Percolation in Complex Networks: Optimal Paths and Optimal Networks

Transcription

1 Percolation in Complex Networs: Optimal Paths and Optimal Networs Shlomo Havlin Bar-Ilan University Israel

2 Complex Networs Networ is a structure of N nodes and 2M lins (or M edges) Called also graph in Mathematics Many examples of networs Internet: nodes represent computers lins the connecting cables Social networ: nodes represent people lins their relations Cellular networ: nodes represent molecules lins their interactions Weighted networs each lin has a weight determining the strength or cost of the lin

3 Outline Percolation Graph Theory: Introduction Degree Distribution, Critical Concentration, Distance, Optimal Distance Complex Networs: Theory vs Experiment Generalized Networs: Broad Degree Distribution Scale Free: Anomalous physics, including percolation Applications: Efficient Immunization Strategy Optimal path- Optimal transport Optimize Networ Robustness References Cohen et al Phys. Rev. Lett. 85, 4626 (2000); 86, 3682 (200) Rozenfeld et al Phys. Rev. Lett. 89, 2870 (2002) Cohen and Havlin Phys. Rev. Lett. 90, (2003) Braunstein et al Phys. Rev. Lett. 9, (2003) Cohen et al Phys. Rev. Lett. 90, (2003)

4 Percolation and Immunization p = 0.3 remove or immune Networ exists P Prob. that a site belongs to a spanning cluster p = 0.5 remove or immune Networ collapse 0 pc p

5 Percolation Theory Sites or bonds are randomly removed from a lattice/graph with probability p. Below threshold Above threshold A phase transition exists at p c. The correlation length scales lie: ξ ν p pc. The size of the spanning cluster near criticality behaves as: ( c ) The number of clusters of size s behaves at criticality as: ns () s τ P p p β

6 Percolation critical exponents p same role as T in thermal phase transitions t P - probability that a site (or bond) belongs to cluster order parameter P ( p p ) β - similar to magnetization c ξ - correlation length mean distance between two sites on the same cluster ξ p p c The average size of finite clusters (analogous to susceptibility) ν S p p γ c P Prob. that a site belongs to a spanning cluster P ~ porder parameter ( ) β c p ν and γ are the same for p>p c and p<p c 0 pc p For ξ and S tae into account all finite clusters β, ν and γ called critical exponents describe critical behavior near the transition The exponents are universal Universality property of second order phase transition (order parameter 0 continuously) All magnets in d=3 have same β independent on the lattice and type of interactions T c depends on details (interactions, lattice) same for p c

7 Applications of Percolation Oil recovery Porous media Gelation Galaxy formation Polymerization Amorphous materials Epidemics and Forest fires

8 Random Graph Theory Developed in the 960 s by Erdos and Renyi. (Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 960). Discusses the ensemble of graphs with N vertices and M edges (2M lins). Distribution of connectivity per vertex is Poissonian (exponential), where is the number of lins : P ( ) c c = e!, c M = = 2 N Distance d=log N -- SMALL WORLD

9 Known Results Phase transition at average connectivity, = : < No spanning cluster (giant component) of order logn > A spanning cluster exists (unique) of order N = The largest cluster is of order N 23 / Size of the spanning cluster is determined by the self-consistent equation: P = e P Behavior of the spanning cluster size near the transition is linear: P ( p p) c β, β =, where p is the probability of deleting a site, pc = /

10 Percolation on a Cayley Tree Contains no loops Connectivity of each node is fixed ( z lins) Critical threshold: pc = z Behavior of the spanning cluster size near the transition is linear: P ( p p), β = c γ =, ν=/2, τ=5/2 β Exactly as for ER networs Mean Field Exponents -- valid for d d c = 6 Upper critical dimension

11 In Real World - Many Networs are non-poissonian P ( ) = e P ( )! λ c m K = 0 otherwise Homogeneous Heterogeneous

12 New Type of Networs Poisson distribution Power-law distribution Exponential Networ Scale-free Networ

13 Networs in Physics

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15 Examples Internet, WWW (Faloutsos et. al., SIGCOMM 99, Border et. al. IBM-Altavista research,albert, Jeong and Barabasi, Nature 2000) Protein Families (Delisi et al. PRL (2000), Unger et al, preprint) Metabolic cellular networs (Barabasi et. al., Nature 2000) Telephone & Power Networs Science collaboration networs (Newman, 200, Barabasi et. al., 200) Ecological Networs (Sole & Montoya, 200) P() P() P()

16 Does Percolation Theory Valid? Stability and Immunization Distribution p c = HOMOGENEOUS Critical concentration 30-70% Critical Infectious disease concentration Malaria 99% Measles 90-95% Whooping cough 90-95% Fifths disease 90-95% Chicen pox 85-90% Mumps 85-90% Rubella 82-87% Poliomyelitis 82-87% Diphtheria 82-87% Scarlet fever 82-87% Smallpox 70-80% INTERNET 99% HETEROGENOUS NEW THEORY IS NEEDED! Generalization of Erdös Theory: Cohen, Erez, ben-avraham, Havlin, PRL 85, 4626 (2000) Epidemiology Theory: Vespignani, Pastor-Satoral, PRL (200), PRE (200) Modelling: Albert, Jeong, Barabasi (Nature 2000)

17 Experimental Data: Virus survival (Pastor-Satorras and Vespignani, Phys. Rev Lett. 86, 3200 (200))

18 Percolation Model I Random Breadown (Immune) The Internet and many other real networs are scale-free networ, where P ( ) λ, 2 λ 3 Nodes are randomly removed (or immune) with probability p Where does the phase transition occur?

19 Percolation Model II Intentional Attac (Immune) The fraction, p, of nodes with the highest connectivity are removed (or immune). Is this fundamentally different from random breadown? We find that not only critical thresholds but also critical exponents are different! THE UNIVERSALITY CLASS DEPENDS ON THE WAY CRITICALITY REACHED

20 Results of Simulations and Theory Random Breadown (Immune) λ λ λ λ > 3 3 λ - no critical threshold a spanning cluster always exists - a critical threshold exists Intentional Attac (Immune) P ( p) λ = λ A critical threshold exist for every λ

21 Experimental Data: Internet Stability (Albert, Jeong and barabasi, Nature 406, 378 (2000))

22 THEORY FOR ANY DEGREE DISTRIBUTION Condition for the Existence of a Spanning Cluster If we start moving on the cluster from a single site, in order that the cluster does not die out, we need that each site reached will have, on average, at least 2 lins (one in and one out ). This means: i i j = ip( i i j) 2, where i j means that site i is i connected to site j. But, by Bayes rule: Pi j Pi ( ji) P ( i) ( i ) = Pi ( j) Exponential graph: We now that Pi j i ( i ) = N and Pi j ( ) = N 2 Combining all this together: κ = 2 (for every distribution) at the critical point. 2 2 = + = Cayley Tree: pc = z = 2 Cohen et al, PRL 85, 4626 (2000): PRL 86, 3862 (200)

23 Percolation for Random Breadown If percolation is considered the connectivity distribution changes according ' to the law: P( ) = P( ) p ( p) > Calculating the change in κ gives the percolation threshold: 2 p c =, κ where 0 κ 0 =. compared to pc = / < 0 > for Erdos Renyi 0 0 For scale-free distribution with lower cutoff m, and upper cutoff K, gives κ 3 λ 3 λ 0 2 λ 2 λ λ 2 λ K m =, K N. 3 λ K m For scale-free graphs with λ 3 the second moment diverges. No critical threshold! Networ is stable (or not immuned) even for p.

24 Percolation for Intentional Attac (Immune) Attac has two inds of influence on the connectivity distribution: Change in the upper cutoff Can be calculated by P ( ) = p, or approximately: K K = K /( λ ) = mp. Change in the connectivity of all other sites due to possibility of a broen lin (which is different than in random breadown). The probability of a lin to be removed can be calculated by: p or approximately: Substituting this into: where 2 α K κ = 3 α K K P( ) =, 0 = K = (2 )/( ). p p λ λ p c = κ, λ m λ m 3 λ 3 2 λ 2, gives the critical threshold. There exists a finite percolation threshold even for networs resilient to random error!

25 Critical Threshold Scale Free Cohen et al PRL (2003); cond-mat/ Efficient Immunization Strategies: Acquaintance Immunization Acquaintance Immunization Random immunization is inefficient in scale free graphs, while targeted immunization requires nowledge 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 nowledge is necessary. ( ) 2 2 c/ p P ()( ) / v e = f c = P ( ) v vp = P( )exp( p/ )/ p c p c

26 p c robust Poor immunization Critical Threshold Random Acquaintance vulnerable Intentional Efficient immunization Cohen et al. Phys. Rev. Lett. 9, 6870 (2003) Scale Free General result: p = κ c p = c For κ = 2 κ 2 0 Poisson = = criticality Efficient Immunization Strategies: Acquaintance Immunization :

27 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 (998) Gen. Func. Newman Callaway PRL(2000), PRE(200)) For random breadown the behavior near criticality in scale-free networs is different than for random graphs or from classical mean field percolation. For intentional attacsame as classical mean-field. 2 Even for networs with 3< λ < 4, where and are finite, the critical exponents differ from the nown mean-field result β =. The order of the phase transition and the exponents 3 are determined by. Size of the infinite cluster: P ~( p p c ) β Distribution of finite clusters at criticality: 2 < λ < 3 3 λ β = 3 < λ < 4 λ 3 λ > 4 (classical mean field, for d>d, and ER) 2λ 3 λ < 4 τ λ 2 n s ~ s τ = 2.5 λ 4 (classical mean field, d>6 or ER) (a) Intentional attac same exponents as for ER, (b) Percolation clusters at criticality are fractals

28 Fractals. Fractal geometry describes Nature better than classical geometry Two types of fractals: deterministic and random. Deterministic fractals and Random fractals Ideal fractals are self-similar. Every small part of the picture when magnified properly, is the same as the whole picture. Koch curve DLA

29 d Definition of fractal dimension MbL ( ) = b f ML ( ) generalization to non-integer dimension d f f Solution: M( L) = AL d Example: Koch curve d f f log4 M L = M( L) = M( L) = or d f = log3 d f - non integer between and 2 dimensions. Koch curve is not a line (d=) but doesn t fill a plane (d=2). d Example: Sierpinsi gaset d f f log3 M L = M( L) = M( L) = or d f = log2 Non integer dimension between and 2 dimensions. d

30 Fractal Dimension at Critical Percolation for d = 2 d 9 f S R, d f = = dl S l, d.68 l 2 d / d 0.95 d f N = R = R, S N N For d = d = 6, N = R, d = 4, d = 2 S = N c 6 2/3 f l

31 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 λ>3. λ 2 λ < 4 λ Largest cluster at criticality Short path dimension: d 3 l = 2 2 λ 4 d 3 l S l S N λ 2 2 λ < 4 λ 3 Scale Free networs Fractal dimension: d f = 4 λ 4 d f d d f dc S R f S R N N λ 2 λ < 4 2 λ 3 Embedding dimension: d c = 3 S N (upper critical dimension) 6 λ 4 The dimensionality of the graphs at criticality depends on the distribution! Random Graphs Erdos Renyi(960) N λ 2 λ = R d λ 4 λ 4

32 λ=2 λ=2.5 λ=5 λ=50 Rozenfeld et al PRL (2002), Manna et al (2002)

33 A 5. D Problem: Optimal Distance 2 C. E 4... w i = weight = price, quality, time.. Wea disorder (WD) all 3 B Path from A to B w i lmin= 2(ACB) lopt= 3(ADEB) w i = minimal optimal path contribute to the sum (narrow distribution) Strong disorder (SD) a single term dominates the sum (broad distribution) THE PATH IN STRONG DISORDER=THE PATH ON A MINIMAL SPANNING TREE (MST) SD example: Broadcasting video over the Internet, a transmission at constant high rate is needed. The narrowest band width lin in the path i between transmitter and receiver controls the rate.

34 Scale Free (Barabasi-Albert) Random Graph (Erdos-Renyi) P ( ) = A λ Small World (Watts-Strogatz) P( ) = e! Z = 4

35 Ultra Small World Shortest Paths in Scale Free Networs Small World d d d d P ( )~ λ = const. λ = 2 = loglog N 2 < λ < 3 log N = λ = 3 loglog N = log N Same as for ER and WS λ > Cohen, Havlin and ben-avraham, in Handboo of Graphs and Networs eds. Bornholdt and Shuster (Willy-VCH, NY, 2002) Chap.4 Cohen, Havlin Phys. Rev. Lett. 90, 5870(2003) 3 (Bollobas, Riordan, 2002) (Bollobas, 985) (Newman, 200) Confirmed also by: Dorogovtsev, Mendes et al (2002), Chung and Lu (2002)

36 Optimal path wea disorder Random Graphs and Watts Strogatz Networs lmin ~ log N l opt ~ log N n 0 pz Typical short range neighborhood Crossover from large to small world For N / n0 < l opt ~ N ~ e log N / n 0

37 Scale Free Optimal Path Wea disorder λ = 2.5K5 min For λ > 3 l ~ A( λ)logn opt l ~ B( λ)logn Thus l ~ l opt min l l opt min Thus For 2 ~ log ~ log log l opt < λ < N N 3 ~ exp( l min )

38 Optimal Path Strong Disorder-Minimal Spanning Trees (MST) CONSTANT SLOPE Random Graphs and Watts Strogatz Networs N total number of nodes l opt ~ N 3 Analytically and Numerically LARGE WORLD!! Compared to the diameter or average shortest path or wea disorder lmin ~ log N (small world) n0 - typical range of neighborhood without long range lins N n 0 - typical number of nodes with long range lins

39 Scale Free Optimal Path λ = 4 Strong Disorder-Minimal Spanning Trees Theoretically + Numerically l opt N ~ N N ( λ 3) /( λ ) 3 3 log N 3 < λ < 4 λ = 4 λ > 4 LARGE WORLD!! λ Numerically ~ log N 2 < λ < 3 l opt SMALL WORLD!! Wea Disorder l opt ~ log N for all λ Braunstein et al, Phys. Rev. Lett. 9, (2003); Cond-mat/ l min Diameter shortest path ~ log N log N / log log log log N N λ > 3 λ = 3 2 < λ < 3

40 Theoretical Approach Strong Disorder (i) Distribute random numbers 0<u< on the lins of the networ. (ii) Strong disorder represented by = exp( au ) (iii) The largest ε with a. i ui in each path between two nodes dominates the sum. (iv) The optimal path is the path with the min-max (v) Percolation exists if we remove all lins with u (vi) The optimal path must therefore be on the percolation cluster at criticality. What do we now about percolation clusters i i > p c A at criticality in networs? B

41 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 λ>3. λ 2 λ < 4 Short path dimension: λ 3 Largest cluster at criticality d l = 2 2 λ 4 dl 3 S l S N λ 2 2 λ < 4 λ 3 Scale Free networs Fractal dimension: d f = 4 λ 4 d f d S R f d f dc λ S R N N 2 λ < 4 λ 3 2 Embedding dimension: d c = 3 S N (upper critical dimension) 6 λ 4 The dimensionality of the graphs depends on the distribution! Random Graphs Erdos Renyi(960) λ 2 λ λ 4 λ 4

42 Theoretical Approach Strong Disorder Conclusions (i) Percolation on random networs is lie percolation in d or. d = d c (ii) Since loops can be neglected the optimal path can be identified with the shortest path on percolation-only a single path exist between any nodes. Calculate the length of shortest path: B A For ER Mass of infinite cluster d c = 6, S From percolation S ~ d f R / 4/ 6 2/3 ~ N d d f = N = N S ~ l 2,( = d l 2) where N ~ d R (also Erdos-Renyi, 960) Thus, for ER, WS and SF with λ > 4 :, d f For SF with 3 < λ < 4 c, l opt / 2 /3 ~ l ~ S ~ N d and d l change due to novel topology: l opt ~ N ( λ 3)/( λ )

43 Optimal Networs Simultaneous waves of targeted and random attacs Bimodal: fraction of (-r) having lins and r having = ( < > r) / r lins 2 + r = from left to right Optimal Bimodal : r ( p t / p ) r=0.00 SF r Bimodal Bimodal p t p r - Fraction of targeted - Fraction of random Condition for connectivity: P() changes: P( ) κ = < < 2 > > K 0 0 = P0 ( ) ( pr ) pr 0 P() changes also due to targeted attac For p, <<, t p r p t / p r f - critical threshold c 2 is the only parameter

44 Given: N=00,, and Specific example: p t / p = 0.05 r = r < >= 2. Optimal Bimodal = 0. i.e, 0 hubs of degree 2 =2 using = ( < > r) / r 2 + Paul et al. Europhys. J. B 38, 87 (2004), (cond-mat/040433) Tanizawa et al. Optimization of Networ Robustness to Waves of Targeted and Random Attacs (Cond-mat/ , Phys. Rev. E (2005)

45 Novel Condition for criticality Conclusions and Applications κ = p = κ κ < > < > 2 0 2, c /( 0 ), 0 / Distance in scale free networs λ<3 : d~loglogn - ultra small world, λ>3 : d~logn. Optimal distance strong disorder Random Graphs and WS (Minimal spanning trees) { λ lopt ~ N for λ > 3 Large World scale free λ lopt ~log N for 2< λ < 3 Small World Transition between wea and strong disorder - in both percolation theory is important Scale Free networs (2<λ<3) are robust to random breadown. Scale Free networs are vulnerable to attac on the highly connected nodes. Efficient immunization is possible without nowledge of topology, using Acquaintance Immunization. The critical exponents for scale-free directed and non-directed networs are different than those in exponential networs different universality class! Large networs can have their topology optimized for maximum robustness to random breadown and/or intentional attac. λ 3 l opt ~ N 3 Large World

46 Generalized random graphs topology. Conclusions and Applications P ( ) λ, λ 4 Erdos-Renyi, λ<4 novel Scale free networs (2<λ<3) are robust to random breadown. Scale free networs are vulnerable to intentional attac on the most highly connected nodes. Efficient immunization is possible without nowledge of topology, using Acquaintance Immunization. The critical exponents for scale-free (λ<4) directed and non-directed networs are different than those in exponential networs different universality class! Moreover, the critical exponents depend on the strategy of node removal! THE UNIVERSALITY CLASS DEPENDS ON THE WAY CRITICALITY IS REACHED! Optimal path and minimal spanning trees on networs can be studied using percolation theory. Networs topology can be optimized for maximum robustness to various scenarios of failures such as random breadown and/or intentional attac-using percolation

47 Size of the Spanning Cluster The spanning cluster size can be determined using either differential equations (Molloy & Reed, Combinatorics, Probability & Statistics, (998)), or by the generating function method (Callaway et. al., Phys. Rev. Lett. 85, 5624 (2000)). Using those methods the spanning cluster size can be shown to be: P = ( p)( P( ) u ), =0 where u is the smallest positive root of: u= p+ p ( ) P( ) u = 0. (Cohen et al., Phys. Rev. Lett. 86, 3682 (200))

48 Connectivity distribution: Probability of reaching: Generating Functions (Newman et al, PRE 64, 0268 (200)) G0 ( x) = P( ) x G = 0 '( x) 0 ( ) ( ) = = P x = 0 G x Branch size: H ( x) = xg ( H ( x)) = P( ) H ( x) = 0 p s = sn s Cluster size: Infinite cluster size: s 0 = s = 0 = s= 0 = 0 H ( x) p x G ( H ( x)) P( ) H ( x) P = H0() With Percolation: (Callaway et al, PRL 85, 5468 (2000)) P ( ) q q P( ) u = = 0 u = q + q = 0 P ( ) u

49 P u Critical exponents Derivation of β - Regular Graphs P ( q qc ) Cayley Tree: ( ) q q P( ) u = = = ε ε = q c 0 q q δ + = q c c + δ + δ = 0 P( )( ε ) ( ) P u = ( ) ε + = 0 2 ( )( 2) +K ε ~ 2 ( ) ( )( 2) 2 ε δ β = q P ~ qc c p ε ~ ( q q ) β z ( ) ( ) P q = q u q z u = q + u z qc + δ ε = q ( ) z c δ + ε z ε ~ 2 δ z 2 β =

50 P ( ) q q P( ) u u = q u P ( q q ) β = ε ε = q = 0 P ( )~ = = q + q c c 0 = 0 = q c P + δ q δ + Critical exponents Derivation of β c ( ) + δ u = 0 ( ) P u ( ) ε + P( )( ε ) q = / ( ) α Using Abelian methods: ( )( 2) (2 ) 2 c α ε + K + Γ α ε c ( ) α α 3 δ 3< α < 4, cγ(2 α ) ε ~ 2 ( ) δ α > 4. ( )( 2) 3< α < 4 β = α 3 α > 4 For 2< α < 3: q c = = 0 0 α 2 ( ) ~ + Γ(2 α ) ε P u c ε = cγ(2 α ) 3 α δ β =, 2 < α < 3 3 α 3 α

51 Critical exponents Derivation of τ n ~ s e s * τ s / s Tauberian theorems: ps = sns = s τ ( ) s H 0 x ps x φ ~ ε y p s ~ s y φ φ = qc + ( ε ) qc qc H ( x) = q+ qx P( ) H ( x) 0 = 0 H x q qx P H x x ( )( 2) Γ(2 ) + + K + c 2 2 α α 2 φ φ τ = 2 + ( ) ( ) ( ) = + 5 = 0, 4 = ε q = qc φ ( ε) H ( ε) y τ = α > 2 2α 3 τ = 2 + =, 2 < α < 4 α 2 α 2

52 Critical exponents Derivation of σ ns s e s q q c * τ s/ s * σ ( ) ~, ~ ( ) Scaling relation in percolation theory: τ = dσν q σ dν τ c ( ) qc ( N ) ~ N = N q c = κ κ 2 3 α α K K m m 3 α 3 α 2 α 2 α /( ) K N α For 3< α < 4: 3 α 3 α α q = q ( ) q ( N) ~ κ ~ K ~ N c c c α 3 σ =,3 < α < 4 α 2 For 2< α < 3: qc ( ) = 0 q ( N)~/ κ ( N)~ K α c 3 α σ =, 2 < α < 3 α 2 3

53 There exists a finite percolation threshold even for networs resilient to random error! Percolation in complex Networs Percolation for Intentional Attac Attac has two inds of influence on the connectivity distribution: Change in the upper cutoff Can be calculated by or approximately: K = K P( ) = p, /( α ) K = mp. Change in the connectivity of all other sites due to possibility of a broen lin (which is different than in random breadown). The probability of a lin to be removed can be calculated by: K p = P( ) 0 = K or approximately: Substituting this into: Where κ = 2 3, α K α K 3 α 2 α ( 2 α ) /( α) p = p. m m 3 α 2 α p c =, κ, gives the critical threshold.

54 The Model Intentional Attac The fraction, p, of nodes with the highest connectivity are removed. Is this fundamentally different from random breadown?

55 THEORY Condition for the Existence of a Spanning Cluster If we start moving on the cluster from a single site, in order that the cluster does not die out, we need that each site reached will have, on average, at least 2 lins (one in and one out ). This means: i i j = ip( i i j) 2, where i site j. i j means that site i is connected to But, by Bayes rule: P( i j) = i We now that Pi ( j) = and Pi ( j) = i Pi ( j) P ( ) N N. i Pi ( j) Combining all this together: 2 κ = 2 (for every distribution) at the critical point. i i Exponential graph: 2 2 = + = Cayley Tree: p = c z = 2

56 Percolation for Random Breadown If percolation is considered the connectivity distribution changes according to the law: ' P P p ' p ( ) = ( ') ( ) ' > Calculating the change in κ gives the percolation threshold: = 0 p c κ, where κ = 0. Plugging in the scale-free distribution with lower cutoff m, and upper cutoff K, gives: 3 α 3 α 2 α K m κ 0 = 2 α 2 α 3 α K m, K ~ N α. For scale-free graphs with α 3 the second moment diverges No critical threshold! Networ is stable even for p.

57 The Model Random Breadown The Internet is believed to be a randomly connected scale-free networ where P( ) α, α 25. Nodes are randomly removed with probability p. Where does a phase transition occur?

58 Results of Simulations and Theory Random Breadown α α > no critical threshold a spanning cluster always exists - a critical threshold exists Intentional Attac P α=2.5 α= A critical threshold exist for every α

59 Critical Exponents Using the properties of power series near a singular point (Abelian methods), the behavior near the critical point can be studied. The behavior near criticality in scale-free networs is different than for exponential ones! 2 Even for networs with 3 < α < 4, in w hich and are finite, the critical exponents change from the nown mean-field result β =. The order of the phase transition and the exponents are 3 determined by. Size of the infinite cluster: 3 α β = P ~( p p c ) β α 3 2 < α < 3 < α < α > Distribution of finite clusters at criticality: 2α 3 α < 4 τ = n s ~ s τ α α 4

60 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. Chemical dimension: d l = α α α < α 4 4 Fractal dimension: d f = α 2 α α < α 4 4 Embedding dimension: d c = α 2 α 3 6 α < α 4 4 The dimensionality of the graphs depends on the distribution!

61 Behavior of the Site to Site Distance near Criticality In standard random graph models the average distance between sites scales as: r ~ log N, where N is the system size. That is, the mass behaves as: M ~ r. From infinite dimensional percolation theory it is nown: This is also true for scale-free networs with α 4. M ~ r 2, or r ~ M at p =. A crossover exists between those behaviors where the correlation length: ξ ~ ( p p c ) p c. M Communication becomes inefficient even before the breadown

62 Conclusions and Applications The Internet is resilient to random breadown. The Internet is sensitive to intentional attac on the most highly connected nodes. Large networs can have their connectivity distribution optimized for maximum resilience to random breadown and/or intentional attac. A virus has a finite probability of infecting a large portion of the Internet, regardless of how low is the probability of infection. However, if a finite fraction of the most highly connected routers of the Internet bloc the virus, it cannot infect a finite portion of the Internet. Even before breadown the diameter of the spanning cluster becomes large maing communication inefficient. The critical exponents for scale-free networs are different than those in exponential networs different universality class!

63 Small World A small world networ is a regular lattice with added random lins. Examples: Movie actors Polymer chains configuration space Acquaintance networs Neural networs Exponents are the same as in mean-field percolation.

64 Results of Simulations and Theory

65 Experimental Data: Virus survival (Pastor-Satorras and Vespignani, Phys. Rev Lett. 86, 3200 (200))

66 Scale Free Random Graph Small World

67 Percolation and Immunization Percolation theory p = 0.3 remove or immune Networ exists P Prob. that a site belongs to a spanning cluster P ~ ( ) β c pporder parameter 0.5 p = remove or immune Networ collapse 0 pc p

68

69 Size of the Spanning Cluster The spanning cluster size can be determined using either differential equations (Molloy & Reed, Combinatorics, Probability & Statistics, (998)), or by the generating function method (Callaway et. al., Phys. Rev. Lett. 85, 5624 (2000)). If After time t, tn lins have been followed, the change in the number of unexposed nodes of connectivity is: dp (, t ) P(, t) = dt 2t. The number of open lins in the cluster is: X() t = 2 t P(,) t. When the entire cluster has been exposed. Solving this gives the cluster size: ( )( P = p P( ) u ), =0 where u is the smallest positive root of: u = p+ p ( ) = 0 P( ) u (Cohen et al., Phys. Rev. Lett. 86, 3682 (200))

70 Critical Threshold Scale Free Cohen et al cond-mat/ Efficient Immunization Strategies: Acquaintance Immunization ( ) 2 2 c/ Acquaintance Immunization Random immunization is inefficient in scale free graphs, while targeted immunization requires nowledge 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 nowledge is necessary. p P ()( ) / v e = f c = P ( ) v vp = P( )exp( p/ )/ p c p c

71 Results of Simulations and Theory λ Efficient Immunization Strategy λ Random immunization is inefficient in scale free graphs, while targeted immunization requires nowledge of the degrees. p c In Acquaintance Immunization one immunizes random neighbors of random individuals.