  Datasets consisting of a network and covariates associated with its vertices
have become ubiquitous. One problem pertaining to this type of data is to
identify information unique to the network, information unique to the vertex
covariates and information that is shared between the network and the vertex
covariates. Existing techniques for network data and vertex covariates focus on
capturing structure that is shared but are usually not able to differentiate
structure that is unique to each dataset. This work formulates a low-rank model
that simultaneously captures joint and individual information in network data
with vertex covariates. A two-step estimation procedure is proposed, composed
of an efficient spectral method followed by a refinement optimization step.
Theoretically, we show that the spectral method is able to consistently recover
the joint and individual components under a general signal-plus-noise model.
  Simulations and real data examples demonstrate the ability of the methods to
recover accurate and interpretable components. In particular, the application
of the methodology to a food trade network between countries with economic,
developmental and geographical country-level indicators as covariates yields
joint and individual factors that explain the trading patterns.
