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.
A fundamental challenge in network science is developing centrality measures that effectively handle both edge importance and directional relationships in complex networks. While traditional approaches excel for undirected graphs, they face limitations with directional information preservation. Currentflow centrality, eigenvector centrality, communicability centrality, and related measures require graph symmetrization for directed networks, potentially losing the asymmetric relationships that define social influence, biological regulation, and information flow systems [Kleinberg, 1999, Freeman, 1978]. Recent work on spectral centralities for directed and higher-order networks further highlights that symmetrization can obscure essential structural information encoded in orientation and flow [Contreras-Aso et al., 2024].
This limitation reveals a theoretical gap. Existing centrality frameworks treat vertex and edge importance as separate problems, requiring distinct computational approaches and often yielding inconsistent results. The lack of mathematical unity between vertex and edge measures can hinder coherent structural interpretation in systems where directionality, hierarchy, and flow are fundamental, such as regulatory, communication, and infrastructure networks [Bonacich, 1987, Newman andGirvan, 2004]. Moreover, edge-centric measures have proved crucial in predictive tasks [Franco et al., 2026].
We propose a unified spectral framework based on Hodge theory. We ground network centrality in the Singular Value Decomposition (SVD) of the incidence matrix, where directional information is naturally preserved through oriented boundary operators. This approach simultaneously derives vertex and edge centralities from the same mathematical foundation, ensuring consistency while maintaining theoretical rigor through connections to electrical network theory and algebraic topology. Related Hodge-theoretic formulations have recently been shown to provide interpretable decompositions of network flows into gradient, cyclic, and harmonic components, emphasizing the importance of orientationpreserving operators for network analysis [Schaub et al., 2020].
For undirected networks, the proposed SVD vertex centrality rankings equal current-flow closeness rankings-validating the electrical network foundations. However, this equivalence breaks down for directed networks, precisely where traditional measures face challenges. SVD centrality preserves directional information through spectral decomposition, enabling natural hub/authority analysis without graph symmetrization and providing a principled alternative to existing directed spectral constructions [Fanuel and Suykens, 2019]. We also provide a natural and direct extension of the framework to hypergraphs.
Our contributions span three key dimensions. Theoretically, we establish rigorous connections between centrality measures and physical observables through electrical resistance analogies and energy landscape interpretations derived from Hodge theory. Methodologically, we develop a unified spectral framework that generates mathematically consistent vertex and edge centralities from a single incidence matrix decomposition. Empirically, we validate the framework across diverse directed network domains, including social, biological, and infrastructure systems, demonstrating how orientationpreserving spectral centrality reveals structural information hidden by symmetrized approaches.
This work explores spectral centrality as an approach for directed networks and hypergraphs analysis, connecting mathematical foundations with practical network science applications.
Centrality measures have long been a key area of research in network science, quantifying the importance of vertices, edges, and hyperedges. Traditional vertex-based approaches like degree centrality [Freeman, 1978] evaluate the importance of a vertex based on its number of connections, a simple yet effective measure. However, such local approaches are insufficient to capture the broader influence of a vertex in the network. More sophisticated methods, such as eigenvector centrality [Bonacich, 1987] and PageRank [Brin, 1998], leverage the network’s global structure. In particular, eigenvector centrality assigns higher scores to vertices connected to other influential vertices, capturing a more nuanced picture of importance. Similarly, PageRank modifies this idea by incorporating directed edges, making it particularly effective for ranking web pages. However, PageRank primarily focuses on vertex centrality and does not directly address edge importance or hypergraph structures [Kucharczuk et al., 2022].
Edge centrality measures like edge betweenness [Newman andGirvan, 2004, Lu andZhang, 2013] emphasize the role of edges that lie on many shortest paths between vertices, highlighting connections in the network and also have hypergraph extensions [Lee and Kim, 2017]. However, betweenness centrality has limitations
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