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Q . ., . ., . ., . . , ~5·103 , . , , , , . , q, . , . , . . Faloutsos 1999 , [1]. , . , , D f = 1.5 ± 0.1 [2]. -- . [2]. 3015 - 4389 -- = 2.1 - 2.2 , 3888 -- = 2.48 , 150000 -- = 2.4 . , C rand 0.0001 . -- . , , , [2]. , . , , , , , . , .

, 5·103 , . , . , , . ( SIS). . , (), c . . , N ~ 5 10 3 . kmax = 112. , . 1. knn(k), . 2.
1 1 10 100 1000

100

0, 1

knn (k )

P (k )

0, 01

10

0,001

0, 0001 k

1 1 10

k

100

1000

. 1. , .. , , k

. 2. knn(k)

, k knn , . b [3]. bi jl i j l ( j l ) , i . i bi = bi jl .
j l

. . 3 , . 4 -- P(b) , .. , , b. P(b) .

100000000 10000000 1000000 100000 b (k )
P (b )

1 1 10 100 1000 10000 100000 1000000 10000000 10000000 0

0,1

10000 1000 100

0,01

0,001

10 1 1 10 k 100 1000
0,0001 b

. 3. b(k)

. 4. P(b)

, C , , . , . , . 5.
1 1 10 100 1000

0,1

P (L )

0,01

0,001

0,0001 L

. 5. ( )

. , : , [2]. . (i) p m . , -- . ,
1 k i = pA + pA N t (i )

ki + 1 k j +1 j

(

)

(1)

N -- . k = 2m , A = m . (ii) q m . :
1 k i = - qB + qB N t (ii )

ki + 1 k j +1 j

(

)

(2)

, , -- , . . , , , B = m . (iii) 1 - p - q , m . ,
k i = (1 - p - q )C t (iii )

ki + 1 . k j +1 j

(

)

(3)

, , , m , , , C = m . (1), (2) (3),
k i k i 1 k k = ( p - q )m + (1 - p - q = + i + i N t t (i ) t (ii ) t (iii )

)

k

ki + 1 k j +1 j

(

)

(4)

N
t N () = m0 + (1 - p - q )t

j

j

-

j

k j = (1 - q )2mt , t

m

, . N () = (1 - p - q )t j (k j + 1) = (1 - q )2mt + N = (1 - q )2mt + (1 - q - p )t , (4) t
0

:
ki + 1 k i 1 = ( p - q )m + (1 - p - q )m . (1 - p - q )t (1 - q )2mt + (1 - q - p )t t

(5)

2m(1 - q ) 2m(1 - q ) + (1 - p - q ) A( p, q, m ) = ( p - q ) . 1 - p - q + 1 B( p, q, m ) = m (5)
k i 1 = [A( p, q, m ) + 1 + k i ] t B( p, q, m )t

(6)

, k i (t ) , k, P[k i () < k ], t P[k i () < k ] = P[t i > C ( p, q, m )t ] , t

t B( t (6) k i () = [A( p, q, m ) + 1 + m] t i

1 p ,q ,m

)

- A( p, q, m ) - 1 .

C ( p, q, m ) =

( A(

p, q, m ) + m + 1)

B ( p ,q ,m

)

(k

+ m + 1)

B ( p ,q ,m

)

.

t i 0 t i t , : (i) C ( p, q, m ) > 1 , P[k i (t ) < k ] = 0 . (ii) 0 < C ( p, q, m ) < 1 , P(k ) . , , 1 C ( p, q, m )t , , t t , P[k i () < k ] = P[t i > C ( p, q, m )t ] = 1 - P[k i () > k ] = 1 - Pi (t i ) = m0 + t m0 + t P[k i (t ) < k ] P (k , t ) = , k
P(k , t ) = D( p, q, m

)

t m0 + t

(k

+ A( p, q, m ) + 1)

-1- B ( p ,q ,m

)

,

D( p, q, m ) = B( p, q, m )[m + A( p, q, m + 1)] . , . -- , , t k , , , t , k [3]. t :
B ( p ,q ,m

)

Pst (k ) = D( p, q, m )(k + A( p, q, m ) + 1)

-1- B ( p ,q ,m

)

.

( p, q, m ) = A( p, q, m ) + 1 ( p, q, m ) = B ( p, q, m ) + 1 ,
Pst (k ) = D( p, q, m )( ( p, q, m ) + k

)-

( p ,q ,m

)

.
- ( p ,q ,m

Pst (k ) = D( p, q, m )( ( p, q, m

))

- ( p , q , m

)

1 +

1
( p,q,m

)

k

)

.

1 1 1 ( p, q, m ) = -(1 - q ) = - ( p, q, m ) . k 0 = k0 1- q ( p, q, m ) ( p, q, m )

k 1 - (1 - q ) . k0 Pst (k ) , Pst (k ) = D( p, q, m )( ( p, q, m

))

- ( p , q , m

)

1 1- q

k 1- q k 1-q 1 Pst (k ) = 1 - (1 - q ) , Z = 1 - (1 - q ) dk . (7) Z k0 k0 0 , [4], q 1 , k .

1

1

. . , . , . S q

[4]:
Sq =

k

q pk - 1

1- q

,

(8)

q -- p k k . (8) (9) pk = 1 ,
k

, P(k )k = , -- P (k )
P(k ) = p c
q k

(10)

k

, c(q) =

q

k

q pk ,

(11)

P(k) -- . (8) (9) (10)
p k

k'

q pk ' - 1

1- q

-

k'

P(k ' )k ' -

k'

-- .
p k = (g

pk ' = 0 ,
1 1- q

)

1 1- q

(1 - q) q

1 q -1

(9)

(1 - q) k 1 - cq

.

g

1 1- q

(

(1 - q )
q

)

1 q -1

k

1 (1 - (1 - q ) k ) cq

1 1- q

=1
1 1- q

=
1 pk =

k

(1 - (1 - q)
1 1- q

c
q

k)

1 1- q

, g

(

(1 - q )
q

)

1 q -1

=

-1

,

1 - (1 - q ) k cq

(11),

P(k ) =

(1
k

- (1 - q )k

)1- )

(1

- (1 - q )k

q q q 1- q

. .

0

P( x )dx = 1

(
0

x - x0 )P( x )dx = 0 .

Sq =

0

p q (x )dx - 1 1- q

.

, ,

P( x) =

(1
0

- (1 - q) x

)1-

q q

dx(1

- (1 - q ) x

)

q 1- q

.

(12)

.
. SIS, , : (S) (I) [5]. , , . , . , S (i ) + I ( j ) I (i ) + I ( j ), I (i ) S (i ), -- . S I

( , ). , c , , < c -- , > c , .. .
1 k( ) kP(k ) 1 + k( ) = 1 , k k =0 k -- , (k ) [6] d d
( ) = 1 k

(13)

k

kP(k )

k( ) . 1 + k( )

P(k ) -- . (13) , c =

k k
2

.

, , p, q m, , . , SIS. (t ) < c 6, > c 7.
0,003

0,002

0,4

(t )

(t )
0,2

0,001

0 0 10 t 20 30

0 0 20 t 40

. 6. < c

. 7. > c

( ) 8.

1

1

0,8

0,8

0,6

0,6 ()

( )

c =0, 06, kmax=112 c =0, 08, kmax=56 c =0, 09, kmax=48 c =0, 11, kmax=36

0,4

0,4

0,2

0,2

0 0 0 ,5

1

0 0 0,2 0,4 0,6 0,8 1

. 8. ( )

. 9. ( ) kmax

, (), c . 9 , () .
. , , , . , : q = 1.1. ,

p (k ) ~ k - . 2.8 . , , . . , , , , , . ( k nn (k ) , ) . , p( L) ~ L- , = 1.2 . [6]. . . , -

: c 0,06 . , , c .
1. Faloutsos M., Faloutsos P., Faloutsos C. On PowerLaw Relationships of the Internet Topology // ACM SIG-COMM '99 Comput. Commun Rev. 1999. Vol. 29. P. 251. 2. Yook S.-H. Jeong H. and BarabАsi A.-L. Modeling the Internet's large-scale topology // Proceedings of the Nat'l Academy of Sciences. 2000. Vol. 99. P. 13382-13386. 3. Albert R., Barabasi A.-L. Statistical Mechanics of Complex Networks // Rev. Mod. Phys. 2002. Vol. 74, 1. P. 4397. 4. Tsallis C. Nonextensive statistics: theoretical, experimental and computational evidences and connections// Braz. J. Phys. 1999. Vol. 29. P. 1-35. 5. Pastor-Satorras R., Vespignani A. Evolution and Structure of the Internet: A Statistical Physics Approach. -- Cambridge: Cambridge University Press, 2004. 267 . 6. Dezzo Z. and BarabАsi A.-L. Halting viruses in scalefree networks // Phys. Rev. E. 2002. Vol. 65. P. 055103.

THE CONTACT PHENOMENA IN NETWORKS WITH Q TYPE STATISTICS Gadjiev B. R., Kalinkina E. A., Kryukov Yu. A., Progulova T. B.

We investigate the computer network consisting of ~5·103 computers which is a subnet of the Internet. On the base of data analysis we determined distribution functions of degree, of betweenness centrality and of distances between nodes. Dependences of an average degree of the nearest neighbours vertices and betweenness centrality on a degree of nodes are constructed. The model of a growing network is introduced and the qtype statistics is derived on its basis. It is shown that the results of this model adequately describe the studied network. According to the offered model we simulated statistically equivalent networks on which computer viruses spreading processes are investigated. We find that the more biased a policy is towards the hubs, the more chance it has to bring the epidemic threshold above the virus' spreading rate