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 ofc(k), the probability that two neighbors of a degree-k node are neighbors themselves. We investigate how the clustering spectrum k →c(k) scales with k in the hidden-variable model and show thatc(k) follows a universal curve that consists of three k ranges wherec(k) remains flat, starts declining, and eventually settles on a power-lawc(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.

Triadic Closure in Configuration Models with Unbounded Degree Fluctuations

“…Hidden-variable models Our results for clustering in the erased configuration model agree with recent results for the hidden-variable model [21]. In the hidden-variable model, every vertex is equipped with a hidden variable w i , where the hidden variables are sampled from a power-law distribution.…”

“…We see that for small values of k, c(k) decays slowly. When k becomes larger, the local clustering coefficient indeed seems to decay as an inverse power of k. Similar behavior has been observed in more real-world networks [21]. The decay of the local clustering coefficient c(k) in k is considered an important empirical observation, because it may signal the presence of hierarchical network structures [19], where high-degree vertices barely participate in triangles, but connect communities consisting of small-degree vertices with high clustering coefficients.…”

“…In this paper we analyze c(k) for networks with a power-law degree distribution with degree exponent τ ∈ (2, 3), the situation that describes the majority of real-world networks [1,8,13,21]. To analyze c(k), we consider the configuration model in the large-network limit, and count the number of triangles where at least one of the vertices has degree k. When the degree exponent satisfies τ > 3, the total number of triangles in the configuration model converges to a Poisson random variable [9,Chapter 7].…”

Variational principle for scale-free network motifs

“…Furthermore, by Lemma 5, the optimal value of (40) is unique if and only if the solution to (32) is unique.…”

“…, h k . The degree of a node is asymptotically Poisson distributed with its hidden variable as mean [32], so (3) can be interpreted as a sum over all possible degree sequences. Therefore, our optimization method then needs to settle the following trade-off, inherently present in powerlaw networks: On the one hand, large-degree vertices contribute substantially to the number of motifs, because they are highly connected, and therefore participate in many motifs.…”