Clustering spectrum of scale-free networks.

Abstract

Real-world networks often have power-law degrees and scale-free properties, such as ultrasmall distances and ultrafast 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 c[over ¯](k), the probability that two neighbors of a degree-k node are neighbors themselves. We investigate how the clustering spectrum k↦c[over ¯](k) scales with k in the hidden-variable model and show that c[over ¯](k) follows a universal curve that consists of three k ranges where c[over ¯](k) remains flat, starts declining, and eventually settles on a power-law c[over ¯](k)∼k^{-α} with α 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.