SVD Incidence Centrality: A Unified Spectral Framework for Node and Edge Analysis in Directed Networks and Hypergraphs

TL;DR

This paper introduces a spectral framework based on SVD of the incidence matrix for analyzing node and edge importance in directed networks and hypergraphs, providing dense, consistent rankings that preserve directionality.

Contribution

It presents a unified spectral approach using pseudoinverse of Hodge Laplacians for centrality, overcoming sparsity and symmetry limitations of traditional measures.

Findings

Effective in real-world networks across domains

Produces dense, well-resolved centrality rankings

Preserves directional information naturally

Abstract

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.