1. On the behaviour of an edge number in a power-law random graph near a critical points E-mail: f_len@mail.ru, pavlov@krc.karelia.ru E. V. Feklistova, Yu. L. Pavlov Institute of Applied Mathematical Research, Karelian Research Centre, Russian Academy of Sciences Petrozavodsk, 15-18 September 2014 RuFiDiM - Third Russian Finnish Symposium on Discrete Mathematics

2. Many papers have been published lately on random graphs that model complex communication networks such as the Internet, mobile communication systems and social networks. We consider a model which as known from the literature and shown in these papers, describes the structure and dynamics of real networks quite well.

3. First this model was considered in these papers: 1.Aiello, W., Chung, F., & Lu, L. A random graph model for massive graphs. Proc. of the 32nd Annual ACM Symposium on Theory of Computing. 2000. P. 171-180. 2.Faloutsos M., Faloutsos P., Faloutsos C. On power-law relationships of the Internet topology. // Computer Communication Rev. 1999. Vol. 29. P. 251-262. 3.Reittu H., Norros I. On the power-law random graph model of massive data networks. // Performance Evaluation. 2004. Vol. 55. P. 3- 23.

4. , (1) where has the value from 1 to, has the value from 1 to infinity, is a positive parameter. We consider power-law random graphs with vertices. The degrees of vertices are independent identically distributed random variables drawn from the following law:

5. Construction of the graph Let stand for the total number of stubs, Construction of the graph consists of two stages. At the first stage degrees of all vertices are determined with the help of distribution (1). It is considered that vertices have outgoing stubs, i.e. edges coming out of a vertex for which the connected vertices are not yet known. All stubs of vertices are numbered in an arbitrary order. At the second stage each stub is connected to another equiprobably to form edges. Such graphs are called power-law random graphs.

6. Construction of the graph 1 2 3 456 0

7. Denote - the value of the Riemann zeta-function at the point Properties of graphs significantly depend not only on the number of vertices, but also on the number of edges. The number of edges in a graph is a random variable and is equal to or depending on whether the amount is even or not, respectively. Thus, the study of the limiting behavior of the number of edges is equivalent to studying the asymptotic behavior of.

8. Y. L. Pavlov, E. V. Feklistova. Limit distributions of the edge number in random configuration graph. European researcher, 48(5-1): 1097-1100, 2013. In this paper local limit theorems were proved for sum when is fixed, tends to infinity.

9. Theorem 1 Let τ > 2, N →∞. Then where

10. Theorem 2 Let τ = 2, N →∞. Then where

11. Theorem 3 Let 1 < τ < 2, N →∞. Then where – the density of stable distributions with characteristic function:

12. Theorem 4 Let τ = 1, N →∞. Then – the density of stable distributions with characteristic function: where

13. Theorem 5 Let τ < 1, N →∞. Then where – the density of stable distributions with characteristic function:

14. From these theorems one can see that the limit distributions of the edge numbers change when passes points and For the limit distribution for the sum of vertex degrees is normal, but for the limit distributions are stable. At the same time, stable limit distributions for and differ from each other. Therefore, points and can be called critical points. The structure and properties of the graph significantly depend on the behaviour of parameter

15. Denote - the variance of random variable We proved local limit theorems for sum in case when depends on and changes in the vicinity of the critical points.

16. Theorem 6 Let N →∞, sequence choose so that: Then

17. Theorem 7 Let N →∞, Then sequence choose so that:

18. Theorem 8 Let N →∞, Then Euler's constant – the density of stable distributions with characteristic function: where

19. Thank you for your attention!