  Dynamic networks are commonly used in applications where relational data is
observed over time. Statistical models for such data should capture not only
the temporal dependencies between networks observed in time, but also the
structural dependencies among the nodes and edges in each network. As a
consequence, effectively making inference on dynamic networks is a
computationally challenging task, and many models established for dynamic
networks are intractable even for moderately sized networks. In this paper, we
propose and investigate a family of dynamic network models, known as
varying-coefficient exponential random graph models (VCERGMs), that
characterize the evolution of network topology through smoothly varying
parameters in an exponential family of distributions. The VCERGM provides an
interpretable dynamic network model that enables the inference of temporal
heterogeneity in a dynamic network. We establish how to fit the VCERGM through
maximum pseudo-likelihood techniques, and thus provide a computationally
tractable method for statistical inference of complex dynamic networks. We
furthermore devise a bootstrap hypothesis testing framework for testing the
temporal heterogeneity of an observed dynamic network sequence. We apply the
VCERGM to the US Congress co-voting network and a resting-state brain
connectivity case study and show that our method provides relevant and
interpretable patterns describing each data set. Comprehensive simulation
studies demonstrate the advantages of our proposed method over existing
methods.
