We consider the problem of interpretable network representation learning for samples of network-valued data. We propose the Principal Component Analysis for Networks (PCAN) algorithm to identify statistically meaningful low-dimensional representations of a network sample via subgraph count statistics. The PCAN procedure provides an interpretable framework for which one can readily visualize, explore, and formulate predictive models for network samples. We furthermore introduce a fast sampling-based algorithm, sPCAN, which is significantly more computationally efficient than its counterpart, but still enjoys advantages of interpretability. We investigate the relationship between these two methods and analyze their large-sample properties under the common regime where the sample of networks is a collection of kernel-based random graphs. We show that under this regime, the embeddings of the sPCAN method enjoy a central limit theorem and moreover that the population level embeddings of PCAN and sPCAN are equivalent. We assess PCAN's ability to visualize, cluster, and classify observations in network samples arising in nature, including functional connectivity network samples and dynamic networks describing the political co-voting habits of the U.S. Senate. Our analyses reveal that our proposed algorithm provides informative and discriminatory features describing the networks in each sample. The PCAN and sPCAN methods build on the current literature of network representation learning and set the stage for a new line of research in interpretable learning on network-valued data. Publicly available software for the PCAN and sPCAN methods are available at https://www.github.com/jihuilee/.

The goal in network state prediction (NSP) is to classify the global state (label) associated with features embedded in a graph. This graph structure encoding feature relationships is the key distinctive aspect of NSP compared to classical supervised learning. NSP arises in various applications: gene expression samples embedded in a protein-protein interaction (PPI) network, temporal snapshots of infrastructure or sensor networks, and fMRI coherence network samples from multiple subjects to name a few. Instances from these domains are typically ``wide'' (more features than samples), and thus, feature sub-selection is required for robust and generalizable prediction. How to best employ the network structure in order to learn succinct connected subgraphs encompassing the most discriminative features becomes a central challenge in NSP. Prior work employs connected subgraph sampling or graph smoothing within optimization frameworks, resulting in either large variance of quality or weak control over the connectivity of selected subgraphs. In this work we propose an optimization framework for discriminative subgraph learning (DSL) which simultaneously enforces (i) sparsity, (ii) connectivity and (iii) high discriminative power of the resulting subgraphs of features. Our optimization algorithm is a single-step solution for the NSP and the associated feature selection problem. It is rooted in the rich literature on maximal-margin optimization, spectral graph methods and sparse subspace self-representation. DSL simultaneously ensures solution interpretability and superior predictive power (up to 16% improvement in challenging instances compared to baselines), with execution times up to an hour for large instances.