Representing Higher-Order Networks with Spectral Moments

TL;DR

This paper introduces a spectral moment-based method for representing higher-order networks, capturing their structural properties and improving graph classification performance.

Contribution

It generalizes spectral analysis to higher-order networks using spectral moments from random walk transition matrices, a novel approach.

Findings

Spectral moments effectively encode higher-order network properties.

The proposed method outperforms existing techniques in graph classification.

Spectral moments relate to key network features like degree and clustering.

Abstract

The spectral properties of traditional (dyadic) graphs, where an edge connects exactly two vertices, are widely studied in different applications. These spectral properties are closely connected to the structural properties of dyadic graphs. We generalize such connections and characterize higher-order networks by their spectral information. We first split the higher-order graphs by their ``edge orders" into several uniform hypergraphs. For each uniform hypergraph, we extract the corresponding spectral information from the transition matrices of carefully designed random walks. From each spectrum, we compute the first few spectral moments and use all such spectral moments across different ``edge orders" as the higher-order graph representation. We show that these moments not only clearly indicate the return probabilities of random walks but are also closely related to various higher-order network properties such as degree distribution and clustering coefficient. Extensive experiments show the utility of this new representation in various settings. For instance, graph classification on higher-order graphs shows that this representation significantly outperforms other techniques.