Frontiers of Network Science Fall 2017 Class 19: Robustness II (Chapter 8 in Textbook) Boleslaw Szymanski based on slides by Albert-László Barabási and Roberta Sinatr a www.BarabasiLab.com
Attack threshold for arbitrary P(k) Attack problem: we remove a fraction f of the hubs. At what threshold f c will the network fall apart (no giant component)? Hub removal changes 1 the maximum degree of the network [K max K’ max ≤ K max ) K ' max = K min f 1 − γ the degree distribution [P(k) P’(k’)] 2 − γ f ' = f 1 − γ A node with degree k will loose some links because some of its neighbors will vanish. Claim: once we correct for the changes in K max 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’ max and f’. < > < > κ 1 2 2 ' k k = − κ = = = ' 1 f ' κ − < > − < > − ' 1 ' ( 1 ) 1 k f k f c c γ > 3 1 − γ = 2 + 2 − γ K 2 − γ 3 − γ min − γ 1 − γ − 1 2 − γ γ − κ = > γ > 3 f c 3 − γ K min f c 2 3 2 K K − γ max min 3 > γ > 2 1 K max Network Science: Robustness Cascades Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).
Attack threshold for arbitrary P(k) Attack problem: we remove a fraction f of the hubs. At what threshold f c will the network fall apart (no giant component)? f c •f c depends on γ; it reaches its max for γ<3 •f c depends on K min (m in the figure) •Most important: f c 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
Application: ER random graphs Consider a random graph with connection probability p such that at least a giant connected component is present in the graph. S surviving giant component Find the critical fraction of removed S nodes such that the giant connected Minimum component is destroyed. damage 1 1 1 = − = − = − l f 1 1 1 c 2 pN k k − 0 0 1 k 0 The higher the original average degree, Empty squares show S the larger damage the network can survive. Filled squares l – avg. distance Q: How do you explain the peak in the average distance? Network Science: Robustness Cascades
Summary: Achilles’ Heel of scale-free networks 1 Attacks Failures γ ≤ 3 : f c =1 S (R. Cohen et al PRL, 2000) f c 0 1 f Network Science: Robustness Cascades Albert, Jeong, Barabási, Nature 406 378 (2000)
Summary: Achilles’ Heel of complex networks failure attack Internet R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000) Network Science: Robustness Cascades
Historical Detour: Paul Baran and Internet 1958 A network of n-ary degree of connectivity has n links per node was simulated The simulation revealed that networks where n ≥ 3 had a significant increase in resilience against even as much as 50% node loss. Baran's insight gained from the simulation was that redundancy was the key. Network Science: Robustness Cascades
Scale-free networks are more error tolerant, but also more vulnerable to attacks • squares: random failure S • circles: targeted attack • S surviving fraction of GC • l average distance l Failures: little effect on the integrity of the network. Attacks: fast breakdown Network Science: Robustness Cascades
Real scale-free networks show the same dual behavior • blue squares: random failure S • red circles: targeted attack • open symbols: S (size of surviving S l component) l • filled symbols: l (average distance) • 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
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
Cascading Failures in Nature and Technology Earthquake Avalanche Blackout Cascades depend on Flows of physical quantities • Structure of the network • congestions • Properties of the flow • instabilities • Properties of the net elements Overloads • • Breakdown mechanism Network Science: Robustness Cascades
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 After the 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
Network Science: Robustness Cascades
Cascades Size Distribution of Blackouts Unserved energy/power magnitude ( S) distribution P ( S ) ~ S − α , 1< α < 2 Source Exponent Quantity Probability of energy North America 2.0 Power unserved during North American blackouts Sweden 1.6 Energy 1984 to 1998. Norway 1.7 Power New Zealand 1.6 Energy China 1.8 Energy I. Dobson, B. A. Carreras, V. E. Lynch, D. E. Newman, CHAOS 17, 026103 (2007) Network Science: Robustness Cascades
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
Failure Propagation Model Initial Setup • Random graph with N nodes • Initially each node is functional. Undercritical <k> Cascade Critical • Initiated by the failure of one node. Overcritical • f i : fraction of failed neighbors of node i . Node i Network falls apart fails if f i is greater than a global threshold φ . (<k>=1) φ =0.4 ● Overcritical f = 1/2 □ Critical f = 1/2 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
Overload Model Initial Conditions Critical • N Components (complete graph) • Each components has random initial load L i drawn at random uniformly from [ L min , 1] . Undercritical Overcritical Cascade • Initiated by the failure of one component. • Component fail when its load exceeds 1 . • When a component fails, a fixed amount P is L min transferred to all the rests. Overcritical L i =0.95 L i =0.8 Critical Undercritical P =0.15 L i =0.7 L i =0.85 L i =0.9 L i =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
Self-organized Criticality (BTW Sandpile Model) Initial Setup Homogenous case • Random graph with N nodes • Each node i has height h i = 0 . Cascade • At each time step, a grain is added at a randomly chosen node i : h i ← h i +1 • If the height at the node i reaches a prescribed threshold z i = k i , then it becomes unstable and all the grains at the node topple to its adjacent nodes: h i = 0 and h j ← h j +1 Scale-free network • if i and j are connected. Homogenous network: < k 2 > converges P ( S ) ~ S −3/2 Scale-free network : p k ~ k - γ (2< γ <3) P ( S ) ~ S − γ /( γ −1) Network Science: Robustness Cascades K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 148701 (2003)
Branching Process Model Branching Process Starting from a initial node, each node in generation t produces k number of offspring nodes in the next t + 1 generation, where k is selected randomly from a fixed probability distribution q k =p k-1 . Fix <k>=1 to be Hypothesis critical power • No loops (tree structure) law P(S) • No correlation between branches Narrow distribution: < k 2 > converged P ( S ) ~ S −3/2 Fat tailed distribution: q k ~ k - γ (2< γ <3) P ( S ) ~ S − γ /( γ −1) Network Science: Robustness Cascades K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 148701 (2003)
Short Summary of Models: Universality Models Networks Exponents Failure Propagation Model ER 1.5 Overload Model Complete Graph 1.5 1.5 (ER) BTW Sandpile Model ER/SF γ /( γ - 1) (SF) 1.5 (ER) Branching Process Model ER/SF γ /( γ - 1) (SF) Universal for homogenous networks P ( S ) ~ S −3/2 Same exponent for percolation too (random failure, attacking, etc.) Network Science: Robustness Cascades
Explanation of the 3/2 Universality Simplest Case: q 0 = q 2 = 1/2 , < k > = 1 S : number of nodes X : number of open branches S = S+ 1 ½ chance S = 2, X = 0 X = X -1 k = 0 S = 2, X = 2 S = 1 S = S+ 1 ½ chance k =2 X = 1 X = X +1 X >0, Branching process stops X when X = 0 Dead S
Explanation of 3/2 Universality X Dead S Equivalent to 1D random walk model , where X and S are the position and time , respectively. Question : what is the probability that X = 0 after S steps? First return probability ~ S -3/2 M. Ding, W. Yang, Phys. Rev. E. 52, 207-213 (1995)
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