SVD Incidence Centrality: A Unified Spectral Framework for Node and Edge Analysis in Directed Networks and Hypergraphs
Identifying influential nodes and edges in directed networks remains a fundamental challenge across domains from social influence to biological regulation. Most existing centrality measures face a critical limitation: they either discard directional information through symmetrization or produce sparse, implementation-dependent rankings that obscure structural importance. We introduce a unified spectral framework for centrality analysis in directed networks grounded in the Singular value decomposition of the incidence matrix. The proposed approach derives both vertex and edge centralities via the pseudoinverse of Hodge Laplacians, yielding dense and well-resolved rankings that overcome the sparsity limitations commonly observed in betweenness centrality for directed graphs. Unlike traditional measures that require graph symmetrization, our framework naturally preserves directional information, enabling principled hub/authority analysis while maintaining mathematical consistency through spectral graph theory. The method extends naturally to hypergraphs through the same mathematical foundation. Experimental validation on real-world networks demonstrates the framework's effectiveness across diverse domains where traditional centrality measures encounter limitations due to implementation dependencies and sparse outputs.