  Latent space models are frequently used for modeling single-layer networks
and include many popular special cases, such as the stochastic block model and
the random dot product graph. However, they are not well-developed for more
complex network structures, which are becoming increasingly common in practice.
Here we propose a new latent space model for multiplex networks: multiple,
heterogeneous networks observed on a shared node set. Multiplex networks can
represent a network sample with shared node labels, a network evolving over
time, or a network with multiple types of edges. The key feature of our model
is that it learns from data how much of the network structure is shared between
layers and pools information across layers as appropriate. We establish
identifiability, develop a fitting procedure using convex optimization in
combination with a nuclear norm penalty, and prove a guarantee of recovery for
the latent positions as long as there is sufficient separation between the
shared and the individual latent subspaces. We compare the model to competing
methods in the literature on simulated networks and on a multiplex network
describing the worldwide trade of agricultural products.
