### 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 |
---|---|

Article number | 295101 |

Journal | Journal of physics a-Mathematical and theoretical |

Volume | 52 |

Issue number | 29 |

DOIs | |

Publication status | Published - 19 Jul 2019 |

### Keywords

- complex networks
- random graphs
- hyperbolic model
- clustering
- COMPLEX NETWORKS
- ORGANIZATION
- DISTANCES
- WEB
- DIAMETER
- INTERNET
- MODELS

### Cite this

*Journal of physics a-Mathematical and theoretical*,

*52*(29), [295101]. https://doi.org/10.1088/1751-8121/ab2269

}

*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; van Der Hofstad, Remco; van Leeuwaarden, Johan S. H.

Research output: Contribution to journal › Article › Scientific › 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 - 2019/7/19

Y1 - 2019/7/19

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 - complex networks

KW - random graphs

KW - hyperbolic model

KW - clustering

KW - COMPLEX NETWORKS

KW - ORGANIZATION

KW - DISTANCES

KW - WEB

KW - DIAMETER

KW - INTERNET

KW - MODELS

U2 - 10.1088/1751-8121/ab2269

DO - 10.1088/1751-8121/ab2269

M3 - Article

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 -