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

Yu. L. Pavlov ; M. M. Stepanov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 212-230

Résumé

We consider a random graph constructed by the configuration model with the degrees of vertices distributed identically and independently according to the law $\mathbf P\{\xi\geq k\}=k^{-\tau}$, $k=1,2,\dots$, with $\tau\in(1,2)$. 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.