### Abstract

Random graphs with power-law degrees can model scale-free networks as sparse topologies with strong degree heterogeneity. Mathematical analysis of such random graphs proved successful in explaining scale-free network properties such as resilience, navigability and small distances. We introduce a variational principle to explain how vertices tend to cluster in triangles as a function of their degrees. We apply the variational principle to the hyperbolic model that quickly gains popularity as a model for scale-free networks with latent geometries and clustering. We show that clustering in the hyperbolic model is non-vanishing and self-averaging, so that a single random graph sample is a good representation in the large-network limit. We also demonstrate the variational principle for some classical random graphs including the preferential attachment model and the configuration model.

Original language | English |
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Article number | 295101 |

Journal | Journal of physics A: mathematical and theoretical |

Volume | 52 |

Issue number | 29 |

Early online date | 17 May 2019 |

DOIs | |

Publication status | Published - 24 Jun 2019 |

Externally published | Yes |

### Keywords

- physics.soc-ph
- cs.SI
- math.PR

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## Cite this

Stegehuis, C., Hofstad, R. V. D., & Leeuwaarden, J. S. H. V. (2019). Scale-free network clustering in hyperbolic and other random graphs.

*Journal of physics A: mathematical and theoretical*,*52*(29), [295101]. https://doi.org/10.1088/1751-8121/ab2269