Consistency and differences between centrality measures across distinct classes of networks

Introduction

• Prior studies found that centrality-measure correlations are typically moderate to high but vary substantially across networks, including closeness–eigenvector correlations of 0.91 in physicist collaborations and −0.04 in a power grid.The introduction says the reasons for these cross-network variations remain unclear.Source: Introduction · p. 3
• Proposed drivers of correlation structure include proximity to threshold graphs, large spectral gaps for walk-based measures, and modularity that can separate local from global centrality.The majorization gap quantifies topological distance from a threshold graph; modular structure can leave a node locally central but globally peripheral, or vice versa.Source: Introduction · p. 4
• Because earlier studies covered limited network classes, sizes, densities, organizational patterns, or centrality measures, the benefits of combining measures to define nodal roles remain insufficiently established.The introduction motivates broad comparative analysis as a more systematic basis for selecting useful measures from a network’s empirical structure.Source: Introduction · pp. 4–5
• The study evaluates 17 centrality measures across 212 networks, relating cross-network correlation variation to global topology and using multivariate centrality profiles to examine node roles.The introduction frames this as a comparative approach to understanding nodal roles across diverse networks.Source: Introduction · p. 5

Centrality Measures

• The analysis compared 17 commonly used or recently studied centrality measures, including measures based on walks, trails, and paths and the participation coefficient.The participation coefficient was included because it is often used to characterize nodal roles in networks with community structure.Source: Methods · p. 6

Network Data

• The dataset comprised 212 networks represented as adjacency matrices, analyzed primarily as unweighted and undirected largest connected components.Weighted analyses were also performed for 39 networks with edge-weight information.Source: Methods · p. 8

Surrogate Networks

• For each empirical network, 100 unconstrained and 100 constrained surrogate networks were generated to test whether size, density, degree sequence, and strength distributions account for correlation patterns.Unconstrained surrogates preserved node and edge counts and edge-weight distributions; constrained surrogates also preserved degree sequence and approximate node strength distributions. Random-walk and communicability betweenness were omitted from surrogate analyses for computational reasons.Source: Methods · pp. 8–10

Centrality Measure Correlations

• Spearman’s ρ quantified correlations between pairs of nodal centrality scores within each network, with mean correlation and standard deviation summarized across networks.Spearman’s ρ was selected because score relationships were often nonlinear but nearly always monotonic, and many metrics were non-Gaussian.Source: Methods · p. 8

Assessing the Relationship Between Network Topology and CMCs

• The study related mean within-network centrality correlations to global topology, including density, assortativity, clustering, global and diffusion efficiency, modularity, majorization gap, and spectral gap.Modularity was quantified with Q using Louvain community detection and consensus clustering; majorization gap represented distance from a threshold graph.Source: Methods · pp. 8–9

Clustering Nodes Based on Their Centrality Profiles

• To explore distinct nodal roles, ranked centrality profiles were clustered hierarchically using Ward’s method and Euclidean distances, with the Davies-Bouldin index guiding dendrogram resolution.The clustering was intended to demonstrate how profiles could identify role subsets, not to establish an optimal or robust clustering solution.Source: Methods · p. 10

• CMCs were mostly high and positive, yet varied considerably across networks without clearly mapping to natural network classes.The distributions shown for five example unweighted and weighted networks illustrate this variation.Source: Results · p. 11, Figure 2 · p. 11
• 97% of unweighted and 80% of weighted mean between-network CMCs exceeded 0.5, with weighted CMCs slightly weaker than unweighted counterparts.In 39 networks with edge-weight information, weighted and unweighted centrality measures were also highly correlated.Source: Results · p. 11, Figure 3 · p. 12
• RWCC and IC correlations ranged from 0.88–1 (mean 0.998) unweighted and 0.937–1 (mean 0.996) weighted, making them practically redundant in most real-world scenarios.These theoretically related measures are also highly correlated across networks.Source: Results · p. 12
• KC and TCC were highly correlated across unweighted networks, with all ρ > 0.98.Source: Results · p. 12
• Participation coefficient and bridging centrality had the lowest average correlations with other measures, while weighted subgraph centrality showed low correlations.These patterns may reflect the conceptual distinctiveness of participation coefficient and bridging centrality, the former's dependence on modular decomposition, and a unique aspect captured by weighted subgraph centrality.Source: Results · p. 12

Network Topology and CMCs

• Higher mean within-network CMCs tracked several global properties in unweighted networks, while weighted networks showed notable exceptions.In unweighted networks, CMCs rose with clustering, density, diffusion and global efficiency, and spectral gap, and fell with assortativity, majorization gap, and modularity. Across both network types, bridging-centrality and participation-coefficient CMCs correlated weakly with nearly all global properties.Source: Figure 4 · p. 13, Network Topology and CMCs · p. 13
• Modularity alone significantly predicted unweighted network CMCs, whereas weighted assortativity explained the most variance and modularity and majorization gap also predicted CMCs.In separate weighted models, modularity accounted for 49% of variance versus 55% for majorization gap.Source: Network Topology and CMCs · pp. 13–14
• The surrogate comparisons matched node and edge counts, with constrained surrogates also matching degree sequence and strength distribution.Unconstrained surrogates additionally matched edge-weight distribution; the comparisons tested whether topology–CMC associations reflected these lower-order features.Source: Network Topology and CMCs · p. 14
• For most networks, matched surrogate networks had mean within-network CMCs at least as high as empirical networks, even when degree sequence fixed the majorization gap.Constrained surrogates shared the empirical networks’ majorization gap, yet empirical networks often had lower CMCs, countering the expected association between higher majorization gap and lower CMCs. Unconstrained surrogates also had higher majorization gaps than empirical networks.Source: Network Topology and CMCs · p. 14, Figure 5 · p. 15, Figure 6 · p. 16

Centrality-Based Clustering of Nodes.

• Hierarchical clustering combined centrality measures to identify node roles, excluding random-walk closeness because it correlated above 0.99 with information centrality.Source: Centrality-Based Clustering of Nodes · p. 16
• In most networks, the Davies-Bouldin criterion favored two clusters, often separating low-centrality nodes from the rest or vice versa.Nearly all networks contained subsets scoring high or low across most measures; these subsets also appeared at finer clustering resolutions.Source: Centrality-Based Clustering of Nodes · p. 16, Figure 7 · p. 17, Figure 8 · p. 18
• Across clustering resolutions, networks showed a high-scoring core and low-scoring periphery, with intermediate nodes forming gradual transitions or semi-discrete clusters.Semi-discrete clusters had distinctive centrality-score profiles; the gradual pattern is illustrated in Fig. 8A, and the semi-discrete pattern in Fig. 7A.Source: Centrality-Based Clustering of Nodes · p. 16, Figure 7 · p. 17, Figure 8 · p. 18
• An intermediate cluster in several networks scored high on closeness and eigenvector-like measures but low on betweenness-based measures.These nodes were positioned within a central core and connected to high-degree nodes, but lacked connections outside the main cluster; other intermediate clusters varied by network and may represent system-specific roles.Source: Centrality-Based Clustering of Nodes · p. 17, Figure 7 · p. 17

• Across 212 networks, 17 centrality measures were generally positively correlated, with weighted-network correlations only slightly weaker; degree tracked most measures, while Leverage, PageRank, bridging, and participation coefficient correlated less broadly.Degree may serve as a useful first approximation of centrality; Leverage and PageRank may align less with other measures because they scale importance relative to immediate neighbours.Source: Discussion · p. 19
• Modularity was the only topological property with a significant unique contribution to mean within-network CMC variation: higher-modularity networks had weaker CMCs than matched surrogates.Modularity can decouple local from global centrality and create bottlenecks separating path-based from degree-based measures, reducing mean CMCs.Source: Discussion · p. 19
• Although density, global efficiency, modularity, majorization gap, and spectral gap correlated with CMCs, majorization gap was not a significant predictor of unweighted mean within-network CMCs.Surrogates with comparable or larger majorization gaps still had CMCs, suggesting the gap may reflect variability in possible centrality ranks rather than their exact ranks.Source: Discussion · p. 19
• The analysis’s global topology measures may only approximate actual network structure, since modularity depends on the decomposition algorithm and spectral and majorization gaps have interpretive limitations.The majorization gap heuristically measures distance from a threshold graph, itself a heuristic for perfect neighbourhood-inclusion preorder.Source: Discussion · p. 19
• Hierarchical clustering found high-scoring core and low-scoring peripheral node groups in nearly all networks, while some networks also had distinct intermediate centrality-profile clusters.These intermediate roles, which may be missed by single measures, occurred especially in networks with higher modularity or ring-and-tendril structure.Source: Discussion · p. 20
• The optimal measure set remains unresolved, and hierarchical clustering may miss individual nodes with distinctive profiles; alternative clustering could better reveal such nodes and their potential role in network dynamics.The study used only a small subset of more than 200 proposed centrality metrics, yet demonstrates that comparative profiling can illuminate node roles.Source: Discussion · p. 20

• S.O, B.F and A.F developed the study concept and design and wrote the main manuscript text.Source: Author Contributions Statement · p. 25
• S.O analysed the data and produced all figures; S.O, A.A, L.P and A.F constructed the methodology for creating the brain networks.Source: Author Contributions Statement · p. 25
• S.O and C.S developed data management and analysis strategies, and all authors reviewed the manuscript.Source: Author Contributions Statement · p. 25

Local connectivity

• Degree/strength counts a node’s connections or sums their weights, while h-index centrality uses its neighbours’ degrees and leverage centrality contrasts its degree with theirs [2][3].Leverage centrality can be negative when a node has fewer connections than its neighbours and positive when it has more.Source: Supplementary Methods · p. 27

Spectral and recursive measures

• Eigenvector centrality rewards connections to highly connected nodes, Katz centrality adds baseline scores to avoid zero-valued nodes [5, 6], and PageRank scales neighbour contributions by degree.Katz used β = 1 and α set to 10% below the inverse largest eigenvalue; PageRank used β = 1 and α = 0.85.Source: Supplementary Methods · p. 28

Proximity measures

• Closeness centrality inverts average shortest-path distance, information centrality aggregates information across paths [11], and random-walk closeness uses mean first-passage time.For weighted networks, shortest-path distance uses inverse edge weights; random-walk transitions use degree or strength-normalized weights.Source: Supplementary Methods · pp. 28–29

Walk-based measures

• Subgraph centrality counts closed walks with longer walks penalized, total communicability counts walks to other nodes as well, and Laplacian centrality reflects energy associated with a node’s removal.The methods distinguish closed from non-closed walks and define Laplacian energy using two-walk contributions.Source: Supplementary Methods · pp. 29–30

Betweenness measures

• Shortest-path betweenness identifies nodes on geodesics between other pairs, whereas random-walk betweenness measures node participation across alternative routes.The random-walk formulation treats edges as resistors and averages current through a node over source–drain pairs.Source: Supplementary Methods · pp. 30–31

Cross-module roles

• Communicability betweenness counts walks involving a node, bridging centrality highlights connectors between communities, and participation coefficient measures how connections span modules.Module membership for participation coefficient was determined by consensus clustering.Source: Supplementary Methods · pp. 31–32

Global network properties

• Assortativity is the Pearson correlation between degrees of connected nodes, while clustering captures how many pairs of a node’s neighbours are connected.For weighted networks, clustering represents how triangle-edge weights compare with the network’s maximum edge weight.Source: Global network properties · p. 32
• Density is the proportion of all possible network connections that exist.Source: Global network properties · p. 33
• Global efficiency is the reciprocal of average shortest-path length, whereas diffusion efficiency measures the average length of random walks between nodes.Source: Global network properties · p. 33
• The majorization gap quantifies similarity to a comparable threshold graph by estimating the number of edges that must be rewired to transform a network into a threshold graph.Source: Global network properties · p. 33
• Modularity Q measures whether connectivity within modules exceeds chance expectations; modules were identified using repeated Louvain runs and consensus clustering.The consensus procedure weighted co-assignment frequencies by modularity quality scores and iterated until node pairs were consistently grouped together or apart.Source: Global network properties · p. 34
• A normalized spectral gap indicates expander-like connectivity, and such networks are associated with high correlations between walk-based centrality measures.The spectral gap is the absolute difference between the two largest eigenvalues of the adjacency matrix; the study normalized it for cross-network comparison.Source: Global network properties · p. 34

Human Brain Network Construction

• Human structural brain networks were created from HCP diffusion-weighted and T1-weighted MRI for 100 unrelated participants aged 22–35 years, with preprocessing segmenting 34 cortical and seven subcortical regions per hemisphere into 82 total nodes.Source: Human Brain Network Construction · pp. 34–35
• The whole-brain parcellation combined 360 cortical regions with seven thalamic and three striatal regions, yielding 380 nodes.Source: Human Brain Network Construction · p. 35
• Tractography generated 10 million streamlines using FACT, anatomically constrained tracking, and dynamic seeding, then reweighted them with SIFT2 to better represent inter-regional connectivity.The streamlines were derived from diffusion tensor eigenvectors and fibre orientation distributions.Source: Human Brain Network Construction · p. 35
• Across 100 participants, the group-average connectome retained the 5.23% most consistent edges, with the threshold selected to match average individual network density.Consistency was estimated from the coefficient of variation across participants for each edge.Source: Human Brain Network Construction · p. 35

Surrogate Networks.

• Unconstrained surrogates match each real network’s size and density, while ensuring connectivity even when density is below the usual random-network threshold.The procedure begins with a uniformly random minimum spanning tree, then adds randomly selected edges until the target edge count is reached.Source: Surrogate Networks · p. 36
• For weighted unconstrained surrogates, original-network edge weights are randomly assigned to edges in the connected surrogate graph.The underlying surrogate is generated by a random minimum spanning tree followed by random edge additions.Source: Surrogate Networks · p. 36
• Constrained unweighted surrogates preserve node and edge counts, degree distribution, and connectivity; weighted surrogates also approximately preserve strength distribution.The surrogates were generated using Brain Connectivity Toolbox functions in MATLAB.Source: Surrogate Networks · p. 36

Supplementary references

• Table S1 describes a general linear model predicting unweighted mean-within CMCs; density and diffusion efficiency were excluded because their relationships were nonlinear.The caption reports significance thresholds of p < .05 and p < .01, but provides no coefficient values in the supplied passage.Source: Table S1 · p. 37
• Table S2 describes a general linear model predicting weighted mean-within CMCs, excluding density, diffusion efficiency, and clustering for nonlinear relationships.The supplied caption identifies significance thresholds but gives no model estimates.Source: Table S2 · p. 37