Evaluating Global Measures of Network Centralization: Axiomatic and Numerical Assessments

TL;DR

This work normalizes 11 measures of network centralization and assesses them systematically using an axiomatic framework and numerical simulations, clarifying conceptual differences among existing measures and offering practical guidance for selecting reliable centralization metrics.

Abstract

Network centralization, driven by hub nodes, impacts communication efficiency, structural integration, and dynamic processes such as diffusion and synchronization. Although numerous centralization measures exist, a major challenge lies in determining measures that are both theoretically sound and empirically reliable across different network contexts. To resolve this challenge, we normalize 11 measures of network centralization and assess them systematically using an axiomatic framework and numerical simulations. Our axiomatic assessment tests each measure against the six postulates of centralization, ensuring consistency with minimal theoretical requirements. In addition, our numerical assessment examines the behavior of normalized centralization measures over different random graphs. Our results indicate major differences among the measures, despite their common aim of quantifying centralization. Together, our assessments point to the relative suitability of three measures: normalized betweenness centralization, normalized closeness centralization, and normalized degree centralization. Applying these three measures to real-world networks from diverse domains reveals meaningful variation in the organization of the networks with respect to hubs. Normalized betweenness centralization highlights path-based dominance; normalized closeness centralization reflects accessibility and efficiency of reach; and normalized degree centralization captures degree-based hub concentration. When used jointly, the three measures demonstrate the required sensitivity to varying levels of centralization and provide complementary aspects of network centralization that no single measure can offer alone. Our dual evaluation framework clarifies conceptual differences among existing measures and offers practical guidance for selecting reliable centralization metrics.

Figures (2)

• Figure 1: Centralization values computed using various measures across different network sizes for three graph configurations: star, ring, and complete graphs (top row), and their perturbed variants with a single edge rewired or removed (bottom row). Each plot shows how each measure responds to changes in network order. Measures that yield constant values of 0 or 1 are not shown and are instead annotated in subplots. Colors and point styles indicate different centralization measures. Abbreviations: ABH, Assortativity-Based Hubness; ECD, Eigenvector Centrality Dispersion; NBC, Normalized Betweenness Centralization; NCC, Normalized Closeness Centralization; NDC, Normalized Degree Centralization; NDE, Normalized Degree Entropy; NDV, Normalized Degree Variance; NGC, Normalized Gini Coefficient; NHD, Normalized Hub Dominance; NHT, Normalized Hub Formation Tendency; NNC, Normalized Natural Connectivity.
• Figure 2: Temporal snapshots of a high-school friendship network across four survey waves. Nodes represent individuals, and links denote reciprocated friendship ties. Node size is proportional to degree, and node color intensity reflects betweenness centrality (darker red = higher values), while node border color reflects closeness centrality (darker blue = higher values). Normalized Betweenness Centralization (NBC), Normalized Closeness Centralization (NCC), and Normalized Degree Centralization (NDC) values are reported below each panel.