TY - JOUR

T1 - Local clustering in scale-free networks with hidden variables

AU - Van Der Hofstad, Remco

AU - Janssen, A. J.E.M.

AU - Van Leeuwaarden, Johan S.H.

AU - Stegehuis, Clara

PY - 2017/2/14

Y1 - 2017/2/14

N2 - 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 hc characterizes the largest degrees in the hidden variable models, and a structural cutoff hs 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 hs and hc. For scale-free networks with exponent 2<τ<3 and the default choices hs∼N1/2 and hc∼N1/(τ-1) this gives C∼N2-τlnN 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=109.

AB - 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 hc characterizes the largest degrees in the hidden variable models, and a structural cutoff hs 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 hs and hc. For scale-free networks with exponent 2<τ<3 and the default choices hs∼N1/2 and hc∼N1/(τ-1) this gives C∼N2-τlnN 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=109.

UR - http://www.scopus.com/inward/record.url?scp=85013806492&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.95.022307

DO - 10.1103/PhysRevE.95.022307

M3 - Article

AN - SCOPUS:85013806492

SN - 2470-0045

VL - 95

JO - Physical Review E

JF - Physical Review E

IS - 2

M1 - 022307

ER -