Title: Clustering Spectrum of scale-free networks,
Abstract: Real-world networks often have power-law degrees and scale-free properties
such as ultra-small distances and ultra-fast information spreading. In this
paper, we study a third universal property: three-point correlations that
suppress the creation of triangles and signal the presence of hierarchy. We
quantify this property in terms of $\bar c(k)$, the probability that two
neighbors of a degree-$k$ node are neighbors themselves. We investigate how the
clustering spectrum $k\mapsto\bar c(k)$ scales with $k$ in the hidden variable
model and show that $c(k)$ follows a {\it universal curve} that consists of
three $k$-ranges where $\bar c(k)$ remains flat, starts declining, and
eventually settles on a power law $\bar c(k)\sim k^{-\alpha}$ with $\alpha$
depending on the power law of the degree distribution. We test these results
against ten contemporary real-world networks and explain analytically why the
universal curve properties only reveal themselves in large networks.