Networks as a tool for Complex systems

Transcription

1 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

2 Weighted networs each lin (node) has a weight determining the strength or cost of the lin (node). Networs as a tool for Complex systems Complex systems are usually composed of many interacting entities. Can be well represented by networs: nodes represent the entities and lins their interactions. Examples: biological systems, social systems, climate, earthquaes, epidemics, transport, economic, etc.

3 Social Networs- Stanley Milgram (1967) Nodes: individuals Lins: social relationship (family/wor/friendship/etc.) John Guare )1992( Six Degrees of Separation

4 Map showing the world-wide internet traffic

5 A snapshot of Internet connectivity.

6 Hierarchical topology of the international web cache

7 Image of Social lins in Canberra, Australia

8 Networ of protein-protein interactions. The color of a node signifies the phenotypic effect of removing the corresponding protein (red, lethal; green, non-lethal; orange, slow growth; yellow, unnown).

9 Complex systems Made of many non-identical elements connected by diverse interactions. NETWORK

10 Networ Properties Degree distribution P() -- - degree of a node Diameter or distance Average distance between nodes--d Clustering Coefficient c()= How many of my friends are also friends? Centrality or Betweeness -- b no. of lins between neighbors ( -1)/2 Number of times a bond or a node is relatively used for the shortest path Critical Threshold: The concentration of nodes that are removed and the networ collapses d( AB) 3 c( D) 2 6 b( BC) N N A D C B

11 Random Graph Theory Developed in the 1960 s by Erdos and Renyi. (Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 1960). 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 M!, c 2 N Distance d=log N -- SMALL WORLD

12 More Results Phase transition at average connectivity, 1: 1 No spanning cluster (giant component) of order logn 1 A spanning cluster exists (unique) of order N 1 The largest cluster is of order N 2/ 3 Behavior of the spanning cluster size near the transition is linear: P ( p pc ), 1 pc 1/, where p is the probability of a site to exist, Size of the spanning cluster is determined by the self-consistent equation: P e P 1

13 Percolation on a Cayley Tree Contains no loops Connectivity of each node is fixed (z connections) Critical threshold: p c Behavior of the spanning cluster size near the transition is linear: P ( p p), 1 c 1 z 1

14 In Real World - Many Networs are non-poissonian P( ) e P ( )! c m K 0 otherwise

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

16 Networs in Physics

17 Faloutsos et. al., SIGCOMM 99 Internet Networ

18 Metabolic networ

19 Jeong et all Nature 2000

20 Jeong et al, Nature (2000)

21 More Examples Trust networs: Guardiola et al (2002) networs: Ebel etal PRE (2002) Trust Trust

22 Erdös Theory is Not Valid Stability and Immunization Distance Distribution q c 1 p 1 1 Critical concentration 30-50% c d ~ log N Critical concentration Infectious disease 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% Generalization of Erdös Theory: Cohen, Erez, ben-avraham, Havlin, PRL 85, 4626 (2000) Epidemiology Theory: Vespignani, Pastor-Satoral, PRL (2001), PRE (2001) Almost constant (Metabolic Networs, Jeong et. al. (Nature, 2000)) Cohen, Havlin, Phys. Rev. Lett. 90, 58701(2003) Modelling: Albert, Jeong, Barabasi (Nature 2000)

23 Experimental Data: Virus survival (Pastor-Satorras and Vespignani, Phys. Rev Lett. 86, 3200 (2001))

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

25 Scale-free model (1) GROWTH : At every time step 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 i of that node ( ) i i j j P() ~ -3 A.-L.Barabási, R. Albert, Science 286, 509 (1999)

26 Shortest Paths in Scale Free Networs P( ) ~ d const. 2 Ultra Small World d log log N log N d log log N (Bollobas, Riordan, 2002) Small World d log N 3 (Bollobas, 1985) (Newman, 2001) Cohen, Havlin Phys. Rev. Lett. 90, 58701(2003) Cohen, Havlin and ben-avraham, in Handboo of Graphs and Networs eds. Bornholdt and Shuster (Willy-VCH, NY, 2002) chap.4 Confirmed also by: Dorogovtsev et al (2002), Chung and Lu (2002)

27 P( ), 2.5 Model of Stability Random Breadown (Immunization) The Internet is believed to be almost randomly connected scale-free networ, where Nodes are randomly removed (or immune) with probability q=1-p Where does the phase transition occur?

28 Model for Stability Targeted Attac (Immunization) The fraction, q, of nodes with the highest degree are removed (or immunized). 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

29 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 i site i is i connected to site j. But, by Bayes rule: P i j P( i j i ) P( i ) ( i ) P( i j) Exponential graph: We now that P i j i ( i ) N 1 and P i j ( ) N 1 2 Combining all this together: 2 (for every distribution) at the critical point Cayley Tree: 1 pc z 1 2 Cohen et al, PRL 85, 4626 (2000): PRL 86, 3862 (2001)

30 Percolation for Random Breadown If percolation is considered the connectivity distribution changes according ' to the law: P( ) P( ) p (1 p) Calculating the change in gives the percolation threshold: 2 1 pc 1 qc, 1 where 0 0. compared to pc 1/ for Erdos Renyi For scale-free distribution with lower cutoff m, and upper cutoff K, gives 2 K 3 K 3 3 m m 1 1, K N. For scale-free graphs where the second moment diverges. No critical threshold! Networ is stable (or not immunized) even for q1

31 p c robust Poor immunization Acquaintance vulnerable Intentional Efficient immunization Critical Threshold Scale Free p For Poisson : p c K K General result: c 1 K Cohen et al. Phys. Rev. Lett. 91, (2003) Efficient Immunization Strategies: Acquaintance Immunization

32 Percolation for Targeted Attac (Immunization) 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 1/(1 ) 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: or approximately: Substituting this into: where 2 K 3 K K 1 p P( ), 0 K (2 )/(1 ) p p. 1 1 p c 1, m m , gives the critical threshold. There exists a finite percolation threshold even for networs robust to random removal

33 p c robust Poor immunization Acquaintance vulnerable Intentional Efficient immunization Critical Threshold Scale Free p For Poisson : p c K K General result: c 1 K Cohen et al. Phys. Rev. Lett. 91, (2003) Efficient Immunization Strategies: Acquaintance Immunization

34 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 breadown the behavior near criticality in scale-free networs is different than for random graphs or from mean field percolation. For intentional attac-same as mean-field. 2 Even for networs with 3 4, where and are finite, the critical exponents change from the nown mean-field result 1. The order of the phase transition and the exponents 3 are determined by. Size of the infinite cluster: P ~ ( p p c ) Distribution of finite clusters at criticality: n s s ~ (nown mean field) (nown mean field)

35 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. 2 Random Graphs Erdos Renyi(1960) 4 Chemical dimension: 3 Largest cluster at criticality d l d S S N Scale Free networs Fractal dimension: d f 4 4 d f 2 S R d f d f dc 1 S R N N Embedding dimension: d c 3 S N 4 (upper critical dimension) 6 4 The dimensionality of the graphs depends on the distribution!