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## Random graphs, models and generators of scale-free graphs.

#### Abstract

In this paper various models of random graphs describing real networks arising in different research fields: biology, computer science, engineering, sociology are considered. Particular attention is paid to models describing social networks. We start with observation of classical Erdos-Renyi model and basic facts concerning properties of random graphs depending on the value p of the probability of appearance each edge in a graph. Then we observe so-called scale-free model of random graphs proposed by Barabassi and Albert and some its generalizations. Some mathematical results about properties of random graphs in such models are described. For one such model of Bollobas-Riordan we describe results concerning the diameter of random graph, the number of verticies of given degree (powerlaw distribution) and other different properties. This class of models can be characterized by the property that so-called power law constant cannot be chosen in advance and has one fixed value. For example in Bollobas-Riordan model this constant is 3. Finely we observe more general models of random graphs such as Aiello-Chung-Lu model where the power law constant can be chosen as a parameter. In such models there are interesting results concerning evolution properties of random graphs depending on the value of parameter of power law constant. The main example of such property is the size of maximum clique in random graph which we consider in this paper.

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#### Edition

Proceedings of the Institute for System Programming, vol. 22, 2012, pp. 419-434.

ISSN 2220-6426 (Online), ISSN 2079-8156 (Print).

DOI: 10.15514/ISPRAS-2012-22-22