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
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, 2000 1. Fractal Approach to Social Network Attack Detection 2. The topology of dark networks Communications of ACM, October 2008 E a parte algoritimica???
Revisão de Artigos The internet s achilles heel: Error and attack tolerance of complex networks R Albert, H Jeong, AL Barabasi Nature, 2000 http://www.nd.edu/~networks/publication%20categories /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 http://www.ct.infn.it/~latora/next_errors.pdf
Ilustração visual das diferenças
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
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
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
Respostas da Rede a Falhas ou Ataques
Mudanças no Diâmetro das Redes
Fragmentação das Redes sob Ataques
Calcanhar de Aquiles das Redes Complexas falha ataque Internet R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)
Estrutura Geral de Ataques a Web 2.0
Próximos Slides estão disponíveis: Efficiency of scale free networks: error and attack tolerance. P. Crucitti, V. Latora, M. Marchiori, A. Rapisarda. www.pv.infn.it/~frontier/2003/talks/crucitti.ppt it/ f ti /2003/t /C itti
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)
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.
Reasons for studying error and attack tolerance Designing robust networks Protecting existing networks
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
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!
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) 198701, 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
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!
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
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
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.
Is this analysis a good representation of what happens in real networks?
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
If today Piazza Emanuele Filiberto is not practicable Peoplehavetofind an alternative path. Load redistribution
Load redistribution can cause traffic in alternative routes. Overload Traffic hold up Degradation in efficiency (times τ ij grow longer)
Traffic hold up leads again to the choice of alternative routes New overload New degradation in efficiency Cascading effect
and the result is
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
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) 278701) 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) 065102) i : i G
Dynamics of the model Crucitti, i Latora, Marchiori, i cond-mat/0309141 (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
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
Time evolution of network efficiency α=13 1.3 α=1.1212 α=1.05 BA scale-free with a random removal for 3 different value of the tolerance parameter
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
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).
Results: BA model Load-based attack Failure Scale-Free (BA model) (Heterogeneous)
A glimpse of distributions: ER and BA models P(LOAD) LOAD
Real networks: Internet Failure Load-based attack
Real networks: US Power Grid Failure Load-based d attack
A glimpse of distributions: Internet and US Power Grid P(LOAD) LOAD
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.