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Class 10: Robustness Cascades

Class 10: Robustness Cascades. Prof. Albert-László Barabási Dr. Baruch Barzel, Dr. Mauro Martino. Network Science: Robustness Cascades March 23, 2011. Thex. A SIMPLE STORY (3):. Network Science: Introduction 2012. Network Science: Introduction January 10, 2011.

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Class 10: Robustness Cascades

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  1. Class 10: Robustness Cascades Prof. Albert-László Barabási Dr. Baruch Barzel, Dr. Mauro Martino Network Science: Robustness Cascades March 23, 2011

  2. Thex • A SIMPLE STORY (3): Network Science: Introduction 2012 Network Science: Introduction January 10, 2011

  3. ROBUSTNESS IN COMPLEX SYSTEMS Complex systems maintain their basic functions even under errors and failures cell mutations There are uncountable number of mutations and other errors in our cells, yet, we do not notice their consequences. Internet  router breakdowns At any moment hundreds of routers on the internet are broken, yet, the internet as a whole does not loose its functionality. Where does robustness come from? There are feedback loops in most complex systems that keep tab on the component’s and the system’s ‘health’. Could the network structure affect a system’s robustness? Network Science: Robustness Cascades March 23, 2011

  4. node failure ROBUSTNESS Could the network structure affect a system’s robustness? How do we describe in quantitave terms the breakdown of a network under node or link removal? ~percolation theory~ Network Science: Robustness Cascades March 23, 2011

  5. PERCOLATION THEORY p= the probability that a node is occupied Increasing p Critical point pc: above pc we have a spanning cluster. Network Science: Robustness Cascades March 23, 2011

  6. PERCOLATION: CRITICAL EXPONENTS p – play the role of T in thermal phase transitions Order parameter: P∞~(p-pc)β probability that a node (or link) belongs to the  cluster P∞ Correlation length: ξ~|p-pc|-ν mean distance between two sites on the same cluster Cluster size: S~|p-pc|-γ Average size of finite clusters • β, ν, γ: critical exponents—characterize the behavior near the phase transition • The exponents are universal(independent of the lattice) • pcdepends on details (lattice) • ν and γare the same for p>pc and p<pc • For ξ and S take into account all finite clusters Network Science: Robustness Cascades March 23, 2011

  7. I: Subcritical <k> < 1 II: Critical <k> = 1 III: Supercritical <k> > 1 IV: Connected <k> > ln N <k> N=100 <k>=1 <k>=3 <k>=5 <k>=0.5

  8. node failure ROBUSTNESS Could the network structure contribute to robustness? How do we describe in quantitave terms the breakdown of a network under node removal? ~percolation theory~ Network Science: Robustness Cascades March 23, 2011

  9. ROBUSTNESS: INVERSE PERCOLATION TRANSITION Remove nodes Remove nodes Unperturbed network Gian Component Persists Network Collapses P∞:probability that a node belongs to the giant component f: fraction of removed nodes. f fc Network Science: Robustness Cascades March 23, 2011

  10. Damage is modeled as an inverse percolation process f= fraction of removed nodes S Component structure Graph fc f (Inverse Percolation phase transition) Network Science: Robustness Cascades March 23, 2011

  11. BOTTOM LINE: ROBUSTNESS OF REGULAR NETWORKS IS WELL UNDERSTOOD f=0: all nodes are part of the giant component, i.e. S=N, P∞=1 fc f 0<f<fc: the network is fragmented into many clusters with average size S~|p-pc|-γ there is a giant component; the probability that a node belongs to it: P∞~(p-pc)β f>fc: the network collapses, falling into many small clusters; giant component disappears Network Science: Robustness Cascades March 23, 2011

  12. ROBUSTNESS: OF SCALE-FREE NETWORKS • The interest in the robustness problem has three origins: • Robustness of complex systems is an important problem in many areas • Many real networks are not regular, but have a scale-free topology • In scale-free networks the scenario described above is not valid Albert, Jeong, Barabási, Nature 406 378 (2000) Network Science: Robustness Cascades March 23, 2011

  13. 1 S 0 1 f ROBUSTNESS OF SCALE-FREE NETWORKS Scale-free networks do not appear to break apart under random failures. Reason: the hubs. The likelihood of removing a hub is small. Albert, Jeong, Barabási, Nature 406 378 (2000) Network Science: Robustness Cascades March 23, 2011

  14. MALLOY-REED CRITERIA: THE EXISTENCE OF A GIANT COMPONENT A giant cluster exists if each node is connected to at least two other nodes. The average degree of a node i linked to the GC, must be 2, i.e. Bayes’ theorem P(ki|i <-> j): joint probability that a node has degree ki and is connected to nodes i and j. For a randomly connected network (does NOT mean random network!) with P(k): i can choose between N-1 nodes to link to, each with probability 1/(N-1). I can try ki times. κ>2: a giant cluster exists κ<2: many disconnected clusters Network Science: Robustness Cascades March 23, 2011 Malloy, Reed, Random Structures and Algorithms (1995); Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).

  15. Apply the Malloy-Reed Criteria to an Erdos-Renyi Network Discrete Formulation -binomial distribution- Continuum Formulation -Poisson distribution- Probability Distribution Function (PDF) Network Science: Robustness Cascades March 23, 2011

  16. Apply the Malloy-Reed Criteria to an Erdos-Renyi Network A giant cluster exists if each node is connected to at least two other nodes. κ>2: a giant cluster exists; κ<2: many disconnected clusters; Malloy-Reed; Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). Network Science: Robustness Cascades March 23, 2011

  17. RANDOM NETWORK: DAMAGE IS MODELED AS AN INVERSE OERCOLATION PROCESS S Component structure Graph f fc kc: <k>=1 <k> f= fraction of removed nodes (Inverse percolation phase transition) Network Science: Robustness Cascades March 23, 2011

  18. BREAKDOWN THRESHOLD FOR ARBITRARY P(k) Problem: What are the consequences of removing a fraction fof all nodes? At what threshold fc will the network fall apart (no giant component)? Random node removal changes the degree of individual nodes [kk’ ≤k] the degree distribution [P(k) P’(k’)] A node with degree k will loose some links and become a node with degree k’ with probability: The prob. that we had a k degree node was P(k), so the probability that we will have a new node with degree k’ : Leave k’ links untouched, each with probability 1-f Remove k-k’ links, each with probability f Let us asume that we know <k> and <k2> for the original degree distribution P(k)  calculate <k’> , <k’2> for the new degree distribution P’(k’). Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). Network Science: Robustness Cascades March 23, 2011

  19. BREAKDOWN THRESHOLD FOR ARBITRARY P(K) Degree distribution after we removed f fraction of nodes. k=[k’, ∞) The sum is done over the triangel shown in the right, so we can replace it with k’ Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). Network Science: Robustness Cascades March 23, 2011

  20. BREAKDOWN THRESHOLD FOR ARBITRARY P(K) Degree distribution after we removed f fraction of nodes. k=[k’, ∞) The sum is done over the triangel shown in the right, i.e. we can replace it with k’ Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). Network Science: Robustness Cascades March 23, 2011

  21. BREAKDOWN THRESHOLD FOR ARBITRARY P(K) Robustness: we remove a fraction f of the nodes. At what threshold fc will the network fall apart (no giant component)? Random node removal changes the degree of individuals nodes [kk’ ≤k) the degree distribution [P(k) P’(k’)] κ>2: a giant cluster exists κ<2: many disconnected clusters S Breakdown threshold: f<fc: the network is still connected (there is a giant cluster) f>fc: the network becomes disconnected (giant cluster vanishes) fc f Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). Network Science: Robustness Cascades March 23, 2011

  22. 1 S 0 1 f ROBUSTNESS OF SCALE-FREE NETWORKS Scale-free networks do not appear to break apart under random failures. Reason: the hubs. The likelihood of removing a hub is small. Albert, Jeong, Barabási, Nature 406 378 (2000) Network Science: Robustness Cascades March 23, 2011

  23. ROBUSTNESS OF SCALE-FREE NETWORKS Scale-free networks do not appear to break apart under random failures. Why is that? Network Science: Robustness Cascades March 23, 2011

  24. ROBUSTNESS OF SCALE-FREE NETWORKS γ>3: κ is finite, so the network will break apart at a finite fc that depens on Kmin γ<3: κ diverges in the N ∞ limit, so fc 1 !!! for an infinite system one needs to remove all the nodes to break the system. For a finite system, there is a finite but large fc that scales with the system size as: Internet: Router level map, N=228,263; γ=2.1±0.1; κ=28 fc=0.962 AS level map, N= 11,164; γ=2.1±0.1; κ=264 fc=0.996 Network Science: Robustness Cascades March 23, 2011

  25. NUMERICAL EVIDENCE Scale-free random graph with Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). Infinite scale-free networks with do not break down under random node failures. Network Science: Robustness Cascades March 23, 2011

  26. SIZE OF THE GIANT COMPONENT DURING RANDOM DAMAGE–WITHOUT PROOF- S: size of the giant component. γ>4: S≈f-fc (similar to that of a random graph) 3>γ>4: S≈(f-fc)1/(γ-3) γ<3: fc =0 and S≈f1+1/(3-γ) R. Cohen, D. ben-Avraham, S. Havlin, Percolation critical exponents in scale-free networks Phys. Rev. E 66, 036113 (2002); See also: Dorogovtsev S, Lectures on Complex Networks, Oxford, pg44 Network Science: Robustness Cascades March 23, 2011

  27. INTERNET’S ROBUSTNESS TO RANDOM FAILURES failure attack Internet R. Albert, H. Jeong, A.L. Barabasi, Nature406 378 (2000) Internet: Router level map, N=228,263; γ=2.1±0.1; κ=28 fc=0.962 AS level map, N= 11,164; γ=2.1±0.1; κ=264 fc=0.996 Internet parameters: Pastor-Satorras & Vespignani, Evolution and Structure of the Internet: Table 4.1 & 4.4 Network Science: Robustness Cascades March 23, 2011

  28. Robust-SF Achilles’ Heel of scale-free networks 1 S 0 1 f Attacks Failures   3 : fc=1 (R. Cohen et al PRL, 2000) fc Network Science: Robustness Cascades March 23, 2011 Albert, Jeong, Barabási, Nature 406 378 (2000)

  29. Attack threshold for arbitrary P(k) Attack problem: we remove a fraction f of the hubs. At what threshold fc will the network fall apart (no giant component)? Hub removal changes the maximum degree of the network [KmaxK’max ≤Kmax) the degree distribution [P(k) P’(k’)] A node with degree k will loose some links because some of its neighbors will vanish. Claim: once we correct for the changes in Kmax and P(k),we are back to the robustness problem. That is, attack is nothing but a robusiness of the network with a new Kmaxand P(k). fc f Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). Network Science: Robustness Cascades March 23, 2011

  30. Attack threshold for arbitrary P(k) Attack problem: we remove a fraction f of the hubs. the maximum degree of the network [KmaxK’max ≤Kmax) ` If we remove an f fraction of hubs, the maximum degree changes: As K’max ≤Kmax we can ignore the Kmax term  The new maximum degree after removing f fraction of the hubs. Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). Network Science: Robustness Cascades March 23, 2011

  31. Attack threshold for arbitrary P(k) Attack problem: we remove a fraction f of the hubs. the degree distribution changes [P(k) P’(k’)] A node with degree kwill loose some links because some of its neighbors will vanish. Let us calculate the fraction of links removed ‘randomly’ , f’, as a consequence of we removing f fraction of hubs. as K’max ≤Kmax For γ2, f’1, which means that even the removal of a tiny fraction of hubs will destroy the network. The reason is that for γ=2 hubs dominate the network Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). Network Science: Robustness Cascades March 23, 2011

  32. Attack threshold for arbitrary P(k) Attack problem: we remove a fraction f of the hubs. At what threshold fc will the network fall apart (no giant component)? Hub removal changes the maximum degree of the network [KmaxK’max ≤Kmax) the degree distribution [P(k) P’(k’)] A node with degree k will loose some links because some of its neighbors will vanish. Claim: once we correct for the changes in Kmax and P(k), we are back to the robustness problem. That is, attack is nothing but a robustness of the network with a new K’maxand f’. Network Science: Robustness Cascades March 23, 2011 Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).

  33. Attack threshold for arbitrary P(k) Attack problem: we remove a fraction f of the hubs. At what threshold fc will the network fall apart (no giant component)? fc • fc depends on γ; it reaches its max for γ<3 • fc depends on Kmin (m in the figure) • Most important: fc is tiny. Its maximum reaches • only 6%, i.e. the removal of 6% of nodes can destroy the network in an attack mode. • Internet: γ=2.1, so 4.7% is the threshold. Figure: Pastor-Satorras & Vespignani, Evolution and Structure of the Internet: Fig 6.12 Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). Network Science: Robustness Cascades March 23, 2011

  34. Application: ER random graphs Consider a random graph with connection probability psuch that at least a giant connected component is present in the graph. Find the critical fraction of removed nodes such that the giant connected component is destroyed. The higher the original average degree, the larger damage the network can survive. Q: How do you explain the peak in the average distance? Minimum damage Network Science: Robustness Cascades March 23, 2011

  35. Robust-SF Achilles’ Heel of scale-free networks 1 S 0 1 f Attacks Failures   3 : fc=1 (R. Cohen et al PRL, 2000) fc Network Science: Robustness Cascades March 23, 2011 Albert, Jeong, Barabási, Nature 406 378 (2000)

  36. Achilles Heel Achilles’ Heel of complex networks failure attack Internet R. Albert, H. Jeong, A.L. Barabasi, Nature406 378 (2000) Network Science: Robustness Cascades March 23, 2011

  37. Achilles Heel Historical Detour: Paul Baran and Internet 1958 Network Science: Robustness Cascades March 23, 2011

  38. Numerical simulations of network resilience • Two networks with equal number of nodes and edges • ER random graph • scale-free network (BA model) • Study the properties of the network as an increasing fraction f of the nodes are removed. • Node selection: random (errors) • the node with the largest number of edges (attack) • Measures: the fraction of nodes in the largest connected cluster, S • the average distance between nodes in the largest cluster, l Network Science: Robustness Cascades March 23, 2011 R. Albert, H. Jeong, A.-L. Barabási, Nature 406, 378 (2000)

  39. Scale-free networks are more error tolerant, but also more vulnerable to attacks • squares: random failure • circles: targeted attack Failures: little effect on the integrity of the network. Attacks: fast breakdown Network Science: Robustness Cascades March 23, 2011

  40. Real scale-free networks show the same dual behavior • blue squares: random failure • red circles: targeted attack • open symbols: S • filled symbols: l • break down if 5% of the nodes are eliminated selectively (always the highest degree node) • resilient to the random failure of 50% of the nodes. Similar results have been obtained for metabolic networks and food webs. Network Science: Robustness Cascades March 23, 2011

  41. Cascades Potentially large events triggered by small initial shocks • Information cascades • social and economic systems • diffusion of innovations • Cascading failures • infrastructural networks • complex organizations Network Science: Robustness Cascades March 23, 2011

  42. Cascading Failures in Nature and Technology Earthquake Avalanche Blackout • Cascades depend on • Structure of the network • Properties of the flow • Properties of the net elements • Breakdown mechanism Flows of physical quantities • congestions • instabilities • Overloads Network Science: Robustness Cascades March 23, 2011

  43. Northeast Blackout of 2003 Origin A 3,500 MW power surge (towards Ontario) affected the transmission grid at 4:10:39 p.m. EDT. (Aug-14-2003) Before the blackout Afterthe blackout Consequences More than 508 generating units at 265 power plants shut down during the outage. In the minutes before the event, the NYISO-managed power system was carrying 28,700 MW of load. At the height of the outage, the load had dropped to 5,716 MW, a loss of 80%. Network Science: Robustness Cascades March 23, 2011

  44. Network Science: Robustness Cascades March 23, 2011

  45. Cascades Size Distribution of Blackouts Unserved energy/power magnitude (S)distribution P(S) ~ S −α, 1< α < 2 Probability of energy unserved during North American blackouts 1984 to 1998. I. Dobson, B. A. Carreras, V. E. Lynch, D. E. Newman, CHAOS 17, 026103 (2007) Network Science: Robustness Cascades March 23, 2011

  46. Cascades Size Distribution of Earthquakes Earthquake size S distribution Earthquakes during 1977–2000. P(S)~ S −α,α≈ 1.67 Y. Y. Kagan, Phys. Earth Planet. Inter. 135 (2–3), 173–209 (2003) Network Science: Robustness Cascades March 23, 2011

  47. Failure Propagation Model • Initial Setup • Random graph with N nodes • Initially each node is functional. • Cascade • Initiated by the failure of one node. • fi : fraction of failed neighbors of node i. Node i fails if fiis greater than a global threshold φ. <k> Undercritical Network falls apart (<k>=1) Critical Overcritical φ=0.4 f = 1/2 ● Overcritical f = 1/2 □ Critical f = 0 f = 1/2 f = 1/3 f = 2/3 Erdos-Renyi network P(S) ~ S −3/2 D. Watts, PNAS 99, 5766-5771 (2002) Network Science: Robustness Cascades March 23, 2011

  48. Overload Model • Initial Conditions • N Components (complete graph) • Each components has random initial load Li drawn at random uniformly from [Lmin, 1]. • Cascade • Initiated by the failure of one component. • Component fail when its load exceeds 1. • When a component fails, a fixed amount P is transferred to all the rests. Undercritical Overcritical Critical Lmin Overcritical Li =0.95 Li =0.8 Critical Undercritical P=0.15 Li =0.7 Li =0.85 Li =0.9 Li =1.05 P(S) ~ S −3/2 I. Dobson, B. A. Carreras, D. E. Newman, Probab. Eng. Inform. Sci. 19, 15-32 (2005) Network Science: Robustness Cascades March 23, 2011

  49. Self-organized Criticality (BTW Sandpile Model) • Initial Setup • Random graph with N nodes • Each node i has height hi = 0. • Cascade • At each time step, a grain is added at a randomly chosen node i: hi ← hi +1 • If the height at the node i reaches a prescribed threshold zi = ki, then it becomes unstable and all the grains at the node topple to its adjacent nodes: hi = 0 and hj ← hj +1 • if i and j are connected. Homogenous case Scale-free network Homogenous network: <k2>converged P(S) ~ S −3/2 Scale-free network :pk~ k-γ (2<γ<3) P(S) ~ S −γ/(γ −1) Network Science: Robustness Cascades March 23, 2011 K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 148701 (2003)

  50. Branching Process Model Branching Process Starting from a initial node, each node in generation t produces knumber of offspring nodes in the next t + 1 generation, where k is selected randomly from a fixed probability distribution qk=pk-1. Fix <k>=1 to be critical  power law P(S) • Hypothesis • No loops (tree structure) • No correlation between branches Narrow distribution: <k2>converged P(S) ~ S −3/2 Fat tailed distribution: qk ~ k-γ (2<γ<3) P(S) ~ S −γ/(γ −1) Network Science: Robustness Cascades March 23, 2011 K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 148701 (2003)

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