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): At a first look the satellite map of Fig. \ref{F-I-Outage} is just like many other night images of the north American continent. Lights shining brightly in highly populated spots, and was dark areas marking deserts, vast uninhabited forests and oceans. Yet, upon closer inspection, something strange becomes apparent: much of the US North East, from DC to New York and Boston, most of state of New York, Pennsylvania and Ohio, are gone. It looks as if something has carved another Gulf Of Mexico out of the US map. Yet, this is not a doctored shot from the next Armageddon movie, but represents a real image of the US on August 14, 2003, the night of a major blackout that left an estimated 45 million people in eight U.S. states and another 10 million in Ontario without power. It illustrates a much-ignored aspect of networks, one that will be an important theme in this book: {\it vulnerability due to interconnectivity}. The 2003 blackout is a typical example of a cascading failure. When a network acts as a transportation system, a local failure shifts loads or responsibilities to other nodes. If the extra load is negligible, the rest of the system can seamlessly absorb it, and the failure remains effectively unnoticed. If the extra load is too much for the neighboring nodes to carry, they will either tip or again redistribute the load to their neighbors. Either way, we are faced with a cascading event, the magnitude and reach of which depend on the centrality and capacity of the nodes that have been removed in the first round. Case in point is electricity: As it cannot be stored, when a line goes down, its power must be shifted to other lines. Most of the time the neighboring lines have no difficulty carrying the extra load. If they do, however, they will also tip and redistribute their increased load to their neighbors. Cascading failures are common in most complex networks. They take place on the Internet, when traffic is rerouted to bypass malfunctioning routers, occasionally creating denial of service attacks on routers that do not have the capacity to handle extra traffic. We witnessed one in 1997, when the International Monetary Fund pressured the central banks of several Pacific nations to limit their credit. And they are behind the 2009-2011 financial meltdown, when the US credit crisis paralyzed the economy of the globe, leaving behind scores of failed banks, corporations and even bankrupt states, like Greece. Cascading failures are occasionally our ally, however. The American effort to dry up the money supply of terrorist organizations is aimed at crippling terrorist networks. And doctors and researchers hope to induce cascading failures to kill cancer cells. The Northeast blackout illustrates an important theme of this class: we must understand how network structure affects the robustness of a complex system. We will therefore develop the quantitative tools to assess the interplay between network structure and the dynamical processes on the networks, and their impact on failures. As chaotic as such a failure may seem, we will learn that in reality it follows rather reproducible laws, that can be quantified and even predicted using the tools of network science. Network Science: Introduction 2012 Network Science: Introduction January 10, 2011

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

ROBUSTNESS node failure Could the network structure affect a system’s robustness? node failure 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

Critical point pc: above pc we have a spanning cluster. 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

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) pc depends on details (lattice) ν and γ are 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

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

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

Gian Component Persists 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. fc f Network Science: Robustness Cascades March 23, 2011

(Inverse Percolation phase transition) 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

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

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

Albert, Jeong, Barabási, Nature 406 378 (2000) 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. 1 S f Albert, Jeong, Barabási, Nature 406 378 (2000) Network Science: Robustness Cascades March 23, 2011

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. A : For a graph having degree distribution P(k) to have a giant component, a node that is reached by following a link from the giant component must have at least one other link on average to allow the component to exist. For this to occur, the average degree must be at least 2 (one incoming and one outgoing link) given that the node i is connected to j: B: κ>2: a giant cluster exists κ<2: many disconnected clusters Malloy, Reed, Random Structures and Algorithms (1995); Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). Network Science: Robustness Cascades March 23, 2011

Continuum Formulation 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

κ>2: a giant cluster exists; κ<2: many disconnected clusters; 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

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

BREAKDOWN THRESHOLD FOR ARBITRARY P(k) Problem: What are the consequences of removing a fraction f of 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’ : Remove k-k’ links, each with probability f Leave k’ links untouched, each with probability 1-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

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

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

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

Albert, Jeong, Barabási, Nature 406 378 (2000) 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. 1 S f Albert, Jeong, Barabási, Nature 406 378 (2000) Network Science: Robustness Cascades March 23, 2011

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

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

Scale-free random graph with 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

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

R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000) INTERNET’S ROBUSTNESS TO RANDOM FAILURES failure attack Internet R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (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

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

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 Kmax and P(k). fc f Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). Network Science: Robustness Cascades March 23, 2011

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

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 k will 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

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’max and f’. Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). Network Science: Robustness Cascades March 23, 2011

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

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. 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

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

Achilles’ Heel of complex networks failure attack Internet R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000) Network Science: Robustness Cascades March 23, 2011

Historical Detour: Paul Baran and Internet Achilles Heel Historical Detour: Paul Baran and Internet While working at the RAND Corporation, Paul Baran was assigned the task of designing a "survivable" communications system that could maintain communication between end points in the face of damage from nuclear attack. Using mini-computer technology of the day, Baran and his team developed a simulation suite to test basic connectivity of an array of nodes with varying degrees of linking. That is, a network of n-ary degree of connectivity would have n links per node. The simulation randomly 'killed' nodes and subsequently tested the percentage of nodes who remained connected. The result of 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. As a result of President Eisenhower's Defense Reorganization Act of 1958, there was a major shift in leadership in the Pentagon around the time Baran's work was accepted by the US Air Force and DoD for implementation and testing. When Baran discovered an older Navy admiral would oversee the project he decided the project would be better off sitting on the shelf as reference material, claiming that an 'old analog guy' couldn't grasp what it was the project aimed to accomplish, and thus would likely fail from lack of understanding. Around the same time when ARPA was developing the idea of an inter-networked set of terminals to share computing resources, among the number of reference materials considered was Paul Baran and the RAND Corporation's On Distributed Communications volumes. The ARPANET was never intended to be a survivable communications network, but some still maintain the myth that it was. Instead, the resilience feature of a packet switched network that uses link-state routing protocols is something we enjoy today in some part from the research done to develop a network that could survive a nuclear attack. (Source: Wikipedia, Paul Baran) 1958 Network Science: Robustness Cascades March 23, 2011

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 R. Albert, H. Jeong, A.-L. Barabási, Nature 406, 378 (2000) Network Science: Robustness Cascades March 23, 2011

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

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

Potentially large events triggered by small initial shocks 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

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

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 March 23, 2011

Network Science: Robustness Cascades March 23, 2011

Cascades Size Distribution of Blackouts Unserved energy/power magnitude (S) distribution P(S) ~ S −α, 1< α < 2 Source Exponent Quantity North America 2.0 Power Sweden 1.6 Energy Norway 1.7 New Zealand China 1.8 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

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

Failure Propagation Model Overcritical Undercritical Critical 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 fi is greater than a global threshold φ. <k> Network falls apart (<k>=1) φ =0.4 □ Critical ● Overcritical f = 1/2 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 March 23, 2011

P(S) ~ S −3/2 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 =0.8 Li =0.95 Critical Undercritical P=0.15 Li =0.7 Li =0.9 Li =0.85 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

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 the avalanche size 𝑆, the number of toppling events in a given avalanche Scale-free network : pk ~ k-γ (2<γ<3) P(S) ~ S −γ/(γ −1) K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 148701 (2003) Network Science: Robustness Cascades March 23, 2011

Branching Process Model 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 qk=pk-1. Hypothesis No loops (tree structure) No correlation between branches Fix <k>=1 to be critical  power law P(S) Narrow distribution: <k2> converged P(S) ~ S −3/2 Fat tailed distribution: qk ~ k-γ (2<γ<3) P(S) ~ S −γ/(γ −1) K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 148701 (2003) Network Science: Robustness Cascades March 23, 2011

Short Summary of Models: Universality Networks Exponents Failure Prorogation Model ER 1.5 Overload Model Complete Graph BTW Sandpile Model ER/SF 1.5 (ER) γ/(γ - 1)(SF) Branching Process Model Universal for homogenous networks P(S) ~ S −3/2 Same exponent for percolation too (random failure, attacking, etc.) Network Science: Robustness Cascades March 23, 2011

Explanation of the 3/2 Universality Simplest Case: q0 = q2 = 1/2, <k> = 1 S: number of nodes X: number of open branches S = 2, X = 0 S = S+1 X = X -1 ½ chance k= 0 S = 2, X = 2 S = S+1 X = X+1 S = 1 X = 1 k=2 ½ chance X X >0, Branching process stops when X = 0 S Dead

Explanation of 3/2 Universality Dead 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)

Size Distribution of Branching Process (Cavity Method) k = 0 k = 1 k = 2 S = 1 S = 1+S1 S = 1+S1+S2 K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Physica A 346, 93-103 (2005) Network Science: Robustness Cascades March 23, 2011

Solving the Equation by Generating Function Definition: GS(x) = ΣS=0 P(S)xS Gk(x) = Σk=0 qkxk Property: GS(1) = Gk(1) = 1 GS’(1) = <S>, Gk’(1) = <k> Phase Transition <S> = GS’(1) = 1+ Gk’(1) GS’(1) = 1 + <k> <S>, then <S> = 1/(1- <k>) The average size <S> diverges at <k>c = 1 Network Science: Robustness Cascades March 23, 2011

Finding the Critical Exponent from Expansion Definition: GS(x) = ΣS=0 P(S)xS Gk(x) = Σk=0 qkxk Theorem: If P(k) ~ k-γ (2<γ<3), then for δx < 0, |δx| << 1 G(1+ δx) = 1 + <k>δx + <k(k-1)/2> (δx)2 + … + O(|δx|γ - 1) P(S) ~ S −α,1< α < 2 GS(1+ δx) ≈ 1 + A|δx|α -1 Homogenous case: <k2> converged <k> = 1, <k2> < ∞ Gk(1+ δx) ≈ 1 + δx + Bδx2 Inhomogeneous case: <k2> diverged <k> = 1, qk ~ k-γ (2<γ<3) Gk(1+ δx) ≈ 1 + δx + B|δx|γ - 1 Network Science: Robustness Cascades March 23, 2011

Critical Exponent for Homogenous Case Gk(1+ δx) ≈ 1 + δx + Bδx2 GS(1+ δx) ≈ 1 + A|δx|α -1 GS(x) = xGk(GS(x)) GS(x) ≈ 1 + A|δx|α -1 xGk(GS(x)) ≈ (1+δx)[1+ (GS(1+δx)-1) + B (GS(1+δx)-1)2] ≈ (1+δx)[1+ A|δx|α -1 + AB|δx|2α -2] = 1 + A|δx|α -1 + AB|δx|2α -2 + δx + O(|δx|α) The lowest order reads AB|δx|2α -2 + δx = 0, which requires 2α -2 = 1and A = 1/B. Or, α = 3/2 Network Science: Robustness Cascades March 23, 2011

Critical Exponent for Inhomogeneous Case Gk(1+ δx) ≈ 1 + δx + B|δx|γ - 1 GS(1+ δx) ≈ 1 + A|δx|α -1 GS(x) = xGk(GS(x)) GS(x) ≈ 1 + A|δx|α -1 xGk(GS(x)) ≈ (1+δx)[1+ (GS(1+δx)-1) + B |GS(1+δx)-1|γ -1] ≈ (1+δx)[1+ A|δx|α -1 + AB|δx|(α -1)(γ -1)] = 1 + A|δx|α -1 + AB|δx|(α -1)(γ -1) + δx + O(|δx|α) The lowest order reads AB|δx|(α -1)(γ -1) + δx = 0, which requires (α -1)(γ -1) = 1and A = 1/B. Or, α = γ/(γ −1) Network Science: Robustness Cascades March 23, 2011

Compare the Prediction with the Real Data Blackout Blackout Source Exponent Quantity North America 2.0 Power Sweden 1.6 Energy Norway 1.7 New Zealand China 1.8 Earthquake α ≈ 1.67 I. Dobson, B. A. Carreras, V. E. Lynch, D. E. Newman, CHAOS 17, 026103 (2007) Y. Y. Kagan, Phys. Earth Planet. Inter. 135 (2–3), 173–209 (2003) Network Science: Robustness Cascades March 23, 2011

The end Network Science: Evolving Network Models February 14, 2011