  Network data are often sampled with auxiliary information or collected
through the observation of a complex system over time, leading to multiple
network snapshots indexed by a continuous variable. Many methods in statistical
network analysis are traditionally designed for a single network, and can be
applied to an aggregated network in this setting, but that approach can miss
important functional structure. Here we develop an approach to estimating the
expected network explicitly as a function of a continuous index, be it time or
another indexing variable. We parameterize the network expectation through low
dimensional latent processes, whose components we represent with a fixed,
finite-dimensional functional basis. We derive a gradient descent estimation
algorithm, establish theoretical guarantees for recovery of the low dimensional
structure, compare our method to competitors, and apply it to a data set of
international political interactions over time, showing our proposed method to
adapt well to data, outperform competitors, and provide interpretable and
meaningful results.
