### 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 |

Number of pages | 20 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 52 |

Issue number | 29 |

DOIs | |

Publication status | Published - 24 Jun 2019 |

### Fingerprint

### Keywords

- clustering
- Complex networks
- hyperbolic model
- random graphs
- complex networks

### Cite this

}

*Journal of Physics A: Mathematical and Theoretical*, vol. 52, no. 29, 295101. https://doi.org/10.1088/1751-8121/ab2269

**Scale-free network clustering in hyperbolic and other random graphs.** / Stegehuis, Clara (Corresponding author); van der Hofstad, Remco; van Leeuwaarden, Johan S.H.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Scale-free network clustering in hyperbolic and other random graphs

AU - Stegehuis, Clara

AU - van der Hofstad, Remco

AU - van Leeuwaarden, Johan S.H.

PY - 2019/6/24

Y1 - 2019/6/24

N2 - 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.

AB - 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.

KW - clustering

KW - Complex networks

KW - hyperbolic model

KW - random graphs

KW - complex networks

UR - http://www.scopus.com/inward/record.url?scp=85069529720&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/ab2269

DO - 10.1088/1751-8121/ab2269

M3 - Article

AN - SCOPUS:85069529720

VL - 52

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 29

M1 - 295101

ER -