  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 eventually for $k=\Omega(\sqrt{n})$ settles on a power
law $c(k)\sim k^{-2(3-\tau)}$ 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.
