title: Local clustering in scale-free networks with hidden variables.
abstract: We investigate the presence of triangles in a class of correlated random graphs in which hidden variables determine the pairwise connections between vertices. The class rules out self-loops and multiple edges. We focus on the regime where the hidden variables follow a power law with exponent τ∈(2,3), so that the degrees have infinite variance. The natural cutoff h_{c} characterizes the largest degrees in the hidden variable models, and a structural cutoff h_{s} introduces negative degree correlations (disassortative mixing) due to the infinite-variance degrees. We show that local clustering decreases with the hidden variable (or degree). We also determine how the average clustering coefficient C scales with the network size N, as a function of h_{s} and h_{c}. For scale-free networks with exponent 2<τ<3 and the default choices h_{s}∼N^{1/2} and h_{c}∼N^{1/(τ-1)} this gives C∼N^{2-τ}lnN for the universality class at hand. We characterize the extremely slow decay of C when τ≈2 and show that for τ=2.1, say, clustering starts to vanish only for networks as large as N=10^{9}.
authors.affiliation: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands.
authors.forename: Remco
authors.initials: R
authors.lastname: van der Hofstad
authors.affiliation: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands.
authors.forename: A J E M
authors.initials: AJ
authors.lastname: Janssen
authors.affiliation: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands.
authors.forename: Johan S H
authors.initials: JS
authors.lastname: van Leeuwaarden
authors.affiliation: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands.
authors.forename: Clara
authors.initials: C
authors.lastname: Stegehuis