  Latent space models are effective tools for statistical modeling and
exploration of network data. These models can effectively model real world
network characteristics such as degree heterogeneity, transitivity, homophily,
etc. Due to their close connection to generalized linear models, it is also
natural to incorporate covariate information in them. The current paper
presents two universal fitting algorithms for networks with edge covariates:
one based on nuclear norm penalization and the other based on projected
gradient descent. Both algorithms are motivated by maximizing likelihood for a
special class of inner-product models while working simultaneously for a wide
range of different latent space models, such as distance models, which allow
latent vectors to affect edge formation in flexible ways. These fitting
methods, especially the one based on projected gradient descent, are fast and
scalable to large networks. We obtain their rates of convergence for both
inner-product models and beyond. The effectiveness of the modeling approach and
fitting algorithms is demonstrated on five real world network datasets for
different statistical tasks, including community detection with and without
edge covariates, and network assisted learning.
