Scale-free network clustering in hyperbolic and other random graphs

Clara Stegehuis; Remco van der Hofstad; Johan S. H. van Leeuwaarden

doi:10.1088/1751-8121/ab2269

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arXiv · 1812.03002

Scale-free network clustering in hyperbolic and other random graphs

Clara Stegehuis·Remco van der Hofstad·Johan S. H. van Leeuwaarden

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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.

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Clara Stegehuis, Remco van der Hofstad, Johan S. H. van Leeuwaarden. 2018-12-07. Scale-free network clustering in hyperbolic and other random graphs. https://doi.org/10.1088/1751-8121%2Fab2269

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