  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.
