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)

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

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

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. 425-437.

Graph construction

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

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.

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

The number of vertices in the second component  

The number of components (S) 

Graph structure  = 1.1  = 1.5  = 1.9

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

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.

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

η1 (%) η2 (%) r (%) η1 (%) η2 (%) r (%)

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

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

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

 = 1.1 0% 0.1% 2% P{A} 6% 8%

 = 1.9 0% 0.1% P{A} 1% 2% 3%

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