  We count the asymptotic number of triangles in uniform random graphs where
the degree distribution follows a power law with degree exponent
$\tau\in(2,3)$. We also analyze the local clustering coefficient $c(k)$, the
probability that two random neighbors of a vertex of degree $k$ are connected.
We find that the number of triangles, as well as the local clustering
coefficient, scale similarly as in the erased configuration model, where all
self-loops and multiple edges of the configuration model are removed.
Interestingly, uniform random graphs contain more triangles than erased
configuration models with the same degree sequence. The number of triangles in
uniform random graphs is closely related to that in a version of the rank-1
inhomogeneous random graph, where all vertices are equipped with weights, and
the edge probabilities are moderated by asymptotically linear functions of the
products of these vertex weights.
