Conditional configuration graphs with discrete power-law distribution of vertex degrees

The paper is concerned with configuration graphs with discrete power-law distribution of vertex degrees. The parameter of this distribution is a random variable, which is unknown except that it imposes relatively weak constraints on the asymptotic behaviour of the probabilities of large values of degrees. For such graphs with the known number of edges, we find the limiting distributions of the maximal degree of a vertex and of the number of vertices of a given degree for various laws of convergence to infinity of the numbers of vertices and edges. The results in the present paper, which are proved using the generalized scheme of allocation of particles to cells, demonstrate the potency of this method in the case of independent random variables with known limiting behaviour of the tail of the distribution.
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