  We study the problem of $k$-way clustering in signed graphs. Considerable
attention in recent years has been devoted to analyzing and modeling signed
graphs, where the affinity measure between nodes takes either positive or
negative values. Recently, Cucuringu et al. [CDGT 2019] proposed a spectral
method, namely SPONGE (Signed Positive over Negative Generalized Eigenproblem),
which casts the clustering task as a generalized eigenvalue problem optimizing
a suitably defined objective function. This approach is motivated by social
balance theory, where the clustering task aims to decompose a given 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 mainly connected by negative edges. Through extensive numerical
simulations, SPONGE was shown to achieve state-of-the-art empirical
performance. On the theoretical front, [CDGT 2019] analyzed SPONGE and the
popular Signed Laplacian method under the setting of a Signed Stochastic Block
Model (SSBM), for $k=2$ equal-sized clusters, in the regime where the graph is
moderately dense.
  In this work, we build on the results in [CDGT 2019] on two fronts for the
normalized versions of SPONGE and the Signed Laplacian. Firstly, for both
algorithms, we extend the theoretical analysis in [CDGT 2019] to the general
setting of $k \geq 2$ unequal-sized clusters in the moderately dense regime.
Secondly, we introduce regularized versions of both methods to handle sparse
graphs -- a regime where standard spectral methods underperform -- and provide
theoretical guarantees under the same SSBM model. To the best of our knowledge,
regularized spectral methods have so far not been considered in the setting of
clustering signed graphs. We complement our theoretical results with an
extensive set of numerical experiments on synthetic data.
