Three hypergraph eigenvector centralities (1807.09644v3)

Abstract: Eigenvector centrality is a standard network analysis tool for determining the importance of (or ranking of) entities in a connected system that is represented by a graph. However, many complex systems and datasets have natural multi-way interactions that are more faithfully modeled by a hypergraph. Here we extend the notion of graph eigenvector centrality to uniform hypergraphs. Traditional graph eigenvector centralities are given by a positive eigenvector of the adjacency matrix, which is guaranteed to exist by the Perron-Frobenius theorem under some mild conditions. The natural representation of a hypergraph is a hypermatrix (colloquially, a tensor). Using recently established Perron-Frobenius theory for tensors, we develop three tensor eigenvectors centralities for hypergraphs, each with different interpretations. We show that these centralities can reveal different information on real-world data by analyzing hypergraphs constructed from n-gram frequencies, co-tagging on stack exchange, and drug combinations observed in patient emergency room visits.

• The paper introduces three tensor-based hypergraph centralities (CEC, ZEC, and HEC) that extend traditional eigenvector centrality to multi-node interactions.
• Methodologies leverage Perron-Frobenius theory and tensor algebra to formulate and compute centralities for complex datasets.
• Computational experiments on real-world data reveal distinct patterns of connectivity and interaction, underscoring each centrality's unique advantages.

An Overview of Hypergraph Eigenvector Centralities

The paper presents an extension of the well-established concept of eigenvector centrality from traditional graphs to uniform hypergraphs. This advancement addresses the need to accurately represent and analyze multi-way interactions observed in complex systems using hypergraphs. The paper introduces three tensor-based centralities for hypergraphs: Clique motif Eigenvector Centrality (CEC), $Z$-eigenvector centrality (ZEC), and $H$-eigenvector centrality (HEC), each derived from different mathematical formulations and offering distinct insights.

Background and Motivation

Eigenvector centrality, traditionally applied to graphs, determines the importance of vertices based on the connectivity to other highly connected nodes, leveraging the Perron-Frobenius theorem to ensure a unique positive eigenvector of an adjacency matrix. However, this binary relationship assumption doesn't suffice for systems with multi-node interactions, such as those found in social networks involving group communication or in chemical reactions with multiple interacting substances. Hypergraphs naturally model these multi-way interactions, but analogous centrality measures were lacking.

Hypergraph Representations and Centralities

Hypergraphs are represented by hypermatrices (or tensors), which enable the extension of eigenvector centrality to multi-way interactions. The paper employs recent developments in Perron-Frobenius theory for tensors to formulate three hypergraph centralities:

1. Clique Motif Eigenvector Centrality (CEC): Extends traditional eigenvector calculations, considering the number of hyperedges containing each pair of nodes. This centrality is accessible via matrix representations, providing a more straightforward computation compared to tensor formulations.
2. $Z$-Eigenvector Centrality (ZEC): Based on a non-linear system where the centrality is proportional to products of neighboring node centralities, inspired by $Z$-eigenpair theory. This measure incorporates more complex interactions but suffers from computational and uniqueness challenges.
3. $H$-Eigenvector Centrality (HEC): Adopts a quadratic scaling approach, where each node's centrality squared is proportional to the product of neighboring centralities. The HEC benefits from the existence of efficient computation algorithms and the assurance of a unique solution.

Computational Analysis on Real-world Data

The paper applies these centralities to datasets including $n$-grams frequencies, Stack Exchange co-tagging, and emergency room drug combinations, revealing the distinct information provided by each method. The CEC tends to favor nodes involved in numerous pairwise interactions, while ZEC and HEC bring out subtleties in multi-node interactions.

For instance, in $n$-grams analysis, while stop words dominate the CEC rankings, ZEC identifies additional significant terms, emphasizing its potential to highlight different aspects of the dataset. Similarly, in the Ask Ubuntu dataset, ZEC's emphasis on Windows-related tags alongside Ubuntu tags suggests its potential to unearth competitive or complementary relationships not as prominent in other centralities.

Theoretical and Practical Implications

The research has notable implications for advancing network science:

• Theoretical Contribution: Provides new mathematical bases and frameworks for analyzing hypergraphs with tensor algebra. It underscores the limitations of prior graph centralities in multi-way edge contexts and proposes concrete alternatives.
• Practical Applications: Facilitates better understanding and feature extraction for systems with inherent multi-node interactions, enhancing machine learning and predictive analyses in these domains.

Considerations and Future Directions

Despite significant advances, questions remain about scalability and computational resource requirements for computing these centralities in particularly large or non-uniform hypergraphs. Potential extensions include exploring optimal hypergraph construction methodologies from diverse datasets, computing efficiencies, and leveraging machine learning to further refine hypergraph analytics. Future work might also investigate relationships among these centralities to converge on integrated measures that capture multilayered system structure accurately.

In summary, this foundational work on hypergraph centralities enriches the toolkit of network analysis for multi-dimensional datasets, laying groundwork for more nuanced exploration and interpretation of complex systems.