Redes Complexas: teoria, algoritmos e aplicações em computação Bloco #7

Transcription

1 Redes Complexas: teoria, algoritmos e aplicações em computação Bloco #7 ``Security and Complex Networks Virgílio A. F. Almeida Outubro de 2009 D d Ciê i d C ã Departamento de Ciência da Computação Universidade Federal de Minas Gerais

2 Security and Complex Networks Análise baseada na teoria das redes 1. Error an attack tolerance of complex networks The internet s achilles heel: Error and attack tolerance of complex networks R Albert, H Jeong, AL Barabasi Nature, Fractal Approach to Social Network Attack Detection 2. The topology of dark networks Communications of ACM, October 2008 E a parte algoritimica???

3 Revisão de Artigos The internet s achilles heel: Error and attack tolerance of complex networks R Albert, H Jeong, AL Barabasi Nature, /03%20Journal%20Articles/Physics/ErrorAttack_Nature%2 0406%20,%20378%20(2000).pdf Efficiency of scale free networks: Error and attack tolerance of complexnetworks networks, Crucitti, V Latora, M Marchiori, A Rapisarda Physica A: Statistical Mechanics and its Applications, 2004 Elsevier

4 Ilustração visual das diferenças

5 Falhas e Ataques a Redes Falha remoção de nodos randômicos da rede Ataque remoção de nodos escolhidos (i.e., importantes da rede). Consequência: perda de integridade da rede, caracterizada pela presença de um giant connected component. Métricas para medir o impacto do ataque Caminho mínimo médio (distância) Tamanho relativo do maior componente conectado Tamanho médio dos componentes conectados, excluindo o gigante

6 Idéia de Robustez Sistemas complexos mantem suas funções básica mesmo em face de erros e falhas 1 S f c 0 1 Fração de nodos removidos, f Falha de nodo

7 Robustez das Redes scale free Falhas Topologia e Tolerância 1 S γ 3 : f c =1 (R. Cohen et al PRL, 2000) 0 f f 1 c Ataques

8 Respostas da Rede a Falhas ou Ataques

9 Mudanças no Diâmetro das Redes

10 Fragmentação das Redes sob Ataques

11 Calcanhar de Aquiles das Redes Complexas falha ataque Internet R. Albert, H. Jeong, A.L. Barabasi, Nature (2000)

12 Estrutura Geral de Ataques a Web 2.0

13 Próximos Slides estão disponíveis: Efficiency of scale free networks: error and attack tolerance. P. Crucitti, V. Latora, M. Marchiori, A. Rapisarda. it/ f ti /2003/t /C itti

14 Efficiency ce cyof Scale-free Scae ee networks: error and attack tolerance. P. Crucitti in collaboration with V. Latora (Dip. di Fisica e Astronomia, Università di Catania and INFN sezione di Catania) MM M. Marchiori i (W3C and Lab. For Computer Science, Massachusetts Institute of Technology) A. Rapisarda (Dip. di Fisica e Astronomia, Università di Catania and INFN sezione di Catania)

15 Overview Whys and wherefores of studying error and attack tolerance. Efficiency of ER random and BA scale-free networks under attacks and errors. A new approach: cascading attacks and errors.

16 Reasons for studying error and attack tolerance Designing robust networks Protecting existing networks

17 Quantities for studying error and attack tolerance Characteristic path length L = N 1 ( N 1) d ij i, j G i j Problem: Because of errors and attacks, networks can become nonconnected d = + L = + d i * j* Possible solution: restrict the analysis to the main connected component But

18 Let us consider two simple graphs G1 and G2, composed by 5 nodes each: L(G 1 )= 13/10 L(G 2 ) = 1 L(G 1 )>L(G 2 ) Considering L, G2 seems to have better structural properties than G1!

19 Efficiency ij 1 E ( G ) = εij N ( N 1) i, j G i j 1 where ε ij = is the efficiency in communication between i and j. d (Latora and Marchiori, PRL 87 (2001) , EPJB 32 (2003) 249) If nodes i and j are not connected, we assume: dij = + ε ij = 0 Efficiency is a well defined quantity also for non-connected networks! In fact

20 E(G 1 1) )=17/20 E(G 2 2) )=3/10 E(G 1 )>E(G 2 ) In perfect agreement with the fact that G1 is better connected than G2!

21 Definitions An attack is: a targeted removal of the most important nodes Now we consider nodes with highest degree Later we will consider nodes with highest h bt betweenness An error is: a removal of randomly selected nodes

22 E Results: global efficiency y( (few removals) N=5000 K=10000 Following the idea of Barabási et al., Nature 406 (2000) 378 Crucitti, Latora, Marchiori, Rapisarda, Physica A 320 (2003) 622 Scale-Free (BA model) (Heterogeneous) Attacks: the removal of a tiny fraction of important nodes (2%) causes the network to lose 50% of its efficiency. Errors: the network is nearly unaffected from the removal of a few nodes Erdös-Rényi Random graph (EXP) (Homogeneous) Attacks & Errors: the network is nearly unaffected from the removal of a few nodes

23 Results: global efficiency y( (many removals) N=5000 K=10000 Scale-Free (BA model) (Heterogeneous) Attacks: global efficiency of the network is completely destroyed, removing 10% of important nodes. Errors: network s efficiency slowly decreases. E Erdös-Rényi Random graph (EXP) Attacks &(Homogeneous) Errors: differences are evident, but less pronounced than in the BA model.

24 Is this analysis a good representation of what happens in real networks?

25 Let us consider a real system: the Pavia road system Nodes = Crossings (thanks to the Comando dei Vigili Urbani of Pavia) Arcs = Streets Arcs weights: τ ij = time spent in order to go from node i to node j

26 If today Piazza Emanuele Filiberto is not practicable Peoplehavetofind an alternative path. Load redistribution

27 Load redistribution can cause traffic in alternative routes. Overload Traffic hold up Degradation in efficiency (times τ ij grow longer)

28 Traffic hold up leads again to the choice of alternative routes New overload New degradation in efficiency Cascading effect

29 and the result is

30 Many real networks show cascading effect: Internet Power grids October 1986: the first documented Internet congestion collapse August 1996: sag of just one electrical line in Oregon August 2003: initial disturbance in Ohio Drop in speed of a factor 100 Blackout for 4 million people in 9 different States Largest blackout in the US s hystory

31 Assumptions and Definitions 1. Each node exchanges information with all the others, using shortest paths. 2. Load of node i at time t: LOAD i ( t ) = Betweenness i ( t ) (Goh, Kahng, Kim, PRL 87 (2002) ) total number of shortest paths passing through i 3. Capacity of node i: C i = α LOAD i ( 0 ) where α>1 is the tolerance parameter of nodes (Motter and Lai, PRE 66 (2002) ) i : i G

32 Dynamics of the model Crucitti, i Latora, Marchiori, i cond-mat/ (2003) a random node (error) Just 1 node is removed a node with high betweenness (attack). Load redistribution Weights Update LOAD i ( t) τ ij ( t + 1) = τ ij (0) if LOAD i ( t) > Ci Ci τ ij ( t 1) = τ ij (0) if LOAD i ( t) C + i j : j G i

33 Problem: not all real networks can be represented with times as arcs weights. Model evolution becomes: Generalization of the model Solution: efficiency e ij of arcs is used. Network-dependent definition of e ij. LOADi ( t ) eij ( t + 1) = eij (0) if LOADi ( t) > Ci Ci e ij ( t 1) = eij (0) if LOADi ( t ) C + i Growing longer of arcs replaced by degradation of efficiency. j : j G i ε hk = inverse of the sum of the inverse of e ij of the best path connecting h and k e.g.: In a road system or in the Internet network e ij 1 = τ and generalization is consistent with previous model ij

34 Time evolution of network efficiency α= α= α=1.05 BA scale-free with a random removal for 3 different value of the tolerance parameter

35 Results: ER and BA models N=2000 K=10000 Load-based attack Failure Erdös-Rényi Random graph (Homogeneous) Scale-Free (BA model) (Heterogeneous) Real networks

36 Differences 1. Homogeneous networks (EXP) are more resistant to cascading failures EXP BA 2. Region of α where the system is stable against errors collapses under attacks is wide for heterogeneous networks (BA).

37 Results: BA model Load-based attack Failure Scale-Free (BA model) (Heterogeneous)

38 A glimpse of distributions: ER and BA models P(LOAD) LOAD

39 Real networks: Internet Failure Load-based attack

40 Real networks: US Power Grid Failure Load-based d attack

41 A glimpse of distributions: Internet and US Power Grid P(LOAD) LOAD

42 Conclusion Simple model Dynamical redistribution of load large but rare cascading effects of real networks. Most failures emerge and dissolve locally A few failures spread over the whole network through an avalanche mechanism.