Network Science: Principles and Applications

Network Science: Principles and Applications CS 695 - Fall 2016 Amarda Shehu,Fei Li [amarda, lifei](at)gmu.edu Department of Computer Science George Mason University

1 Outline of Today s Class 2 Robustness Cascades:Percolation Theory Robustness in Complex Systems Robustness of Regular Networks Robustness of Scale-free Networks Molloy-Reed Criterion Application of Molloy-Reed Criterion Attack Tolerance Cascading Failures Modeling Cascading Failures Building Robustness Summary: Achilles Heel Amarda Shehu,Fei Li () Outline of Today s Class 2

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 components and the systems health Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 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 components and the systems health Could the network structure affect a system s robustness? Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 3

Robustness Could the network structure affect a system s robustness? Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 4

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? Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 4

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 Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 4

Percolation Theory p = the probability that a node is occupied Critical point p c : above p c we have a spanning/percolating cluster (has at least one site on each of the four boundaries of a lattice) Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 5

Percolation Theory Key Questions What is the expected size of the largest cluster? What is the average cluster size? Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 6

Percolation Theory Key Questions What is the expected size of the largest cluster? What is the average cluster size? The higher p, the larger are the clusters Key prediction: cluster size does not change gradually with p For a wide range of p, the lattice is populated with numerous tiny clusters If p approaches a critical value p c, these small clusters grow and coalesce, leading to the emergence of a large cluster at p c The percolating cluster as it reaches the end of the lattice phase transition at p c: from many small clusters to a percolating cluster that percolates the whole lattice Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 6

Percolation Theory Key Questions What is the expected size of the largest cluster? What is the average cluster size? The higher p, the larger are the clusters Key prediction: cluster size does not change gradually with p For a wide range of p, the lattice is populated with numerous tiny clusters If p approaches a critical value p c, these small clusters grow and coalesce, leading to the emergence of a large cluster at p c The percolating cluster as it reaches the end of the lattice phase transition at p c: from many small clusters to a percolating cluster that percolates the whole lattice Three critical quantities: Order Parameter, Average Cluster Size, and Correlation Length Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 6

Percolation: Critical Exponents p plays the role of T in thermal phase transitions Order parameter: P (p p c ) β probability that a node (or link) belongs to cluster Correlation length: η p p c ν mean distance between two sites/pebbles on same cluster Cluster size: S p p c γp Average size of finite clusters β, ν, γ p : critical exponentscharacterize the behavior near the phase transition The exponents are universal (independent of the lattice) p c depends on details (lattice) ν and γ are the same for p > p c and p < p c For η and S take into account all finite clusters Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 7

Percolation Theory and Phase Transitions Site or bond percolation in 2D lattice: simplest system that exhibits characteristics of a sharp second order phase transition in the limit of infinite size The size of the system is the number of sites (or bonds) in the lattice There is a single control variable, the probability p that a lattice site is occupied A second order phase transition has characteristic singular behavior in the limit of infinite size These singularities appear in the equations of state at the critical value of the control parameter and are characterized by numbers called critical exponents The critical exponents are very useful and interesting because they appear to take the numerical same values for systems with the same dimensionality and symmetries, and are independent of the microscopic details of individual systems Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 8

Order Parameter P Convenient order parameter: the fraction of occupied sites in the spanning cluster: nr. sites in spanning cluster P = total nr. occupied sites This is also the probability that a randomly chosen pebble belongs to the largest cluster. As N, P = constant (p p c) β for p p c and 0 otherwise (β = 5/36 in 2D square lattice) Figure: Schematic illustration of P belongs to the largest connected component: For p < p c, all components are small, so P = 0. Once p reaches p c, a giant component emerges. Beyond p c, there is a finite probability that a node belongs to the largest component. Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 9

From Cluster Size Distribution to Average Cluster Size The cluster size distribution of a lattice configuration is defined to be: nr. clusters with s sites n s (p) = nr. sites in lattice By convention, spanning clusters are not included in calculating this distribution The probability w s that an occupied site belongs to a cluster of s sites is: w s = s ns s s ns The average or mean cluster size in a lattice configuration is then: S = s s w s s = s2 n s s s ns Here again, spanning clusters are excluded from this calculation so the average cluster size is finite above the percolation threshold In the limit of infinite lattice size, the average cluster size becomes singular near the percolation threshold S p p c γ where γ = 43/18 for percolation on 2D square lattice. Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 10

Average Cluster Size S S p p c γp Figure: As we approach p c from below, numerous small clusters coalesce and S diverges. Same divergence is observed above p c, where to calculate S we remove the percolating cluster from the average. Same exponent γ p characterizes divergence on both sides of the critical point p c. Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 11

Correlation Length Two sites on the lattice are said to be correlated or connected if they are both occupied and belong to the same cluster The correlation length η of a lattice configuration is the average distance between connected sites By convention, spanning clusters are not included in calculating the correlation length so that correlation length remains finite in the limit of infinite lattice size Even so, the correlation length diverges near the percolation phase transition as in: η p p c ν where ν = 4/3 for percolation on 2D square lattice Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 12

Universality and Scaling Laws p c dependent of lattice type and dimensionality d (e.g., p c 0.593 for 2D square lattice) Critical exponents only dependent of d Theoretical and experimental indications that the critical exponents that characterize second-order phase transitions are connected by scaling laws The scaling law which appears to apply to systems with transitions similar to percolation is 2β + γ = νd Sharp phase transitions, critical singularities and exponents, and scaling laws are all properties of infinitely large systems Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 13

Role of Percolation in Understanding 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 Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 14

Robustness: Inverse Percolation Transition Phenomenon of interest: impact of node failures on integrity of network Percolation theory helps us understand this process Treat square lattice as network whose nodes are intersections Randomly remove f fraction of nodes and ask how this impacts integrity of lattice If f is small, damage should be minimal Increasing f should isolate chunks of nodes from giant component For sufficiently large f, giant component breaks into tiny disconnected components Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 15

Damage is Modeled as an Inverse Percolation Process Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 16

Robustness: Inverse Percolation Transition Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 17

Bottom Line: Robustness of Regular Networks is Well Understood! Breakdown under random node removal is not gradual: once fraction of removed nodes reaches a critical threshold f c, network breaks into disconnected components Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 18

Robustness of Scale-free Networks 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 Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 19

Robustness of Scale-free Networks Scale-free networks do not appear to break apart under random failures Reason: the hubs (likelihood of removing a hub is small) Figure: y-axis: relative size of giant component. Have to remove almost all nodes to break down network f c 1. Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 20

Molloy-Reed Criterion High f c characteristic of scale-free networks Calculate f c for network with arbitrary degree distribution For a network to have a giant component, most nodes that belong to it must be connected to at least two other nodes Molloy-Reed criterion: a randomly-wired network has a giant component if: K = k2 k > 2 Networks with K < 2 lack a giant component, being fragmented into many disconnected components Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 21

Derivation of Molloy-Reed Criterion Each node in a giant component must be connected to 2 other nodes on average Denote with P(k i i j) the conditional probability that a node i with degree k i is connected to a node j that is part of the giant component Then, expected degree of such a node i is: k i i j = k i k i P(k i i j) Condition for node i to also be part of the giant component: k i i j 2 Equivalently: k i k i P(k i i j) 2 Goal: rewrite inequality in terms of first ( k ) and second momenta ( k 2 ) of p k We can rewrite the conditional probability P(k i i j) using Bayes formula: P(k i i j) = P(k i,i j) P(i j) = P(i j k i )p(k i ) P(i j) Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 22

Derivation of Molloy-Reed Criterion Continued For network with degree distribution p k, in absence of degree correlations, we can write: P(i j) = 2L N(N 1) = k N 1 We can also write: P(i j k i ) = k i N 1 The above expresses the fact that we can choose between N 1 nodes to link to, each with probability 1/(N 1) and we can try this k i times So, now we return to expected degree of node i in giant component: k i k i P(k i i j) = k i k i 1 k k i N 1 p(k i ) k N 1 = k i k 2 i p(k i ) = k2 k So we arrive at the Molloy-Reed criterion providing condition for giant component: K = k2 k > 2 2 is the criticality point Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 23

Application of Molloy-Reed Criterion: Predictive Power on Random Network In a random network, the degree distribution follows the Poisson distribution with peak at k and standard deviation σ = k 1 2 so, σ 2 = k Normally, we write σ in terms of the spread, so σ = ( k 2 k 2 ) 1 2 Equivalently, σ 2 = k 2 k 2 It follows then that in a random network, k = k 2 k 2 So, we have k 2 = k + k 2 = k (1 + k ) Then, K = k2 k = k (1+ k ) k = 1 + k > 2 Or, equivalently: k > 1 condition for giant component to emerge The Molloy-Reed-derived criterion coincides with that derived by Erdos and Renyi in their classical 1959 paper (i.e.; there is a giant component iff each node has on average more than one link) Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 24

Application of Molloy-Reed Criterion: Prediction of Giant Component Molloy-Reed criterion tells us that a network will have a giant component as long as k 2 diverges Recall that k n = k max k min k n P(k)dk In scale-free networks, k n = C kn γ+1 max k n γ+1 min n γ+1 C collects constant terms: Barabasi 4.20 All moments larger than γ 1 diverge (n > γ 1) Example: k 2 diverges in scale-free network with γ < 3 Since in most scale-free networks 2 < γ < 3, k 2 diverges in most scale-free networks Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 25

Critical Threshold f c At what threshold will a network lose its giant component? After the random deletion of a fraction f of nodes, a vertex with former connectivity k will now be attached to a number k of existing nodes. This number k is distributed binomially: P(k ) = k=k P(k)( k k ) (1 f ) k f k k Substituting the new k 2 and k in the criterion for criticality k2 k one obtains: 1 f c = 1 k 2 k 1 Barabasi 8.C pg. 42 Note: f c depends on k 2 and k, which are determined by degree distribution p k Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 26

Critical Threshold f c So, we have: f c = 1 1 k 2 k 1 On scale-fre networks: k 2 diverges for most scale-free networks, where 2 < γ < 3 So f c 1 On random networks: k 2 = k ( k + 1): So, fc ER = 1 1 k Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 27

Critical Threshold f c So, we have: f c = 1 1 k 2 k 1 On scale-fre networks: k 2 diverges for most scale-free networks, where 2 < γ < 3 So f c 1 On random networks: k 2 = k ( k + 1): So, fc ER = 1 1 k What do these derivations mean? On scale-free graphs, one has to remove almost all nodes in order for the giant component to break Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 27

Critical Threshold f c So, we have: f c = 1 1 k 2 k 1 On scale-fre networks: k 2 diverges for most scale-free networks, where 2 < γ < 3 So f c 1 On random networks: k 2 = k ( k + 1): So, fc ER = 1 1 k What do these derivations mean? On scale-free graphs, one has to remove almost all nodes in order for the giant component to break On random networks, f c is finite and inversely related to density the denser the random network, the higher its f c Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 27

Criticality Threshold in Degree-correlated Graphs The above derivation of f c assumes no degree-degree correlations In the case of degree-degree correlations, it is non-trivial to derive f c Consider the case of the giant component in a disassortative network Disassortative correlations compete with the formation of the giant component In the presence of disassortative correlations, divergence of k 2 is not a sufficient condition to get a robust graph with f c = 1 A detailed treatment in section 3 in Boccaletti 2006 review What is f c in real networks, and what is its dependence on γ? Just obtain k2 in terms of γ k Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 28

Criticality Threshold in Scale-free Networks Recall that k n = k max k min And that f c = 1 1 k 2 k 1 k n P(k)dk = C kn γ+1 max After some manipulations, one obtains: k n γ+1 min n γ+1 1 1, 2 < γ < 3 γ 2 3 γ f c = kγ 2 min k3 γ max 1 1 1, γ > 3 γ 2 γ 3 k min 1 What does this mean? How does f c change with respect to γ? Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 29

Enhanced Robustness in Scale-Free Networks For scale-free networks with γ < 3, second moment k 2 diverges as N So, if k 2, f c 1 To fragment a scale-free network, almost all nodes have to be removed 1 f c = 1 1 γ 2 3 γ kγ 2 min k3 γ 1 γ 2 max 1, 2 < γ < 3 γ 3 k min 1, γ > 3 For γ > 3, f c depends only on γ and k min So, f c is independent of network size N This network behaves like a random network; it falls apart once a finite fraction of its nodes are removed For γ < 3, k max diverges for large N, so f c 1 (dependence on N is hidden in k max) Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 30

Robustness of Finite Networks Above equation predicts that for a scale-free network, f c converges iff k max which corresponds to N Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 31

Robustness of Finite Networks Above equation predicts that for a scale-free network, f c converges iff k max which corresponds to N What about finite networks? Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 31

Robustness of Finite Networks Above equation predicts that for a scale-free network, f c converges iff k max which corresponds to N What about finite networks? If we insert k max = k min N γ 1 1 (Barabasi 4.18) into the above equation, we obtain: f c 1 C 3 γ with C collecting all terms that do not depend on N N γ 1 Above again shows that the larger the network, the closer f c is to 1 Enhanced Robutness Displayed by a network if its breakdown threshold deviates from the random network prediction: f c > fc ER All we need for it is deviation of k 2 from expected value for a random network of same size (not even a strict power law in degree distribution) Emerges under link removal, as well Scale-free property changes not only f c but also critical exponents γ, β, ν Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 31

Breakdown Thresholds Under Random Failures and Attacks Estimated f c for random node failures (column 2) and attacks (column 4) for ten reference networks. Power grid (p k is exponential), citation, and actor networks (high k ) lack enhanced robustness. Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 32

Attack Tolerance What if we do not remove nodes at random, but instead target hubs? Attack: remove nodes in order of highest to lowest degree? Attack assumes a detailed knowledge of the network topology in the ability to target hubs Impact of Hub Removal: A scale-free networks extreme fragility to attacks: f c is small, implying that the removal of only a few hubs can disintegrate the network. The initial network has degree exponent γ = 2.5, k min = 2 and N = 10, 000. Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 33

Critical Threshold Under Attack An attack on a scale-free network has two consequences: The critical threshold f c < 1, indicating that under attacks a scale-free network can be fragmented by the removal of a finite fraction of its hubs. The observed f c is remarkably low, indicating that we need to remove only a tiny fraction of the hubs to cripple the network! How do we analytically calculate f c for a scale-free network under attack? We need two observations on how hub removal changes the network: It changes the maximum degree of the network from k max to k max, as all nodes with degree larger than k max have been removed Degree distribution also changes from p k to p k, as nodes connected to the removed hubs will lose links, changing the degrees of the remaining nodes Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 34

Critical Threshold Under Attack An attack on a scale-free network has two consequences: The critical threshold f c < 1, indicating that under attacks a scale-free network can be fragmented by the removal of a finite fraction of its hubs. The observed f c is remarkably low, indicating that we need to remove only a tiny fraction of the hubs to cripple the network! How do we analytically calculate f c for a scale-free network under attack? We need two observations on how hub removal changes the network: It changes the maximum degree of the network from k max to k max, as all nodes with degree larger than k max have been removed Degree distribution also changes from p k to p k, as nodes connected to the removed hubs will lose links, changing the degrees of the remaining nodes Gist of idea: view attack as random node removal from network with adjusted k max and p k Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 34

Critical Threshold Under Attack Solution to equation fc 2 γ 1 γ = 2 + 2 γ 3 γ k min(f 3 γ 1 γ c 1) (Barabasi 8F) f c for attacks is non-monotonic, may increase for small γ and decrease for large γ f c for attacks < than f c for random failures Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 35

Summary: Critical Threshold Under Attack For large γ, scale-free network behaves like a random network, with the impact of an attack being similar to the impack of random node removal Failure and attack thresholds converge to each-other for large γ If γ, p k δ(k k min ), meaning all nodes have the same degree k min Random failures and attacks become indistinguishable in the γ limit, obtaining f c 1 1 k min 1 Summary While random node failures do not fragment a scale-free network, an attack that targets the hubs can easily destroy such a network This fragility is bad news for the Internet, as it indicates that it is inherently vulnerable to deliberate attacks It can be good news in medicine, as the vulnerability of bacteria to the removal of their hub proteins offers avenues to design drugs that kill unwanted bacteria Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 36

Cascading Failures Assumed that nodes fail independently of each-other In reality, activity of each node depends on activity of neighboring nodes Failure of a node can induce failures of neighboring nodes Examples: Blackouts (power grid) shifting power to other lines upon a node/generator failure may overwhelm capacities and cause more nodes to fail Denial of Service Attacks (DoS, internet) a failed router increases traffic on other routers, potentially cascading DoS attacks Financial Crises drop in the house prices in 2008 in the U.S. has spread along the links of the financial network, inducing a cascade of failed banks, companies and even nations Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 37

Empirical Patterns Governing Cascading Failures: Blackouts Frequently recorded measured of blackout size is energy unserved Electrical engineers approximate the probability distribution p(s) of energy unserved in North America with the power law p(s) s α α is the avalance exponent, and most blackouts are small but coexist with occasional major blackouts (millions of consumers lose power) Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 38

Empirical Patterns Governing Cascading Failures: Information Cascades As users frequently share web-content using URL shorteners, one can also track each spreading/sharing process Twitter study tracking 74 million such events over two months followed the diffusion of each URL from a particular seed node through reposts Size distribution of observed cascades follows power law, as well, with avalanche exponent α 1.75 (vast majority of posted URLS do not spread; few reposted thousands of times) Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 39

Empirical Patterns Governing Cascading Failures: Earthquakes Seismologists approximate the distribution of earthquake amplitudes with the power law with α 1.67 The cumulative distribution of earthquake amplitudes recorded between 1977 and 2000 The dashed lines indicate the power law fit used by seismologists to characterize the distribution The earthquake magnitude shown on the horizontal axis is the logarithm of s, which is the amplitude of the observed seismic waves Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 40

Modeling Cascading Failures Size distribution of the observed avalanches is universal, being independent of the particularities of the system Goal: understand the mechanisms governing cascading phenomena and to explain the power-law nature of the avalanche size distribution Many models have been proposed to capture the dynamics of cascading events They indicate that systems that develop cascades share three key ingredients: The system is characterized by some flow over a network, like the flow of electric current in the power grid or the flow of information in communication systems Each component has a local breakdown rule that determines when it contributes to a cascade, either by failing (power grid, earthquakes) or by choosing to pass on a piece of information (Twitter) Each system has a mechanism to redistribute the traffic to other nodes upon the failure or the activation of a component Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 41

Modeling Cascading Failures Size distribution of the observed avalanches is universal, being independent of the particularities of the system Goal: understand the mechanisms governing cascading phenomena and to explain the power-law nature of the avalanche size distribution Many models have been proposed to capture the dynamics of cascading events They indicate that systems that develop cascades share three key ingredients: The system is characterized by some flow over a network, like the flow of electric current in the power grid or the flow of information in communication systems Each component has a local breakdown rule that determines when it contributes to a cascade, either by failing (power grid, earthquakes) or by choosing to pass on a piece of information (Twitter) Each system has a mechanism to redistribute the traffic to other nodes upon the failure or the activation of a component Let s see two such models Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 41

Failure Propagation Model Introduced to model spread of ideas and opinions Consider a network with an arbitrary degree distribution, where each node contains an agent. An agent i can be in the state 0 (active or healthy) or 1 (inactive or failed), and is characterized by a breakdown threshold φ i = φ for all i All agents are initially in the healthy state 0 At time t = 0 one agent switches to state 1, corresponding to an initial component failure or to the release of a new piece of information In each subsequent time step, we randomly pick an agent and update its state following a threshold rule: If the selected agent i is in state 0, it inspects the state of its k i neighbors and adopts state 1 (i.e. it also fails) if at least a fraction of its k i neighbors are in state 1, otherwise it retains its original state 0 If selected agent i is in state 1, it does not change its state Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 42

Illustration of Failure Propagation Model Development of a cascade in a small network in which each node has the same breakdown threshold φ = 0.4 Initially all nodes are in state 0, shown as green circles After node A changes its state to 1 (purple), its neighbors B and E will have a fraction f = 1/2 > 0.4 of their neighbors in state 1 Consequently they also fail, changing their state to 1 In the next time step C and D will also fail, as both have f > 0.4 Consequently the cascade sweeps the whole network, reaching a size s = 5 One can check that if we initially flip node B, it will not induce an avalanche Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 43

Regimes in Failure Propagation Model Subcritical Regime If k is high, flipping a node is unlikely to flip other nodes, as healthy nodes already have many healthy neighbors Cascades in this regime die out quickly, and sizes follow an exponential distribution (no global cascades) Supercritical Regime If k is small, flipping single nodes can put several of neighbors over the threshold, triggering global cascade Perturbations induce major breakdowns Critical Regime At the boundary of the subcritical and supercritical regime, the avalanches have widely different sizes Numerical simulations indicate that in this regime the avalanche sizes s follows the power law p(s) s α with α = 3/2 if underlying network is random Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 44

Branching Model Simplest model that still captures the basic features of a cascading event Builds on the observation that each cascading failure follows a branching process Call the node whose initial failure triggers the avalanche the root of the tree Branches of the tree are the nodes whose failure was triggered by this initial failure The breakdown of node A starts the avalanche, hence A is the root of the tree The failure of A leads to the failure of B and E, representing the two branches of the tree Subsequently, E induces the failure of D and B leads to the failure of C Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 45

Branching Model Starts with a single active node In next time step, each active node produces k offspring, where k is selected from p k distribution If a node selects k = 0, that branch dies out If it selects k > 0, it will have k new active sites Size of avalanche corresponds to size of tree when all active sites die out Branching model predicts same phases as those observed in cascading failures model, but now phases are determined only by the p k distribution Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 46

Regimes in Branching Model Subcritical Regime For k < 1, on average each branch has less then one offspring Consequently each tree will terminate quickly In this regime, avalanche sizes follow an exponential distribution Supercritical Regime For k > 1, on average each branch has more than one offspring Consequently, tree will grow indefinitely, with all avalanches being global Critical Regime For k = 1, on average each branch has exactly one offspring Consequently, some trees are large and others die out shortly Numerical simulations indicate that in this regime the avalanche size distribution follows the power law Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 47

Size Distribution in Branching Model The branching model can be solved analytically, allowing us to determine the avalanche size distribution for an arbitrary p k If p k is exponentially-bounded, e.g. it has an exponential tail, the calculations predict α = 3/2 If, however, p k is scale-free, then the avalanche exponent depends on the power-law exponent γ, following: { 3/2, γ 3 α = γ/(γ 1), 2 < γ < 3 Indeed, empirically-observed avalanche exponents are all between 1.5 and 2, as predicted by the branching model Many other models in literature: overload model to capture power grid failures, sandpile model to capture cascading failures in critical regimes; other models can account for different traffic capacities in nodes and links Yet all predict existence of a critical state in which avalanche size follows the power law, suggesting the underlying phenomenon is universal Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 48

Building Robustness Can we enhance robustness? We can, if we connect peripheral nodes so that the removal of hubs does not fragment the network Price to pay: doubling number of links Cost: proportional to k Can we maximize the robustness of a network to both random failures and targeted attacks without changing the cost? Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 49

Low-cost, Enhanced Robustness A network s robustness against random failures is captured by its percolation threshold f c So, increase f c to increase robustness f c depends only on k 2 and k Consequently, the degree distribution which maximizes f c needs to maximize k 2 if we wish to keep the cost k fixed Achieved by binomial distribution corresponding to a network with only two kinds of nodes, with degrees k min and k max If we wish to simultaneously optimize the network topology against both random failures and attacks, we search for topologies that maximize the sum fc tot = fc rand + fc targ Combination of analytical arguments and numerical simulations indicate that this is best achieved by the bimodal degree distribution p k = (1 r) δ(k k min ) + rδ(k k max), where r fraction of nodes have degree k max and 1 r have degree k min Halting cascading failures: actually achieved via selectice node and link removal Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 50

Summary: Achilles Heel The remarkable robustness of real networks represents good news for most complex systems Indeed, there are uncountable errors in our cells, from misfolding proteins to the late arrival of a transcription factor Yet, the robustness of the underlying cellular network allows our cells to carry on their normal functions Network robustness also explains why we rarely notice the effect of router errors on the Internet or why the disappearance of a species does not result in an immediate environmental catastrophe Topological robustness comes with a price, vulnerability to attacks Bad news for the internet, economic systems, but good news for drug design Each complex system has its Acchilles heel: the networks behind them are simultaneously robust to random failures but vulnerable to attacks Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 51

Summary: Scale-free Property and Robustness Are complex networks robust because they are scale-free or are they scale-free because they are robust? If real systems have evolved to be scale-free to satisfy need for robustness, then one should obtain a scale-free network with robustness as optimization criterion However, the degree distribution in a network with maximal robustness is bimodal rather than a power law Robustness is not the principle that drives the development of scale-free networks Rather, networks are scale-free due to preferential (proportional) attachment It just so happens that scale-freeness also results in enhanced robustness but not maximal robustness Amarda Shehu,Fei Li () Robustness Cascades:Percolation Theory 52