Johan van Leeuwaarden
Real-world networks often have power-law degrees and scale-free properties such as ultra-small distances and ultra-fast information spreading. We provide evidence of 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(k), the probability that two neighbors of a degree-k node are neighbors themselves. We investigate how c(k) scales with k and discover a universal curve that consists of three k-ranges where c(k) remains flat, starts declining, and eventually settles on a power law with an exponent that depends 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.