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 ln N 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 .

Clustering spectrum of scale-free networks

“…Therefore, we expect the ECM and the hidden-variable model to have similar properties (see, e.g., Ref. [31]) when we choose Figure 5 illustrates how both null models generate highly similar spectra, which provides additional support for the claim that the clustering spectrum is a universal property of simple scale-free networks. The ECM is more difficult to deal with compared to the hidden-variable models since edges in the ECM are not independent.…”

“…In particular, here the local clustering does not depend on the degree and in fact corresponds with the large-N behavior of the global clustering coefficient [31,32]. Note that the interval [0,β(τ )] diminishes when τ is close to 2, a possible explanation for why the flat range associated with Range I is hard to recognize in some of the real-world data sets.…”

“…Figure 4 also brings to bear a potential pitfall when the goal is to obtain statistically accurate estimates for the slope of c(h). Observe the extremely slow convergence to the limiting curve for N = ∞; a well documented property of certain clustering measures [31,32,37,38]. In Appendix B we again use the integral expression (4) ; this also is apparent from visual inspection of Fig.…”

“…We therefore work with (although many asymptotically equivalent choices are possible; see Ref. [31] and Appendix A)…”

“…We are interested in computing the value of σ N (t) for large values of N . Adopting the standard choices [31],…”