Advances in Spectral Graph and Hypergraph Learning: Towards More Expressive Laplacian Matrices

Stefano Coniglio (Università di Bergamo)

Abstract: Graphs and hypergraphs have gained increasing attention in deep learning due to their effectiveness in modeling complex relationships across various domains, including chemistry, biology, and social network analysis. Hypergraphs, in particular, are crucial to capture real-world phenomena that involve polyadic (many-to-many) relationships, extending beyond the diadic (pairwise) connections typically represented by (standard) graphs.
In this presentation, we will provide an overview of the Graph Neural Network (GNN) and Graph Convolutional (Neural) Network (GCN) literature, with a focus on the construction of spectral graph-convolutional operators that are grounded in graph signal theory. Next, we will introduce some recent advancements we made in this area, achieved through a series of novel graph Laplacian matrices of increasing generality. Specifically, we will present:
– The Sign-Magnetic Laplacian and SigMaNet, a generalized GCN capable of processing both directed and undirected graphs with unrestricted edge weights.
– A quaternion-valued extension of the Sign-Magnetic Laplacian, designed for graphs with digons (antiparallel edges) and asymmetric weights, along with its corresponding GCN, QuaterGCN.
– The Generalized Directed Laplacian and GeDi-HNN, a Hypergraph Neural Network (HNN) tailored for learning tasks where hyperedge directionality is essential.
– The Directed Line Graph Laplacian and DLGNet, an HNN designed for chemical-reaction classification, where input directed hypergraphs are transformed into directed line graphs with complex-valued edge weights.