  We investigate signed networks with community structure with respect to their
spectrum and their evolution under a dynamical model of structural balance, a
prominent theory of signed social networks. The spectrum of the adjacency
matrix generated by a stochastic block model with two equal size communities
shows detectability transitions in which the community structure becomes
manifest when its signal eigenvalue appears outside the main spectral band. The
spectrum also exhibits "sociality" transitions involving the homogeneous
structure representing the average tie value. We derive expressions for the
eigenvalues associated with the community and homogeneous structure as well as
the transition boundaries, all in good agreement with numerical results. Using
the stochastically-generated networks as initial conditions for a simple model
of structural balance dynamics yields three outcome regimes: two hostile
factions that correspond with the initial communities, two hostile factions
uncorrelated with those communities, and a single harmonious faction of all
nodes. The detectability transition predicts the boundary between the
assortative and mixed two-faction states and the sociality transition predicts
that between the mixed and harmonious states. Our results may yield insight
into the dynamics of cooperation and conflict among actors with distinct social
identities.
