1. On the two aspects of configuration random graphs’ robustness Institute of Applied Mathematical Research Karelian Research Centre RAS (Petrozavodsk, Russia) Marina Leri, Yury Pavlov

2. Complex networks

3. Approaches to the study of random graph robustness: 1.Link saving = preserving graph connectivity 2. Node survival = forest fire modeling Random graph robustness

4. – i.i.d. random variables, possessing natural values N – the number of nodes, numbered from 1 to N. that are equal to the degrees of the nodes 1, 2,…, N, respectively. 1. Model description: power law (1) h(k) – a slowly varying function, τ - the parameter of node degree distribution. for large values of k, Reittu H., Norros I. On the power-law random graph model of massive data networks. Performance evaluation, 55, 2004. Faloutsos C., Faloutsos P., Faloutsos M. On power-law relationships of the Internet topology. Computer Communications Rev., 29, 1999. Pavlov Yu. L. The limit distribution of the size of a giant component in an Internet-type random graph. Discrete Mathematics and Applications, 17 (5), 2007.

5. Configuration model (Bollobas, 1980): graph construction

6. 3 1 1 9 2 1 2 2 1 1 2 1 1 5 1 3 2 2 1 11

7. 3 1 1 9 2 1 2 2 1 1 2 1 1 5 1 3 2 2 1 11

8. 2 3 1 1 9 3 1 2 2 1 1 2 1 1 5 1 3 2 2 1 11

9. Configuration graph structure: the giant component Power-law random graphs with τ  (1, 2) have only one giant component.

10. Approaches to the study of random graph robustness: 1.Link saving “Random” breakdown “Target attack” 2. Node survival = forest fire modeling Configuration graph robustness

11. 1. Power-law random graph robustness: link saving Cohen R., Erez K., Ben-Avraham D., Havlin S. Resilience of the Internet to Random Breakdowns. Phys. Rev. Lett. Vol. 85. 2000. Durrett R. Random Graph Dynamics. Cambridge: Cambridge Univ. Press, 2007. Reittu H., Norros I. Attack resistance of power-law random graphs in the finite mean, infinite variance region. Internet Mathematics. Vol. 5, N 3. 2008. Bollobas B., Riordan O. Robustness and vulnerability of scale-free random graphs. Internet Mathematics. Vol. 1, N 1. 2004.

12. – the size of the giant component (%) – the size of the second largest component (%) – the number of components Considered graph characteristics: Tangmunarunkit H., Govindan R., Jamin S. et al. Network topology generators: degree- based vs. structural. Proceedings of the SIGCOMM'02. Pittsburgh, USA. 2002. Matsumoto M., Nishimura T. Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Trans. on Modeling and Computer Simulation. Vol. 8. N 1. 1998. How these structure characteristics change with the removal of graph nodes? How these changes depend on the graph size (N) and the parameter τ? 1. Link saving: simulation modeling

13. estimated probability of graph destruction P{A}P{A} r (%)   P{A}P{A} The criteria of graph destruction: “Target attack” “Random” breakdown 1. Link saving: the main result: r – the % of removed nodes

14. r (%) P{A}P{A} “Target attack” 0,010,050,10,50,90,950,99 1,12,22,42,74,97,17,47,6 1,22,02,22,44,46,46,76,9 1,31,82,02,24,05,86,06,2 1,41,61,82,03,65,35,55,6 1,5 1,61,83,34,84,95,1 1,61,31,41,63,04,34,54,6 1,71,21,31,52,73,94,04,2 1,81,11,21,32,43,53,73,8 1,91,01,11,22,23,23,33,4 P{A}P{A} 1. Link saving: the threshold value of graph destruction

15. r (%) P{A}P{A} “Random” breakdown 0,010,050,10,50,90,950,99 1,135,738,942,665,181,683,584,9 1,234,237,340,862,478,279,981,3 1,332,835,839,259,975,176,878,1 1,431,634,537,857,772,474,075,3 1,530,633,336,555,869,971,572,7 1,629,632,335,454,067,769,270,4 1,728,731,334,352,465,767,268,3 1,827,930,433,350,963,865,366,4 1,927,129,632,449,662,163,564,6 P{A}P{A} 1. Link saving: the threshold value of graph destruction

16. Approaches to the study of random graph robustness: 1.Link saving “Random” breakdown “Target attack” 2. Node survival = forest fire modeling Configuration graph robustness

17. 2. Configuration graph robustness: “Forest fire model” Annakov B. B. Bank crisis and forest fires – What's in common? 2008. Available at: http://www.empatika.com/blog/agent_modeling _forest_fire#more-403. http://www.empatika.com/blog/agent_modeling _forest_fire#more-403 Bertoin J. Burning cars in a parking lot. Commun. Math. Phys. Vol. 306. 2011. Bertoin J. Fires on trees. Annales de l'Institut Henri Poincare Probabilites et Statistiques. Vol. 48(4). 2012. Arinaminparty N., Kapadia S., May R. Size and complexity model financial systems. Proceedings of the National Academy of Sciences of the USA. Vol. 109. 2012. Pavlov Yu. L., Khvorostyanskaya E. V. Whether the tree will burn in a fire in random forest? Proc. of the Karelian Research Centre of the RAS. Ser. Mathematical modeling and information technologies. Iss. 3. № 5. 2012.

18. – i.i.d. random variables, possessing natural values N – the number of nodes, numbered from 1 to N. that are equal to the degrees of the nodes 1, 2,…, N, respectively. 2. Configuration random graphs: power-law and Poisson (3) (2)

19. 2. Forest graph topology m – an average inner node degree

20. 2. Forest graphs: τ of N and λ of N relations m = λτ : ζ(τ)=mN 1.016.753350 1.212.963600 1.332.533750 1.422.323900 1.52.194000 1.62.054200 21.735000 2.661.494780 31.424489 41.295578 51.236700 5.331.217500 61.188350 71.168911 81.1310000 Lattice size: 100  100

21. 2. Configuration graph: a destruction process 3 1 1 9 3 1 2 2 1 1 2 1 1 5 1 3 2 2 1 11 0 < p  1 – the probability of link destruction

22. 0 1 0 0 0 1 0 1 2 1 11 2. Configuration graph: a destruction process

23. Approaches to the study of random graph robustness: 1.Link saving “Random” breakdown “Target attack” 2. Node survival = forest fire modeling “Random” fire start “Target” fire start Configuration graph robustness

24. 2. Node survival: simulation modeling Power-law simulation models: Poisson simulation models: N  [3000, 10000], p  (0, 1] The aim: to compare power-law and Poisson graph topologies. Which of then would ensure a better (more) node survival in case of a fire for the same graph size (N) and the probability of link destruction (p)?

25. Random fire start 2. Node survival: power-law vs. Poisson g – the number of survived (remained in a graph) nodes.

26. Target fire start 2. Node survival: power-law vs. Poisson g – the number of survived (remained in a graph) nodes.

27. Random fire start 2. Node survival: power-law vs. Poisson Vertical axes (positive direction) – % of cases where g power-law > g Poisson. Vertical axes (negative direction) – % of cases where g power-law < g Poisson.

28. Target fire start 2. Node survival: power-law vs. Poisson Vertical axes (positive direction) – % of cases where g power-law > g Poisson. Vertical axes (negative direction) – % of cases where g power-law < g Poisson.

29. 2. Node survival: power-law or Poisson? Target startRandom start – g power-law > g Poisson – g power-law < g Poisson

30. 2. Node survival: power-law or Poisson? Poisson power-law Which graph topology will ensure a better survival of nodes? Target fire start