Table 1.
Compliance of centralization measures with the six axioms proposed by Palak and Nguyen (2021) for evaluating network centralization.
A check mark (✓) indicates that the measure satisfies the corresponding axiom, whereas a cross (✓) indicates a violation. The final column reports the total number of satisfied axioms (maximum = 6). 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.
Fig 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.
Table 2.
Performance of centralization measures across canonical graph topologies and their perturbed variants. Measures were evaluated on star, ring, and complete graphs, as well as perturbed versions with a single edge rewired or removed.
A check mark (–) indicates expected behavior for the given topology, whereas a cross (‗) indicates deviation from expectations. The final column reports the number of topologies for which each measure performed as expected (maximum = 6). 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.
Table 3.
Normalized Betweenness Centralization (NBC), Normalized Closeness Centralization (NCC), and Normalized Degree Centralization (NDC) for diverse real-world networks.
For each network, the category, description, number of nodes (n), and number of links (m), as well as minimum, average, and maximum degrees () are provided. For multi-component networks, values are computed for the largest connected component of the network.
Fig 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.