayer
graph, and a regularization on the (single-layer) graph induced by the
embedding. The fidelity term uses the contrastive loss to properly aggregate
the observed layers into a representative embedding. The regularization pushes
for a sparse and community-aware graph, and it is based on a measure of graph
sparsification called "effective resistance", coupled with a penalization of
the first few eigenvalues of the representative graph Laplacian matrix to favor
the formation of communities. The proposed optimization problem is nonconvex
but fully differentiable, and thus can be solved via the descent gradient
method. Experiments show that our method leads to a significant improvement
w.r.t. state-of-the-art multilayer graph clustering algorithms.
