[Paper Review] A matrix analysis of different centrality measures.

Michele Benzi, Christine Klymko|arXiv (Cornell University)|Dec 23, 2013

Complex Network Analysis Techniques44 references22 citations

TL;DR

This paper provides a matrix-based analysis of node centrality measures—degree, eigenvector, Katz, and subgraph centrality—showing how parameterized measures like Katz and subgraph centrality interpolate between degree and eigenvector centrality. It demonstrates that these measures reflect local versus global influence via graph walks, with the spectral gap of the adjacency matrix playing a key role in ranking stability and parameter selection.

ABSTRACT

Node centrality measures including degree, eigenvector, Katz and subgraph centralities are analyzed for both undirected and directed networks. We show how parameter-dependent measures, such as Katz and subgraph centrality, can be tuned to interpolate between degree and eigenvector centrality, which appear as limiting cases of the other measures. We interpret our finding in terms of the local and global influence of a given node in the graph as measured by graph walks of different length through that node. Our analysis gives some guidance for the choice of parameters in Katz and subgraph centrality, and provides an explanation for the observed correlations between different centrality measures and for the stability exhibited by the ranking vectors for certain parameter ranges. The important role played by the spectral gap of the adjacency matrix is also highlighted.

Motivation & Objective

• To analyze the mathematical relationships between major node centrality measures using matrix theory.
• To understand how parameter-dependent centrality measures (Katz and subgraph centrality) behave as they interpolate between degree and eigenvector centrality.
• To provide guidance on selecting optimal parameters for Katz and subgraph centrality in network analysis.
• To explain observed correlations and ranking stabilities across different centrality measures.
• To highlight the role of the spectral gap in the adjacency matrix for centrality measure behavior.

Proposed method

• Formalizing centrality measures using matrix functions of the adjacency matrix, particularly the resolvent and matrix exponential.
• Using graph walk interpretations to link centrality values to the number of walks of varying lengths through a node.
• Analyzing limiting behaviors of parameterized measures as parameters approach zero or infinity to recover degree and eigenvector centrality.
• Employing spectral decomposition to relate centrality rankings to eigenvalues and eigenvectors of the adjacency matrix.
• Investigating the influence of the spectral gap (difference between the largest and second-largest eigenvalues) on stability and convergence of centrality rankings.
• Deriving conditions under which parameter choices yield stable and interpretable centrality rankings.

Experimental results

Research questions

• RQ1 How do Katz and subgraph centrality measures relate to degree and eigenvector centrality in terms of matrix formulations?
• RQ2 What is the role of walk length and graph structure in shaping centrality values across different measures?
• RQ3 How does the spectral gap of the adjacency matrix affect the stability of centrality rankings?
• RQ4 What parameter ranges in Katz and subgraph centrality lead to stable and meaningful node rankings?
• RQ5 Why do different centrality measures often correlate strongly in practice?

Key findings

• Katz and subgraph centrality can be tuned to smoothly interpolate between degree and eigenvector centrality as their parameters vary.
• The limiting cases of these parameterized measures correspond exactly to degree centrality (short walks) and eigenvector centrality (long walks).
• Centrality values are interpreted as weighted sums of walks through a node, with weights depending on walk length and measure-specific parameters.
• The spectral gap of the adjacency matrix determines the rate of convergence and stability of centrality rankings across parameter ranges.
• Parameter ranges with large spectral gaps exhibit more stable centrality rankings, explaining empirical observations of robustness.
• The analysis provides a principled method for selecting parameters in Katz and subgraph centrality based on desired balance between local (degree-like) and global (eigenvector-like) influence.