1. PROBABILISTIC DESCRIPTION of NETWORK
BEHAVIOR UNDER RANDOM ATTACK on ITS
NODES
Ilya B. Gertsbakh
Department of Mathematics, Ben-Gurion University
P. O. Box 653, Beer-Sheva, 84105, Israel,
Yoseph Shpungin
Software Engineering Department
Sami Shamoon College of Engineering, Beer Sheva Israel,

2. This work compares the probabilistic behavior of networks
having different structure when their
nodes fail in random order

3. The five-dimensional cube H-5
All nodes have
d=5 1

4. The structure of 5-dim cube H-5
NUMBER nodes from to and connect the “binary” neighbors
00000
10000
00010
00100
01000
00001
The structure of 5-dim cube H-5
00011

5. One hub with d= 15, one with d=10, many nodes with d=4-5
32 nodes, 80 links,
average d=5
222
One hub with d= 15, one with d=10, many nodes with d=4-5

6. From “Linked” of A. Barabasi: The Birth of scale-free topology… When deciding where to link new nodes prefer to attach to the more connected nodes. Thanks to growth and preferential attachment, a few highly connected hubs emerge
3/8
2/8
1/8
2/8
New Node:

7. Source: “Linked”, Barabasi
9/11/2001 terrorist network
Source: “Linked”, Barabasi

8. Node degrees are more dispersed that in PREFNET
Originally, 34 nodes, 91 links,
Slightly modified to 32 nodes and 80 links
Average degree d=5,
Two hubs with d=16 and , d=15, several nodes with
small degree 1-3
Node degrees are more dispersed that in PREFNET

9. MAX becomes smaller or equal 5 on 5-th step of destruction
D-spectrum 1 6 8 3 1 9 5 7 2 4 10
1->2->3->4->5->6->7->8->9 -> 10: Nodes fail in random order
MAX= (10) | 9, 7, 7,6, 5, 2, 2, 2, 1, 1
MAX becomes smaller or equal 5 on 5-th step of destruction

10. NODE FAILURE : ALL INCIDENT LINKS FAIL, node remains intact

11. F(x) = Pr (DOWN after x components have failed)
Network failure: Maximal connected component has 5 or less nodes
f(i)= Pr ( Network fails on the i-th step of destruction)
f(1), f(2), f(3),…,f(n) –Signature or D-spectrum
F(x)=f(1)+f(2)+…+f(x), x=1,2,…,n is called
Cumulative signature or cumulative D-spectrum
F(x) = Pr (DOWN after x components have failed)

12. The cumulative D-spectrum is our main formal tool to investigate
the network probabilistic behavior
in the destruction process

13. Monte Carlo procedure to estimate the D-spectrum:
1. Define network UP state and its complement DOWN state. Number the network components as 1,2,…,n and generate a random permutation
\pi=[i(1),i(2),…,i(n)] of these numbers.
2. Assume that all components are initially up and erase
(destruct) one after another moving along the permutation from left to right.
3. Check the state of the network on each stage of the destruction process and remember the position j of the component i(k) whose failure coincides with the network failure.
Repeat this procedure M times by generating M random permutations.
Let N(j) be the number of permutations, out of M, such that the system failed
on the j-th step of the destruction process. As the Monte Carlo estimate of F(x), take the quantity
F*(x) = [N(1) + ::: + N(x)]/ M:

14. Internal Distribution
Signature:
System has n components with I.I.D lifetimes X(1),X(2),…,X(n). Let X(1:n), X(2:n),…,X(n:n) be the corresponding order statistics. System failure time coincides with X(k:n) with probability s_k.
The collection S= (s_1,s_2,…,s_n) is called signature.
By Total Probability Law, the system CDF G_sys(t) can
be presented as
Samaniego (1985),
Elperin et al (1991)-
Internal Distribution
Important:The signature S coincides with the D-spectrum: s_k=f(k), k=1,2,…,n
Import

15. Denote: P(X <= t_0)=q. F(x)- cumulative D-spectrum.
After some simple algebra, obtain:
Network is DOWN (at t_0)
Component is down (at t_0)
The particular form CDF of component G(t)=P(X <=t) is not important,
Important is only its value q at t_0

16. C(x)=F(x) n!/[ x! (n-x)!] Number of system failure sets of size x
Important combinatorial property of
the cumulative D-spectrum F(x):
C(x)=F(x) n!/[ x! (n-x)!]
Number of system failure sets of size x

17. F(x)=Pr(the network has MAX less or equal 5 after x nodes have failed)
F(x) is a CDF 1
F(x) x

18. D-spectra for H-5, max=5, max=4, max=3
f (x) - discrete density x
D-spectra for H-5, max=5, max=4, max=3

19. Cumulative D-spectra for H-5: max=12, max=10,
max=5, max=4, max=3 , from left to right

20. 11 nodes or more 10 or less nodes Network DOWN state:
FOR all three networks the UP state is defined as follows:
The size of MAXIMAL connected component has
11 nodes or more
Network DOWN state:
Maximal connected component has
10 or less nodes

21. Which of the three networks
What do you think:
Which of the three networks
is the most resilient ?

22. F(x) curves for H-5, PREFNET and TERNET
5-dim Cube
TERNET
Prefnet
F(x) curves for H-5, PREFNET and TERNET

23. TERNET
H_5
F(x) curves for H-5, PREFNET and TERNET

24. Conclusion : Regular networks are more resilient to the random
“attack” on their nodes
than the scale –free networks

25. Regular network H-5 has the smallest DOWN Probability
Network DOWN probability
Node down probability
q P(DOWN; q) H5 P(DOWN; q) Prefnet P(DOWN; q) Ternet
Regular network H-5 has the smallest DOWN Probability

26. REFERENCES
Barabasi, A-L. (2003). Linked. Penguin Books, Ltd.
Barabasi, A. and Albert, R. (1999). Emergence of scaling in random networks, Science, V. 286, No.5439, pp
Elperin,T., Gertsbakh, I. and Lomonosov, M. (1991). Estimation of network reliability using graph evolution models. IEEE Transactions on Reliability, TR-40,
Gertsbakh, I. and Shpungin, Y. (2009). Models of Network Reliability: Analysis, Combinatorics and Monte Carlo. CRC Press.
Mitzenmacher, M. and Upfal, E. (2005). Probability and Computing. Cambridge University Press.
Samaniego, F. (1985). On the closure of the IFR under formation of coherent systems, IEEE Trans. TR-34,69-72.
Samaniego, F. (2007). System Signatures and Their Application in Engineering Reliability. Springer.

27. Thank you for attention