Daniel ben - Avraham Clarkson University Boston Universtiy Reuven Cohen Tomer Kalisky Alex Rozenfeld Bar-Ilan University Eugene Stanley Lidia Braunstein Sameet Sreenivasan

References Cohen et al Phys. Rev. Lett. 85, 4626 (2000); 86, 3682 (2001) Rozenfeld et al Phys. Rev. Lett. 89, (2002) Cohen and HavlinPhys. Rev. Lett. 90, (2003) Cohen et al Phys. Rev. Lett. 91, (2003) Phys. Rev. Lett. 91, (2003)Braunstein et al Sreenivasan et al Phys. Rev. E Submitted (2004)

Percolation theory

New Type of Networks Poisson distribution Exponential Network Power-law distribution Scale-free Network Airlines

Networks in Physics

Internet Network Faloutsos et. al., SIGCOMM ’99

Metabolic network

More Examples Trust networks: Guardiola et al (2002) networks: Ebel etal PRE (2002) Trust -2.9

Cohen, Havlin, Phys. Rev. Lett. 90, 58701(2003) Infectious disease Malaria 99% Measles 90-95% Whooping cough 90-95% Fifths disease 90-95% Chicken pox 85-90% Mumps 85-90% Rubella 82-87% Poliomyelitis 82-87% Diphtheria 82-87% Scarlet fever 82-87% Smallpox 70-80% INTERNET 99% Stability and Immunization Critical concentration 30-50%

Erdös-Rényi model (1960) - Democratic - Random Pál Erdös Pál Erdös ( ) Connect with probability p p=1/6 N=10  k  ~ 1.5 Poisson distribution

Scale-free model (1) GROWTH : A t every timestep we add a new node with m edges (connected to the nodes already present in the system). (2) PREFERENTIAL ATTACHMENT : The probability Π that a new node will be connected to node i depends on the connectivity k i of that node A.-L.Barabási, R. Albert, Science 286, 509 (1999) P(k) ~k -3 BA model

Shortest Paths in Scale Free Networks Cohen, Havlin Phys. Rev. Lett. 90, 58701(2003) Cohen, Havlin and ben-Avraham, in Handbook of Graphs and Networks eds. Bornholdt and Shuster (Willy-VCH, NY, 2002) chap.4 Confirmed also by: Dorogovtsev et al (2002), Chung and Lu (2002) Small World (Bollobas, Riordan, 2002) (Bollobas, 1985) (Newman, 2001)

We find that not only critical thresholds but also critical exponents are different ! THE UNIVERSALITY CLASS DEPENDS ON THE WAY CRITICALITY REACHED

Efficient Immunization Strategies: Acquaintance Immunization Critical Threshold Scale Free Cohen et al. Phys. Rev. Lett. 91, (2003) Random Acquaintance Intentional General result: robust vulnerable Poor immunization Efficient immunization

. Optimal Distance - Disorder l min = 2(ACB) l opt = 3(ADEB) D A E B C = weight = price, quality, time….. minimal optimal path Weak disorder (WD) – all contribute to the sum (narrow distribution) Strong disorder (SD)– a single term dominates the sum (broad distribution) SD – example: Broadcasting video over the Internet, a transmission at constant high rate is needed. The narrowest band width link in the path between transmitter and receiver controls the rate. Path from A to B

Random Graph (Erdos-Renyi)Scale Free (Barabasi-Albert) Small World (Watts-Strogatz) Z = 4

Optimal path – weak disorder Random Graphs and Watts Strogatz Networks Typical short range neighborhood Crossover from large to small world For

Scale Free – Optimal Path – Weak disorder

Optimal path – strong disorder Random Graphs and Watts Strogatz Networks CONSTANT SLOPE - typical range of neighborhood without long range links - typical number of nodes with long range links Analytically and Numerically LARGE WORLD!! Compared to the diameter or average shortest path or weak disorder (small world) N – total number of nodes

Scale Free – Optimal Path Theoretically + Numerically Strong Disorder Weak Disorder Diameter – shortest path LARGE WORLD!! SMALL WORLD!! Braunstein, Buldyrev, Cohen, Havlin, Stanley, Phys. Rev. Lett. 91, (2003); Cond-mat/

Theoretical Approach – Strong Disorder (i) Distribute random numbers 0<u<1 on the links of the network. (ii) Strong disorder represented by with (iii) The largestin each path between two nodes dominates the sum. (v) The optimal path must therefore be on the percolation cluster at criticality. (vi) Percolation on random networks is like percolation in or (iv) The lowest are on the percolation cluster where (vii) Since loops can be neglected the optimal path can be identified with the shortest path. Mass of infinite cluster where Thus, (see also Erdos-Renyi, 1960) From percolation Thus, for ER, WS and SF with For SF withand change due to novel topology:

Transition from weak to strong disorder For a given disorder strength Sreenivasan et al Phys. Rev. E Submitted (2004) For details see POSTER for

Conclusions and Applications Distance in scale free networks 3 : d~logN. Optimal distance – strong disorder – Random Graphs and WS scale free Transition between weak and strong disorder Scale Free networks (2<λ<3) are robust to random breakdown. Scale Free networks are vulnerable to attack on the highly connected nodes. Efficient immunization is possible without knowledge of topology, using Acquaintance Immunization. The critical exponents for scale-free directed and non-directed networks are different than those in exponential networks – different universality class! Large networks can have their connectivity distribution optimized for maximum robustness to random breakdown and/or intentional attack.  Large World Small World Large World

Conclusions and Applications Distance in scale free networks 3 : d~logN. Optimal distance – strong disorder – Random Graphs and WS scale free Scale Free networks (2<λ<3) are robust to random breakdown. Scale Free networks are vulnerable to attack on the highly connected nodes. Efficient immunization is possible without knowledge of topology, using Acquaintance Immunization. The critical exponents for scale-free directed and non-directed networks are different than those in exponential networks – different universality class! Method for embedding Scale Free Networks embedded in Euclidean space - d min < 1 Large networks can have their connectivity distribution optimized for maximum resilience to random breakdown and/or intentional attack.  Large World Small World Large World

Conclusions and Applications Generalized random graphs, >4 – Erdos-Renyi, <4 – novel topology. Distance in scale free networks: d~loglogN - ultra small world Scale free networks (2<λ<3) are resilient to random breakdown. Scale free networks are sensitive to attacks on the most highly connected nodes. Efficient immunization is possible without knowledge of topology, using Acquaintance Immunization. The optimal distance makes random graphs large worlds while scale-free networks are still small worlds! Large networks can have their connectivity distribution optimized for maximum resilience to random breakdown and/or intentional attack.

Scale Free – Optimal Path – Weak disorder

2.8 A critical threshold exist for every Intentional Attack (Immune) no critical threshold – a spanning cluster always exists - a critical threshold exists Random Breakdown (Immune) Results of Simulations and Theory N=500000

Conclusions and Applications Generalized random graphs, >4 – Erdos-Renyi, <4 – novel topology. Distance in scale free networks: d~loglogN - ultra small world Scale free networks (2<λ<3) are resilient to random breakdown. Scale free networks are sensitive to intentional attack on the most highly connected nodes. Efficient immunization is possible without knowledge of topology, using Acquaintance Immunization. The critical exponents for scale-free directed and non-directed networks are different than those in exponential networks – different universality class! Large networks can have their connectivity distribution optimized for maximum resilience to random breakdown and/or intentional attack.

Conclusions and Applications Generalized random graphs, >4 – Erdos-Renyi, <4 – novel topology –novel physics. Distance in scale free networks 3 : d~logN (small world). Scale free networks (2<λ<3) are robust to random breakdown. Scale free networks are vulnerable to intentional attacks on the most highly connected nodes. Efficient immunization is possible without knowledge of topology, using Acquaintance Immunization. The critical exponents for scale-free directed and non-directed networks are different in 4 – different universality classes! THE UNIVERSALITY CLASS DEPENDS ON THE WAY CRITICALITY REACHED! Optimize networks to have maximum robustness to both random breakdown and intentional attacks.

Scale Free – Optimal Path Theoretically + Numerically Strong Disorder Weak Disorder Diameter – shortest path LARGE WORLD!! SMALL WORLD!! For Braunstein, Buldyrev, Cohen, Havlin, Stanley, Phys. Rev. Lett. 91, (2003); Cond-mat/

K=A/B

Critical Threshold Binomial Distribution Two Gaussians SIR (Susceptible-Infected-Removed) model Scale Free d – distance between Gaussians one at k=3 second at k=3+d Top- random immunization Bottom- Acquaintance immunization variance = 2 variance = 8 r – the infection rate  - infection time λ = 2.5 λ = 3.5 Top – random immunization Bottom – Acquaintance Immunization

Percolation in Directed Networks General example. (2D) :, - the in degree of the node - the out degree of the node. [ Newman et al PRE (2001) Dorogortev & Mendes PRE (2001) General condition for giant component ]

For every graph we start with For Scale-Free we have: and In the fraction BA, we consider the relationship So that.  This correlation retains the original distributions. Percolation in Scale Free Directed Networks Critical concentration:

Directed SF networks without correlation Directed SF networks with correlation Non-Directed SF networks Directed Scale Free Critical Exponents:

Scale Free Networks in Euclidean Space Principle: Minimum total length of links Rozenfeld, Cohen, ben-Avraham, Havlin Phys. Rev. Lett. 89, (2002)

λ=2 λ=2.5 λ=5 λ=50

Shortest path For d>1 d min <1 For all

We find that not only critical threshold but also critical exponents are different ! THE UNIVERSALITY CLASS DEPENDS ON THE WAY CRITICALITY REACHED

Random Graphs – Erdos Renyi(1960) Largest cluster at criticality Scale Free networks Fractal Dimensions From the behavior of the critical exponents the fractal dimension of scale-free graphs can be deduced. Far from the critical point - the dimension is infinite - the mass grows exponentially with the distance. At criticality - the dimension is finite for >3. Chemical dimension: Fractal dimension: Embedding dimension: (upper critical dimension) The dimensionality of the graphs depends on the distribution!

Random Graphs – Erdos Renyi(1960) Largest cluster at criticality Scale Free networks Fractal Dimensions From the behavior of the critical exponents the fractal dimension of scale-free graphs can be deduced. Far from the critical point - the dimension is infinite - the mass grows exponentially with the distance. At criticality - the dimension is finite for > l d             Chemical dimension: Fractal dimension: Embedding dimension: (upper critical dimension) The dimensionality of the graphs depends on the distribution!

Cohen et al, PRL 85, 4626 (2000): PRL 86, 3862 (2001)

Critical Exponents Using the properties of power series (generating functions) near a singular point (Abelian methods), the behavior near the critical point can be studied. (Diff. Eq. Melloy & Reed (1998) Gen. Func. Newman Callaway PRL(2000), PRE(2001)) For random breakdown the behavior near criticality in scale-free networks is different than for random graphs or from mean field percolation. For intentional attack-same as mean-field. (known mean field) Size of the infinite cluster: (known mean field) Distribution of finite clusters at criticality: Even for networks with, where and are finite, the critical exponents change from the known mean-field result. The order of the phase transition and the exponents are determined by.

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Percolation theory

For Barabasi-Albert scale free networks:

Conclusions and Applications Distance in scale free networks: d~loglogN - ultra small world Scale free networks (2<λ<3) are resilient to random breakdown. Scale free networks are sensitive to intentional attack on the most highly connected nodes. Efficient immunization is possible without knowledge of topology, using Acquaintance Immunization. The critical exponents for scale-free directed and non-directed networks are different than those in exponential networks – different universality class! Large networks can have their connectivity distribution optimized for maximum resilience to random breakdown and/or intentional attack.

Efficient Immunization Strategies: Acquaintance Immunization Random immunization is inefficient in scale free graphs, while targeted immunization requires knowledge of the degrees. In Acquaintance Immunization one immunizes random neighbors of random individuals. One can also do the same based on n neighbors. The threshold is finite and no global knowledge is necessary. Critical Threshold Scale Free Cohen et al cond-mat/

Percolation and Immunization of Complex Networks Shlomo Havlin Daniel ben -Avraham Clarkson University Laszlo Barabasi Notre Dame University Reuven Cohen Tomer Kalisky Alex Rozenfeld Nehemia Schwartz Bar-Ilan University  Eugene Stanley Boston University Armin Bunde Giessen University

P(k) DNA

Complex system NETWORK New York Times

For every graph we start with For Scale-Free we have: and In the fraction BA, we consider the relationship So that.  This correlation retains the original distributions. Percolation in Scale Free Directed Networks Critical concentration:

Directed SF networks without correlation Directed SF networks with correlation Non-Directed SF networks Directed Scale Free Critical Exponents: Schwartz, Cohen, ben-Avraham Barabasi, Havlin PRE RC (2002)

Infectious disease Malaria 99% Measles 90-95% Whooping cough 90-95% Fifths disease 90-95% Chicken pox 85-90% Mumps 85-90% Rubella 82-87% Poliomyelitis 82-87% Diphtheria 82-87% Scarlet fever 82-87% Smallpox 70-80% INTERNET 99% Stability and Immunization Critical concentration 30-50%

Distance in Scale Free Networks Cohen, Havlin, Phys. Rev. Lett. (2003); Cond-mat/ (2002) Small World

Percolation theory

c p Results of Simulations and Theory

2.8 A critical threshold exist for every Intentional Attack (Immune) no critical threshold – a spanning cluster always exists - a critical threshold exists Random Breakdown (Immune) Results of Simulations and Theory

Networks in Physics

Efficient Immunization Strategy Theory : Acquaintance Immunization Random immunization is inefficient in scale free graphs, while targeted immunization requires knowledge of the degrees. In Acquaintance Immunization one immunizes random neighbors of random individuals. One can also do the same based on n neighbors. The threshold is finite and no global knowledge is necessary.

Conclusions and Applications Distance in scale free networks: d~loglogN - ultra small world Scale free networks (2<λ<3) are resilient to random breakdown. Scale free networks are sensitive to intentional attack on the most highly connected nodes. Efficient immunization is possible without knowledge of topology, using Acquaintance Immunization. The critical exponents for scale-free directed and non-directed networks are different than those in exponential networks – different universality class! Method for embedding Scale Free Networks in Euclidean space - d min < 1 Large networks can have their connectivity distribution optimized for maximum resilience to random breakdown and/or intentional attack.

Conclusions and Applications Generalized random graphs, >4 – Erdos-Renyi, <4 – novel topology. Distance in scale free networks 3 : d~logN. Scale free networks (2<λ<3) are resilient to random breakdown. Scale free networks are sensitive to intentional attack on the most highly connected nodes. Efficient immunization is possible without knowledge of topology, using Acquaintance Immunization. The critical exponents for scale-free ( <4) directed and non-directed networks are different than those in exponential networks – different universality class! Large networks can have their connectivity distribution optimized for maximum resilience to random breakdown and/or intentional attack.

Conclusions and Applications Scale Free network – Tomography - Distance in scale free networks: d~loglogN - ultra small world Optimal distance – strong disorder – Random Graphs and WS Scale free Scale Free networks (2<λ<3) are resilient to random breakdown. Scale Free networks are sensitive to intentional attack on the most highly connected nodes. Efficient immunization is possible without knowledge of topology, using Acquaintance Immunization. The critical exponents for scale-free directed and non-directed networks are different than those in exponential networks – different universality class! Method for embedding Scale Free Networks embedded in Euclidean space - d min < 1 Large networks can have their connectivity distribution optimized for maximum resilience to random breakdown and/or intentional attack.  Large World Small World Large World

Conclusions and Applications The Internet is resilient to random breakdown. The Internet is sensitive to intentional attack on the most highly connected nodes. Distance in scale free networks: d~loglogN - ultra small world Efficient immunization is possible without knowledge of topology, using a strategy of Acquaintance Immunization. The critical exponents for scale-free directed and non-directed networks are different than those in exponential networks – different universality class! Method for embedding Scale Free Networks in Euclidean space - d min < 1 Large networks can have their connectivity distribution optimized for maximum resilience to random breakdown and/or intentional attack.

Armin Bunde Giessen University

Percolation and Immunization of Complex Networks Shlomo Havlin Daniel ben - Avraham Clarkson University Laszlo Barabasi Notre Dame University Reuven Cohen Tomer Kalisky Alex Rozenfeld Nehemia Schwartz Bar-Ilan University   Lidia Braunstein Sergey Buldyrev Eugene Stanley Boston University

Armin Bunde Giessen University

Armin Bunde Giessen University

Analytically and Numerically LARGE WORLD!! Compared to the diameter or average shortest path or weak disorder (small world)