1. On the stopping of destruction process in the Internet-type random graphs
Marina Leri
Institute of Applied Mathematical Research
Karelian Research Centre
of the Russian Academy of Sciences
(Petrozavodsk)

2. Complex networks Telecommunication Social networks
Index of citation and co-authorship
Internet

3. AS-Graph Nodes (vertices) – Autonomous Systems
Edges – direct connection between Autonomous Systems

4. Power-law random graph
Random graph of Internet type
Power-law random graph
N – the number of vertices, numbered from 1 through N
– i.i.d. random variables, possessing natural values
that are equal, respectively, to the degrees of the vertices 1, 2,…, N
(1)
In the case when τ  (1, 2) vertex degrees distribution (1) has finite expectation and infinite variance.
Reittu H., Norros I. On the power-law random graph model of massive data networks. Performance evaluation, 2004, 55, p.3-23.
Pavlov Yu. L. The limit distribution of the size of a giant component in an Internet-type random graph. Discrete Mathematics and Applications, 2007, vol. 17, iss. 5, p

5. Graph construction

6. The giant component – a connected set of vertices the number of which has an expectation c·N:
Denote:
– sizes of graph
components
in decreasing order

7. Simulation modeling Considered graph characteristics:
– the size of the giant component
– the size of the second biggest component
– the number of components
Design of experiment:
N: 103, 3·103, 5·103, 104, 5·104, 105
τ  (1, 2): 1.1, 1.2, . . ., 1.9
For each pair (N, τ) there were generated 100 graphs.

8. The number of vertices in the giant component (%)
Estimated value of constant c

9. The number of vertices in the second component 

10. The number of components (S)

11. Graph structure
 = 1.1
 = 1.5
 = 1.9

12. Graph destruction as a “targeted attack” on the vertices with the highest degrees.

13. Graph destruction Consider the following event:
The occurrence of event A – the criteria of graph destruction.
Design of experiments:
N: 103, 3·103, 5·103, 104
τ  (1, 2): 1.1, 1.2, . . ., 1.9
For each pair (N, τ) there were generated 100 graphs.

14. Volumes of the giant (η1) and the second biggest components (η2)r r
Where r is the % of vertices removed from the graph.

15. η1 (%)
η2 (%)
r (%)
η1 (%)
η2 (%)
r (%)

16. The number of components
Graph volume (%)
r (%)

17. The probability of graph destruction
P{A}
r (%)
where

18. The threshold value of graph destruction
P{A}
Probability of graph destruction P{A} τ
0,01
0,05
0,1
0,5
0,9
0,95
0,99
1,1
2,2
2,4
2,7
4,9
7,1
7,4
7,6
1,2
2,0
4,4
6,4
6,7
6,9
1,3
1,8
4,0
5,8
6,0
6,2
1,4
1,6
3,6
5,3
5,5
5,6
1,5
3,3
4,8
5,1
3,0
4,3
4,5
4,6
1,7
3,9
4,2
3,5
3,7
3,8
1,9
1,0
3,2
3,4

19.  = 1.1 0%
0.1% 2%
P{A} 6% 8%

20.  = 1.9 0%
0.1%
P{A} 1% 2% 3%

21. Thank you!