Limit distributions of the number of loops in a random configuration graph

We consider a random graph constructed by the configuration model with the degrees of vertices distributed identically and independently according to the law P(ξ≥k), k = 1, 2, …, with τ ∈ (1, 2). Connections between vertices are then equiprobably formed in compliance with their degrees. This model admits multiple edges and loops. We study the number of loops of a vertex with given degree d and its limiting behavior for different values of d as the number N of vertices grows. Depending on d = d(N), four different limit distributions appear: Poisson distribution, normal distribution, convolution of normal and stable distributions, and stable distribution. We also find the asymptotics of the mean number of loops in the graph.