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, elyager@bezeqint.net Yoseph Shpungin Software Engineering Department Sami Shamoon College of Engineering, Beer Sheva 84100 Israel, yosefs@sce.ac.il

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

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

The structure of 5-dim cube H-5 NUMBER nodes from 00000 to 11111 and connect the “binary” neighbors 00000 10000 00010 00100 01000 00001 The structure of 5-dim cube H-5 00011

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

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:

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

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

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

NODE FAILURE : ALL INCIDENT LINKS FAIL, node remains intact

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)

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

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:

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

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

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

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

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

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

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

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

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

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

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

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 0.40 0.00227 0.053 0.134 0.45 0.070 0.126 0.245 0.50 0.170 0.250 0.395 0.55 0.332 0.424 0.567 0.60 0.538 0.619 0.733 0.65 0.740 0.793 0.846 0.70 0.889 0.914 0.947 0.75 0.967 0.975 0.986 0.80 0.994 0.996 0.998 0.85 0.9996 0.9999 1.0 Regular network H-5 has the smallest DOWN Probability

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 509-512. Elperin,T., Gertsbakh, I. and Lomonosov, M. (1991). Estimation of network reliability using graph evolution models. IEEE Transactions on Reliability, TR-40, 572-581. 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.

Thank you for attention