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Latent space models are effective tools for statistical modeling andexploration of network data. These models can effectively model real worldnetwork characteristics such as degree heterogeneity, transitivity, homophily,etc. Due to their close connection to generalized linear models, it is alsonatural to incorporate covariate information in them. The current paperpresents two universal fitting algorithms for networks with edge covariates:one based on nuclear norm penalization and the other based on projectedgradient descent. Both algorithms are motivated by maximizing likelihood for aspecial class of inner-product models while working simultaneously for a widerange of different latent space models, such as distance models, which allowlatent vectors to affect edge formation in flexible ways. These fittingmethods, especially the one based on projected gradient descent, are fast andscalable to large networks. We obtain their rates of convergence for bothinner-product models and beyond. The effectiveness of the modeling approach andfitting algorithms is demonstrated on five real world network datasets fordifferent statistical tasks, including community detection with and withoutedge covariates, and network assisted learning.

Exploration of Large Networks with Covariates via Fast and Universal Latent Space Model Fitting