  We introduce a principled and theoretically sound spectral method for $k$-way
clustering in signed graphs, where the affinity measure between nodes takes
either positive or negative values. Our approach is motivated by social balance
theory, where the task of clustering aims to decompose the network into
disjoint groups, such that individuals within the same group are connected by
as many positive edges as possible, while individuals from different groups are
connected by as many negative edges as possible. Our algorithm relies on a
generalized eigenproblem formulation inspired by recent work on constrained
clustering. We provide theoretical guarantees for our approach in the setting
of a signed stochastic block model, by leveraging tools from matrix
perturbation theory and random matrix theory. An extensive set of numerical
experiments on both synthetic and real data shows that our approach compares
favorably with state-of-the-art methods for signed clustering, especially for
large number of clusters and sparse measurement graphs.
