Shlomo Havlin } Anomalous Transport in Scale-free Networks, López, et al,prl (2005) Bar-Ilan University. Reuven Cohen Tomer Kalisky Shay Carmi

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1 Anomalous Transport in Complex Networs Reuven Cohen Tomer Kalisy Shay Carmi Edoardo Lopez Gene Stanley Shlomo Havlin } } Bar-Ilan University Boston University Anomalous Transport in Scale-free Networs, López, et al,prl (2005)

2 Important Function of Networs -- Transport a) Transport: s, viruses over Internet, epidemics in social networs, passengers in airline networs, etc. b) Main past focus: studies of static properties of networs. Robustness, shortest paths, degree distribution, growth models, etc. c) No general theory of transport properties on networs. Some results-not yet a global picture.

3 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 (degree distribution) is Poissonian (exponential), where is the number of lins : c c P ( ) = e!, c M = = 2 pc N Distance d=log N -- SMALL WORLD ; = 1 1/ < > ;MF critical exponents

4 In Real World - Many Networs are non-poissonian P ( ) = e P ( )! λ c m K = 0 otherwise Erdos-Renyi (1960) Barabasi-Albert (1999)

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

6 Networs in Physics

7 WWW-Networ Barabasi et al (1999)

8 Barabasi and Albert Emergence of scaling in random networs Science 286, (1999).

9 Faloutsos et. al., SIGCOMM 99 Internet Networ

10 Jeong, Tombor, Albert, Barabasi, Nature (2000)

11 Distance almost constant does not depend on N Jeong, Tombor, Albert, Barabasi, Nature (2000) Many Social networs are also found to be scale free

12 New models based on preferential attachment (Barabasi, 2000) Anomalous Mean Distance in Scale Free Networs Ultra Small World Small World P ( )~ λ l = const. λ = 2 l = log log N 2 < λ < 3 log N l = loglog N λ = 3 l = log N λ > 3 (Bollobas, Riordan, 2002) (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 (2003), Chung and Lu (2002)

13 p c More anomalies: Percolation on Scale Free robust Poor immunization vulnerable Efficient immunization Random Acquaintance Intentional Cohen et al. Phys. Rev. Lett. 85, 4626 (2000); 86, 3682 (2001); 91, (2003) p = 1 K c 0 For K General result: 0 p = 1 c K Poisson = = 1 Efficient Immunization Strategie: Acquaintance Immunization SF new topology critical exponents are different and anomalous (not regular MF)! THE UNIVERSALITY CLASS DEPENDS ON THE WAY CRITICALITY REACHED :

14 Scale Free is not sufficient: Fractal and Non Fractal Networs Box counting method Generate boxes where all nodes are within a distance Calculate number of boxes, n() l, of size needed to cover the networ l We obtain for WWW, social networs, cellular networs, etc. l N B M N / N ~ l 2< d < 5 d B () l or d B B B B l Self similarity Chaoming Song, SH, Hernan Mase, Nature. 433, 392 (2005); cond-mat/

15 Renormalization of WWW networ with l B = 3 l

16 SOME REAL NETWORKS ARE FRACTALS AND SOME NOT!!

17 Important Function of Networs -- Transport a) Transport: s, viruses over Internet, epidemics in social networs, passengers in airline networs, etc. b) Main past focus: studies of static properties of networs. c) No general theory of transport properties on networs. Some important results-not yet a global picture.

18 Transport in Complex Networs Transport on regular networs and fractals: 2 R < >= Dt ρ L 2 d Anomalous transport 2/ d 2 w < R >= L ζ ; At d = d + ζ ρ w f Einstein Relation Bunde-Havlin, Fractals and Disordered Systems, Springer (1996) Klafter et al, Levy wals < 2 Ben-avraham and SH, Diffusion on Fractals and Disordered Systems, Cambridge Univ. Press (2000) d w

19 Transport in Complex Networs Some important results-not yet a global picture: 1. Bolt and ben-avraham (NJOP, 2005): transit time faster as the SF networ grows; wals are recurrent despite the infinite dimension. 2. Lasaros Gallos (PRE, 2004): super diffusion with d w < 2 depending on λ numerically. < l > 3. Noh and Rieger (PRL, 2004): exact expression for MFPT, n 2/ d 2 w (2 ) p( τ) τ λ 4. Lopez et al (PRL, 2005): Broad distribution of conductances and diffusion constants (depending on degrees) heterogeneous transport of SF networs

20 Anomalous Transport in Complex Networs G conductance between two nodes A and B (G) cumulative distribution (V=1) A B (V=0) each lin unit resistor solving Kirchhoff Eqs G ~ D (Einstein relation) Simulations- N=10,000 Power law tail Scale free improve transport

21 Origin of power law? λ = A 2.5, 1000 Strong correlations between conductance and degree of nodes A B P ( G A, B ) * G - distribution of G given A and - most probable conductance B * Large A and B dominate the high conductanc e regime * Many parallel paths reduce dramatically the conductance

22 1 G B * large For A 1 2 ) ( ) ( ) ( ) ( + = = Φ Φ λ B A A B B d P P G B 1 2 where ) ( Thus = Φ λ G g g G G G Supported by simulations Origin of power law?

23 Simulations scale free F(G) where = G g G Φ( G) dg G = 2λ 1 g G + 1 slope 2λ - 2 In good agreement with theory E. Lopez et al, Anomalous transport in complex networ, PRL (2005)

24 Scaling laws of resistance and diffusion for fractal and non-fractals SF networs R( l;, ) = l ξ F( 1, l d l d t (; l, ) = l dw D( 1, 2 ) wal 1 2 l d l d ) In regular homogeneous fractals D and F are constants

25 Conclusions and Applications Scale Free - p ( ) λ : * Anomalous properties- d=loglog N, diff. percolation * Rich topology: Fractal-Nonfractal real networs * Generalization of ER: λ > 4 ER, Infinite dimension, regular MF Anomalous Transport: * Broad distribution of diffusion constants or conductances * Heterogeneous {Dij} depending on nodes (i,j)-mainly on degreedue to heterogeneous topology. Applications: * Optimize topology of networs against various types of failures * Optimize transport, searching and navigating in networs

26 Simple Physical Picture A Transport Bacbone B Networ can be seen as series circuit. A B Conductance G* is related to node degrees A and B through a networ dependent parameter c. To first order (conductance of transport bacbone >> c A B ) G * = c A + A B B