Abstract
The configuration graph where vertex degrees are independent identically distributed random variables is often used for modeling of complex networks such as the Internet. We consider a random graph consisting of N vertices. The random variables η 1 ,….,η N are equal to the degrees of vertices with the numbers 1,… ,N The probability P{η i =k}, i=1,…, N, k=1,2,…., is equivalent to h(k)/k τ as k→∞, where h(x) is a slowly varying function integrable in any finite interval, τ>1. We obtain the limit distributions of the maximum vertex degree and the number of vertices with a given degree under the condition that the sum of degrees is equal to n and N,n→∞.
Highlights
The study of random graphs has been causing growing interest in connection with the wide use of these models for the description of complex networks. Such models can be used to adequate by describe the topology of transport, electricity, social, telecommunication networks and global Internet
Observations on real networks showed that their topology can be described by random graphs with vertex degrees being independent identically distributed random variables with power-law distribution
That is why in [11] it was suggested that the distribution of the vertex degree η is
Summary
The configuration graph where vertex degrees are independent identically distributed random variables is often used for modeling of complex networks such as the Internet. We consider a random graph consisting of N vertices. ΗN are equal to the degrees of vertices with the numbers 1, . We obtain the limit distributions of the maximum vertex degree and the number of vertices with a given degree under the condition that the sum of degrees is equal to n and N, n → ∞. ПРЕДЕЛЬНЫЕ РАСПРЕДЕЛЕНИЯ СТЕПЕНЕЙ ВЕРШИН В УСЛОВНОМ КОНФИГУРАЦИОННОМ ГРАФЕ. Конфигурационный граф, степени вершин которого являются независимыми одинаково распределенными случайными величинами, часто используют для моделирования сложных сетей, таких как Интернет. Найдены предельные распределения максимальной степени вершин и числа вершин заданной степени при условии, что сумма степеней равна n при N, n → ∞.
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