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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.
Pi  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
  EDD  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
Pi  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
Pi  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
 O1,
   is the Rimann’ s zeta - function.


P  N  m N  d 
m
m!
e

1  o1,
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 
 o1.
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
pk  m d i 1 , d i   Pk   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  o1
p1 m d i , d i 1  
exp 

.
m
N 
m!d i  m !d i 1  m !N 
 N
Corollary1 . Let N   then
p1 0 d i , d i 1   1  o1.
Corollary2 . Let N , d i , d i 1   then m  0,1,2,
p1 m d i , d i 1
m

1  o1  d i d i 1 
 d i d i 1 

exp 
.
m!


 N 



N 
Theorem2. Let N   then for m  0,1,2,

1  o1 2
P1  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
 o1,
d i d i 1
 O1,
  N
Then

1  o1  d i d i 1 


p2  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.
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