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Consistency and differences between centrality measures across distinct classes of networks
Stuart Oldham, Ben Fulcher, Linden Parkes, Aurina Arnatkeviciute, Chao Suo, Alex Fornito
TL;DR
Whether multiple centrality measures provide redundant or unique information about node roles across network types remains unclear. The paper compares 17 measures across 212 networks, relates their correlations to network topology, and clusters nodes by multivariate centrality profiles. Measures are generally positively correlated but vary across networks, modularity predicts much of this variation, and profiling reveals recurring cores and peripheries alongside network-specific intermediate roles.
Problem
The study addresses limited systematic evidence about how centrality-measure correlations vary across diverse networks and whether combining measures improves identification of nodal roles.
Method
The authors calculate correlations among 17 centrality measures across 212 networks, relate them to global topology, and hierarchically cluster nodes using ranked multivariate centrality profiles.
Results
Centrality measures show moderate-to-high positive correlations across most networks, modularity is the strongest predictor of mean correlation variability in unweighted networks, and most networks contain consistently high-scoring cores and low-scoring peripheries.
Takeaways & Limitations
Comparing centrality measures can inform interpretation of nodal roles, especially in networks where modular structure produces distinct intermediate centrality profiles.
Takeaways & Limitations
The study examines only a small subset of the more than 200 proposed centrality metrics, and hierarchical clustering may miss individual nodes with distinctive profiles.
Abstract
from arXiv · showhide
The roles of different nodes within a network are often understood through centrality analysis, which aims to quantify the capacity of a node to influence, or be influenced by, other nodes via its connection topology. Many different centrality measures have been proposed, but the degree to which they offer unique information, and such whether it is advantageous to use multiple centrality measures to define node roles, is unclear. Here we calculate correlations between 17 different centrality measures across 212 diverse real-world networks, examine how these correlations relate to variations in network density and global topology, and investigate whether nodes can be clustered into distinct classes according to their centrality profiles. We find that centrality measures are generally positively correlated to each other, the strength of these correlations varies across networks, and network modularity plays a key role in driving these cross-network variations. Data-driven clustering of nodes based on centrality profiles can distinguish different roles, including topological cores of highly central nodes and peripheries of less central nodes. Our findings illustrate how network topology shapes the pattern of correlations between centrality measures and demonstrate how a comparative approach to network centrality can inform the interpretation of nodal roles in complex networks.
Introduction
The paper systematically compares centrality-measure correlations across diverse networks and relates their variation to global topology, while testing whether multivariate centrality profiles identify distinct nodal roles.
- Motivation: Prior studies found that centrality-measure correlations are often moderate to high but can vary substantially between networks.For example, closeness and eigenvector centrality correlated at r=0.91 in one collaboration network but r=−0.04 in a power-grid network.
- Approach: The study evaluates 17 centrality measures across 212 networks and examines associations between correlation structure, global topology, and multivariate nodal profiles.The analyses include unweighted networks and a separate weighted analysis for 39 networks with edge-weight information.
- Centrality correlations: Most centrality measures were positively correlated, but correlation patterns remained heterogeneous across networks and did not clearly follow natural network categories.Mean correlations exceeded 0.5 for 97% of measure pairs in unweighted networks and 80% in weighted networks; weighted correlations were slightly weaker.
- Measure relationships: Some measure pairs were practically redundant, whereas participation coefficient, bridging centrality, and weighted subgraph centrality often captured more distinctive information.Random-walk closeness and information centrality had mean correlations of 0.998 unweighted and 0.996 weighted; Katz and total communicability correlations exceeded 0.98 in unweighted networks.
- Topology and correlations: Modularity and other global properties were associated with mean within-network correlation, with higher modularity corresponding to lower centrality-measure correlations.In unweighted networks, higher correlation was also associated with higher density, clustering, diffusion efficiency, global efficiency, and spectral gap, and lower assortativity and majorization gap.
- Nodal roles: Centrality-profile clustering usually identified high-scoring cores and low-scoring peripheries, with intermediate structures ranging from gradual transitions to distinct clusters.Distinct intermediate profiles varied across networks and may represent roles specific to particular systems.
Author Contributions Statement
The study concept and design were developed by S.O, B.F, and A.F, with contributions spanning methodology, analysis, data management, writing, and manuscript review.
- S.O, B.F, and A.F developed the study concept and implemented its design.
- S.O analysed the data and produced all figures.
- Methodology for creating brain networks was constructed by S.O, A.A, L.P, and A.F.
- S.O and C.S developed data management and analysis strategies, while S.O, B.F, and A.F wrote the main manuscript text.
- All authors reviewed the manuscript.
Title: Consistency and differences between centrality measures across distinct classes of networks
The authors are affiliated with Monash University in Australia, with one author also affiliated with the University of Sydney. The paper reports no conflicts of interest.
- The authors are affiliated with Monash University's Brain and Mental Health Research Hub and related psychological sciences institutes.
- One author is additionally affiliated with the School of Physics at the University of Sydney.
- All authors report no conflict of interest.
Supplementary Methods
The paper defines centrality measures from network adjacency or weight matrices, covering local connectivity, path-based access, walks, diffusion, community structure, and neighbour influence. These measures quantify distinct structural aspects of nodes, while the majorization gap assesses how closely a network resembles one where all centrality rankings agree.
- Network representation: Centrality measures are computed from an N×N adjacency matrix, with weighted versions generally substituting edge weights for binary connections.The adjacency matrix records whether nodes are connected, while the weighted matrix encodes edge weights.
- Local connectivity: Degree or strength centrality counts a node’s attached edges, or sums their weights in weighted networks.Strength is the weighted analogue of degree.
- Neighbour-based measures: Eigenvector, Katz, and PageRank centrality incorporate the centrality or degree of neighbouring nodes, with Katz penalizing distant dependencies and PageRank scaling neighbour contributions by node degree.The Katz parameters α and β control distant dependencies and baseline contributions, while PageRank uses α = 0.85 and β = 1 in these analyses.
- Path and diffusion measures: Closeness, information, and random-walk closeness centrality quantify accessibility through shortest paths, all possible paths, or mean first-passage times.Closeness uses inverse average minimum path length; information centrality weights overlapping paths by their information; random-walk closeness uses inverse average mean-first-passage time.
- Walk and energy measures: Subgraph and total communicability centrality count weighted walks, respectively emphasizing closed walks or walks to all other nodes, while Laplacian centrality measures effects related to network energy.Walks are weighted by the inverse factorial of their length, and subgraph centrality gives greater importance to smaller closed-walk-defined subgraphs.
- Topology and ranking agreement: The majorization gap measures the distance from a threshold graph in which all centrality measures rank nodes identically, estimated by the number of edges requiring rewiring.Threshold graphs have a complete neighbourhood-inclusion preorder, with undominated nodes ranked most centrally.