Interpretable Network Representation Learning with Principal Component Analysis

TL;DR

This paper introduces PCAN and sPCAN, interpretable algorithms for low-dimensional network representations based on subgraph counts, enabling visualization, clustering, and classification of network data with theoretical guarantees.

Contribution

The paper proposes novel interpretable network embedding methods, PCAN and sPCAN, with theoretical analysis and practical algorithms for network data analysis.

Findings

sPCAN enjoys a central limit theorem under certain regimes

Population embeddings of PCAN and sPCAN are equivalent

Algorithms effectively classify and visualize real-world network data

Abstract

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/.