# Clustering Spectrum of scale-free networks

Clara Stegehuis, Remco van der Hofstad, Johan S. H. van Leeuwaarden, A. J. E. M Janssen

Research output: Contribution to journalArticleAcademicpeer-review

10 Citations (Scopus)

## Abstract

Real-world networks often have power-law degrees and scale-free properties such as ultra-small distances and ultra-fast information spreading. In this paper, we study a third universal property: three-point correlations that suppress the creation of triangles and signal the presence of hierarchy. We quantify this property in terms of $\bar c(k)$, the probability that two neighbors of a degree-$k$ node are neighbors themselves. We investigate how the clustering spectrum $k\mapsto\bar c(k)$ scales with $k$ in the hidden variable model and show that $c(k)$ follows a {\it universal curve} that consists of three $k$-ranges where $\bar c(k)$ remains flat, starts declining, and eventually settles on a power law $\bar c(k)\sim k^{-\alpha}$ with $\alpha$ depending on the power law of the degree distribution. We test these results against ten contemporary real-world networks and explain analytically why the universal curve properties only reveal themselves in large networks.
Original language English 042309 Physical review E: covering statistical, nonlinear, biological, and soft matter physics 96 https://doi.org/10.1103/PhysRevE.96.042309 Published - 26 Oct 2017 Yes

## Keywords

• cs.SI
• math.PR
• physics.soc-ph

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