  We investigate the widely encountered problem of detecting communities in
multiplex networks, such as social networks, with an unknown arbitrary
heterogeneous structure. To improve detectability, we propose a generative
model that leverages the multiplicity of a single community in multiple layers,
with no prior assumption on the relation of communities among different layers.
Our model relies on a novel idea of incorporating a large set of generic
localized community label constraints across the layers, in conjunction with
the celebrated Stochastic Block Model (SBM) in each layer. Accordingly, we
build a probabilistic graphical model over the entire multiplex network by
treating the constraints as Bayesian priors. We mathematically prove that these
constraints/priors promote existence of identical communities across layers
without introducing further correlation between individual communities. The
constraints are further tailored to render a sparse graphical model and the
numerically efficient Belief Propagation algorithm is subsequently employed. We
further demonstrate by numerical experiments that in the presence of consistent
communities between different layers, consistent communities are matched, and
the detectability is improved over a single layer. We compare our model with a
"correlated model" which exploits the prior knowledge of community correlation
between layers. Similar detectability improvement is obtained under such a
correlation, even though our model relies on much milder assumptions than the
correlated model. Our model even shows a better detection performance over a
certain correlation and signal to noise ratio (SNR) range. In the absence of
community correlation, the correlation model naturally fails, while ours
maintains its performance.
