🤖 AI Summary
This work addresses the representation learning challenge for higher-order networks (e.g., hypergraphs). We propose a unified spectral-moment-based framework: higher-order graphs are decomposed by edge order into uniform hypergraphs; order-specific random walk transition matrices are defined; and low-order moments of their spectral densities are extracted as joint features. This constitutes the first systematic extension of spectral moment theory to higher-order networks, establishing analytical connections between spectral moments and higher-order structural properties—such as higher-order degree distributions and hyperedge clustering coefficients—thereby overcoming the expressive limitations of conventional pairwise graph spectral methods. By fusing multi-order spectral moments, our representation achieves significant performance gains over state-of-the-art methods on higher-order graph classification tasks. Experiments demonstrate that the proposed representation accurately captures walk return probabilities and diverse higher-order topological characteristics.
📝 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.