  Network data are commonly collected in a variety of applications,
representing either directly measured or statistically inferred connections
between features of interest. In an increasing number of domains, these
networks are collected over time, such as interactions between users of a
social media platform on different days, or across multiple subjects, such as
in multi-subject studies of brain connectivity. When analyzing multiple large
networks, dimensionality reduction techniques are often used to embed networks
in a more tractable low-dimensional space. To this end, we develop a framework
for principal components analysis (PCA) on collections of networks via a
specialized tensor decomposition we term Semi-Symmetric Tensor PCA or SS-TPCA.
We derive computationally efficient algorithms for computing our proposed
SS-TPCA decomposition and establish statistical efficiency of our approach
under a standard low-rank signal plus noise model. Remarkably, we show that
SS-TPCA achieves the same estimation accuracy as classical matrix PCA, with
error proportional to the square root of the number of vertices in the network
and not the number of edges as might be expected. Our framework inherits many
of the strengths of classical PCA and is suitable for a wide range of
unsupervised learning tasks, including identifying principal networks,
isolating meaningful changepoints or outlying observations, and for
characterizing the "variability network" of the most varying edges. Finally, we
demonstrate the effectiveness of our proposal on simulated data and on an
example from empirical legal studies. The techniques used to establish our main
consistency results are surprisingly straightforward and may find use in a
variety of other network analysis problems.
