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.
|Journal||Journal of physics A: mathematical and theoretical|
|Early online date||17 May 2019|
|Publication status||Published - 24 Jun 2019|
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), . https://doi.org/10.1088/1751-8121/ab2269