Representing Higher-Order Networks: A Survey of Graph-Based Frameworks

Higher-order models (e.g., hypergraph neural networks) often outperform lower-order baselines on hypergraph learning benchmarks, and their advantages are commonly attributed to their ability to exploit higher-order information. However, better performance alone does not establish this explanation. We therefore ask: Do higher-order models win for higher-order reasons? To investigate this question, we introduce a controlled performance-attribution framework that perturbs higher-order information while preserving the lower-order, i.e., pairwise, information. Across 25 commonly used hypergraph learning benchmarks spanning three tasks, we frequently observe an intriguing pattern: higher-order models originally outperform lower-order baselines, yet retain most of their advantage after perturbation. This suggests that much of the observed advantage remains achievable without the higher-order information. We then investigate potential lower-order explanations for these remaining gaps. We find that simple additions to a lower-order baseline, e.g., richer pairwise weighting, more steps of pairwise feature propagation, and normalization, reduce the remaining performance gaps, supporting lower-order explanations for part of the observed advantage. Our analysis calls for the hypergraph learning community to rethink performance attribution by distinguishing performance gains from their explanations, adopt stronger lower-order baselines, and use suitable benchmarks that better test the value of higher-order information.

Neural networks are fundamental tools of modern machine learning. The standard paradigm assumes binary interactions (across feedforward linear passes) between inter-tangled units, organized in sequential layers. Generalized architectures have been also designed that move beyond pairwise interactions, so as to account for higher-order couplings among computing neurons. Higher-order networks are however usually deployed as augmented graph neural networks (GNNs), and, as such, prove solely advantageous in contexts where the input exhibits an explicit hypergraph structure. Here, we present Spectral Higher-Order Neural Networks (SHONNs), a new algorithmic strategy to incorporate higher-order interactions in general-purpose, feedforward, network structures. SHONNs leverages a reformulation of the model in terms of spectral attributes. This allows to mitigate the common stability and parameter scaling problems that come along weighted, higher-order, forward propagations.

This book presents a comprehensive and systematic survey of graph theory under uncertainty, with particular emphasis on the unifying role of the uncertain graph framework. It reviews fundamental concepts, structural properties, graph classes, and graph parameters within fuzzy, neutrosophic, and related models, while also introducing a wide range of extensions such as uncertain digraphs, hypergraphs, superhypergraphs, and dynamic graphs. In addition to theoretical developments, the book explores practical applications, including uncertain molecular graphs, decision-making systems, graph neural networks, knowledge graphs, and cognitive maps. By organizing diverse uncertainty-aware graph models within a common perspective, this work provides a coherent framework for understanding their relationships, capabilities, and applications in complex systems.