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

2026-02-17

AI Summary

Existing centrality measures for directed networks often discard directional information through symmetrization or yield sparse, implementation-dependent rankings that inadequately capture the structural importance of nodes and edges. This work proposes a unified spectral framework that integrates singular value decomposition (SVD) of the incidence matrix with the pseudoinverse of the Hodge Laplacian—a novel combination that enables joint computation of node and edge centralities in both directed graphs and hypergraphs. By preserving the original directional structure, the method produces dense, high-resolution rankings, effectively overcoming the sparsity and implementation sensitivity inherent in conventional metrics. Extensive experiments on multiple real-world networks demonstrate the approach’s effectiveness and robustness.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Computational social science
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• directed networks
• centrality
• influential nodes
• influential edges
• sparsity

Innovation

• Singular Value Decomposition
• Incidence Centrality
• Directed Networks
• Hodge Laplacian
• Hypergraphs