Abstract
The configuration model generates random graphs with any given degree distribution, and thus serves as a null model for scale-free networks with power-law degrees and unbounded degree fluctuations. For this setting, we study the local clustering c(k), i.e., the probability that two neighbors of a degree-k node are neighbors themselves. We show that c(k) progressively falls off with k and the graph size n and eventually for k=varOmega (sqrt{n}) settles on a power law c(k)sim n^{5-2tau }k^{-2(3-tau )} with tau in (2,3) the power-law exponent of the degree distribution. This fall-off has been observed in the majority of real-world networks and signals the presence of modular or hierarchical structure. Our results agree with recent results for the hidden-variable model and also give the expected number of triangles in the configuration model when counting triangles only once despite the presence of multi-edges. We show that only triangles consisting of triplets with uniquely specified degrees contribute to the triangle counting.
Highlights
Random graphs can be used to model many different types of networked structures such as communication networks, social networks and biological networks
A well-known characteristic of many real-world networks is that the degree distribution follows a power law
A second measure of clustering is the local clustering coefficient, which measures the fraction of triangles that arise from one specific node
Summary
Random graphs can be used to model many different types of networked structures such as communication networks, social networks and biological networks. When the power-law degree exponent τ is close to two, the exponent γ approaches two, a considerable difference with the preferential attachment model with triangles or several fractal-like random graph models that predict c(k) ∼ k−1 [7,14,19]. Related to this result on the c(k) fall-off, we show that for every node with fixed degree k only pairs of nodes with specific degrees contribute to the triangle count and local clustering. The remaining sections prove all the main results, and in particular focus on establishing Propositions 1 and 2 that are crucial for the proof of Theorem 1
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