  Spectral estimators have been broadly applied to statistical network analysis
but they do not incorporate the likelihood information of the network sampling
model. This paper proposes a novel surrogate likelihood function for
statistical inference of a class of popular network models referred to as
random dot product graphs. In contrast to the structurally complicated exact
likelihood function, the surrogate likelihood function has a separable
structure and is log-concave yet approximates the exact likelihood function
well. From the frequentist perspective, we study the maximum surrogate
likelihood estimator and establish the accompanying theory. We show its
existence, uniqueness, large sample properties, and that it improves upon the
baseline spectral estimator with a smaller sum of squared errors. A
computationally convenient stochastic gradient descent algorithm is designed
for finding the maximum surrogate likelihood estimator in practice. From the
Bayesian perspective, we establish the Bernstein--von Mises theorem of the
posterior distribution with the surrogate likelihood function and show that the
resulting credible sets have the correct frequentist coverage. The empirical
performance of the proposed surrogate-likelihood-based methods is validated
through the analyses of simulation examples and a real-world Wikipedia graph
dataset. An R package implementing the proposed computation algorithms is
publicly available at https://fangzheng-xie.github.io./publication/ .
