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  • Author: YU. L. PAVLOV x
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Abstract

We consider the random graphs modelling the structure of large data transmission networks including Internet. We investigate the subset of such graphs consisting of N vertices under the condition that the number of edges is equal to n. We obtain the limit distributions of the maximum degree of vertices and the number of vertices of a given degree as N, n → ∞ so that n/N → λ, where λ is a positive constant.

We find limit distributions of the maximum size of a tree and of the number of trees of given size in an unlabelled random forest consisting of N rooted trees and n non-root vertices provided that N, n → ∞ so that 0 < C 1N / √nC 2 < ∞. With the use of these results, for the unlabelled graph of a random single-valued mapping of the set {1, 2, . . .,n} into itself we prove theorems on the limit behaviour of the maximum tree size and of the number of trees of size r as n → ∞ in the cases of fixed r and r/n 1/3C 3 > 0.

Abstract

We consider conditional random graphs of Internet type under the condition that the number of edges of the graph is known and the degrees of the vertices have no mathematical expectation. We prove limit theorems for the maximum vertex degree and the number of vertices of a given degree.