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Limit theorems for the number of multiple edges in the configuration graph Irina Cheplyukova Karelian Research Centre of Russian Academy of Sciences [email protected] CONFUGURATION MODEL B,Bollobas (1980). A Probabilistic proof of an asymptotic formula for the number of labeled regular graphs. European Journal of Combinatorics. Vol.1. P. 311-316. • model with fixed degree sequence M.Molloy and B.Reed (1995). A critical point for random graphs given Random Structures and Algorithms. Vol.6. P.161-179. degree sequence. • model with independent identically distributed vertex degrees M.E.J. Newman, S.H. Strogatz, D.J. Watts.(2001) Random graphs with arbitrary degree distribution and their applications, Phys. Rev. E 64 026118. A.-L. Barabasi, R. Albert.(1999) Emergence of scaling in random network, Science 286, P.509-512. Faloutsos C., Faloutsos P.,Faloutsos M. (1999) On power-law relationships of the internet topology. Computer Communications. Rev. 29. P. 251−262. Pi k pk , k 1,2,; i 1,2,, N . 2 3 1 6 4 5 H. Reittu, I. Norros (2004). On the power-law random graph model of massive data networks. Performance Evaluation. 55, 3-23. 0 • Erdös P., Rényi A. (1960) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. Vol.5. P. 17−61. • Hofstad R., Hooghiemsra G., Znamenski D. (2007) Distances in random graphs with finite mean and infinite variance degrees. Electronic Journal of Probability. Vol.12. P.703−766. • Janson S., Luczak T., Rucinski A. (2000) Random graphs. New York: Wiley, 348p. • Pavlov Yu.L. (2007) On power-law random graphs and branching processes. Proceedings of the Eight International Conference CDAM. Minsk: Publishing center BSU. Vol.1. P. 92−98. • Bollobas B.(1980) A probabilistic proof of an asymptotic formula for the number of labeled European Journal of Combinatorics. Vol.1. P. 311−316. regular graphs. Hofstad R. Random graphs and complex networks. 2011. N , DN D, lim EDN ED, lim EDN2 ED 2 EDD 1 ED . The number of loops and number of multiple edges in graph asymptotic ally has Poisson' s distributi on with parameters 2 and 2 4 by the correspond ingly. The first configuration graph Let 1 , , N be random variables equal to the degrees of vertices 1, , N which have a binomial distributi on with parameters (N,p) N k N k Pi k p 1 p , i 1, , N , k where parameter p p(N) is determined by Np , N ,0 . Power-law random graph Aiello W., Chung F., Lu L. A random graph model for power-law graphs. Experiment Math., 10, 1, 2001, 53-66. Newman M.E.J., Strogats S.H., Wats D.J. Random graphs with arbitrary degree distribution and their appliсations. Phys. Rev. E., 64, 026118, 2000. Fundamenta l property of many real networks is that the number of nodes with degree k is close to proportion al to k , k , where is positive constant. This is called a ' power - law degree sequence' and correspond ing graph is also called the power - law random graph. The second random graph Pi k k , k 1,2, ., 1,2. Faloutsos C., Faloutsos P., Faloutsos M. (1999) On power - law relationsh ips of the internet t opology. Computer Communicat ions. Rev. 29. P. 251 - 262. Yu.Pavlov, M.Stepanov. Limit distribution of the number of loops in a random graph. (2013) Proceeding of Steklov Institute of Mathematics. Volume 282. Issue 1. Pp.209-219. N , d2 2 N O1, is the Rimann’ s zeta - function. P N m N d m m! e 1 o1, m 0,1, Yu.Pavlov, M.Stepanov. Limit distribution of the number of loops in a random graph. (2013) Proceeding of Steklov Institute of Mathematics. Volume 282. Issue 1. Pp.209-219. Theorem. Let N P N xN 1 e x o1. where N max 1 , , N . Let us choose two vertices of the random graph, for example, vertices with numbers i 1 and i . Let d i 1 and d i be the degrees of these vertices. Assume that d i d i 1. Let ( k ) , k 1,2 be random variable be equal to the number of edges joining vertices with numbers i 1 and i . Let pk m d i 1 , d i Pk m i 1 d i 1 ,i d i , m 0,1,2, , k 1,2. Theorem1. Let N then for m 0,1,2, d i2 d i d i 1 d i !d i 1!1 o1 p1 m d i , d i 1 exp . m N m!d i m !d i 1 m !N N Corollary1 . Let N then p1 0 d i , d i 1 1 o1. Corollary2 . Let N , d i , d i 1 then m 0,1,2, p1 m d i , d i 1 m 1 o1 d i d i 1 d i d i 1 exp . m! N N Theorem2. Let N then for m 0,1,2, 1 o1 2 P1 m m , m m!N where E m , m 1,2, m 1, m 0; E E 1 m 1. m Theorem3. d i2 N Let N , d i , d i 1 such that o1, d i d i 1 O1, N Then 1 o1 d i d i 1 p2 m d i , d i 1 m! N m d i 1 O N 2 3 . d i d i 1 exp . N Thanks for your attention.