Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bringing order to network centrality measures

G. Exarchakos, R. van der Hofstad, O. Nagy, M. Pandey

TL;DR

The paper introduces the Centrality Comparison Curve (CCC), a rank-based, monotone-invariant tool to compare arbitrary centrality measures on the same graph by examining the overlap of top-ranked vertices. It establishes fundamental properties (symmetry, monotonicity, identity when identical, and a $x^2$-type benchmark for independence) and demonstrates how CCC can diagnose when centrality measures are similar, different, or independent. The work shows practical utility through plots on synthetic and real networks, revealing relationships such as PageRank with in-degree and the alignment between closeness and harmonic centralities, while also offering a path to reduce computation by substituting expensive measures with cheaper, CCC-approved equivalents. Finally, it analyzes convergence of CCC in large graph limits, proving results for local convergence in sparse graphs and graphon-based convergence in dense graphs, and outlines open problems and future directions for empirical mapping and rigorous convergence theory.

Abstract

We introduce a quantitative method to compare arbitrary pairs of graph centrality measures, based on the ordering of vertices induced by them. The proposed method is conceptually simple, mathematically elegant, and allows for a quantitative restatement of many conjectures that were previously cumbersome to formalize. Moreover, it produces an approximation scheme useful for network scientists. We explore some of these uses and formulate new conjectures that are of independent interest.

Bringing order to network centrality measures

TL;DR

The paper introduces the Centrality Comparison Curve (CCC), a rank-based, monotone-invariant tool to compare arbitrary centrality measures on the same graph by examining the overlap of top-ranked vertices. It establishes fundamental properties (symmetry, monotonicity, identity when identical, and a -type benchmark for independence) and demonstrates how CCC can diagnose when centrality measures are similar, different, or independent. The work shows practical utility through plots on synthetic and real networks, revealing relationships such as PageRank with in-degree and the alignment between closeness and harmonic centralities, while also offering a path to reduce computation by substituting expensive measures with cheaper, CCC-approved equivalents. Finally, it analyzes convergence of CCC in large graph limits, proving results for local convergence in sparse graphs and graphon-based convergence in dense graphs, and outlines open problems and future directions for empirical mapping and rigorous convergence theory.

Abstract

We introduce a quantitative method to compare arbitrary pairs of graph centrality measures, based on the ordering of vertices induced by them. The proposed method is conceptually simple, mathematically elegant, and allows for a quantitative restatement of many conjectures that were previously cumbersome to formalize. Moreover, it produces an approximation scheme useful for network scientists. We explore some of these uses and formulate new conjectures that are of independent interest.
Paper Structure (45 sections, 11 equations, 25 figures)

This paper contains 45 sections, 11 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Background.
  3. Main innovation of this paper
  4. Results
  5. The centrality comparison curve
  6. Properties of CCC
  7. Symmetry of the CCC
  8. Monotonicity of the CCC
  9. Invariant under monotone increasing transformation
  10. CCC is identity curve when $R=S$.
  11. Bounded above by the identity curve
  12. Lower bound for maximally different centrality measures
  13. Using CCC for testing independence of Centrality Measures
  14. Summary of comparisons
  15. Putting the centrality comparison curve to practice
  16. ...and 30 more sections

Figures (25)

  • Figure 1: CCC for PageRank versus in-degree, and PageRank versus out-degree, for the "hep-ph" citation network GehrkeGinspargKleinberg2003.
  • Figure 2: CCC for closeness versus harmonic centrality, for the "hep-ph" citation network GehrkeGinspargKleinberg2003 and the artificial network.
  • Figure 3: CCC for betweenness versus load centrality, for the "hep-ph" citation network GehrkeGinspargKleinberg2003 and the artificial network.
  • Figure 4: CCC for PageRank versus in-degree, for the "hep-ph" citation network GehrkeGinspargKleinberg2003 and the artificial network.
  • Figure 5: CCC for PageRank with damping factors 0.5 versus 0.9, and 0.3 versus 0.9, respectively, for the artificial network.
  • ...and 20 more figures

Theorems & Definitions (2)

  • Definition 1: Induced vertex ordering
  • Definition 2: Centrality Comparison Curve