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Complex networks A. Barrat, LPT, Université Paris-Sud, France

Complex networks A. Barrat, LPT, Université Paris-Sud, France. I. Alvarez-Hamelin (LPT, Orsay, France) M. Barthélemy (CEA, France) L. Dall’Asta (LPT, Orsay, France) R. Pastor-Satorras (Barcelona, Spain) A. Vespignani (LPT, Orsay, France). http://www.th.u-psud.fr/. Plan of the talk.

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Complex networks A. Barrat, LPT, Université Paris-Sud, France

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  1. Complex networksA. Barrat, LPT, Université Paris-Sud, France I. Alvarez-Hamelin (LPT, Orsay, France) M. Barthélemy (CEA, France) L. Dall’Asta (LPT, Orsay, France) R. Pastor-Satorras (Barcelona, Spain) A. Vespignani (LPT, Orsay, France) http://www.th.u-psud.fr/

  2. Plan of the talk • Complex networks: examples • Small-world networks • Scale-free networks: evidences, modeling, tools for characterization • Consequences of SF structure • Perspectives: weighted complex networks

  3. Examples of complex networks • Internet • WWW • Transport networks • Protein interaction networks • Food webs • Social networks • ...

  4. Social networks:Milgram’s experiment Milgram, Psych Today2, 60 (1967) Dodds et al., Science 301, 827 (2003) “Six degrees of separation”

  5. Small-world properties:also in the Internet Distribution of chemical distances between two nodes Average fraction of nodes within a chemical distance d

  6. Usual random graphs: Erdös-Renyi model (1960) N points, links with proba p: static random graphs Poisson distribution (p=O(1/N)) short distances (log N) BUT...

  7. n 3 Higher probability to be connected 2 1 # of links between 1,2,…n neighbors C = n(n-1)/2 Clustering coefficient Clustering: My friends will know each other with high probability! (typical example: social networks)

  8. Asymptotic behavior Lattice Random graph

  9. In-between: Small-world networks N nodes forms a regular lattice. With probability p, each edge is rewired randomly =>Shortcuts N = 1000 • Large clustering coeff. • Short typical path Watts & Strogatz, Nature393, 440 (1998)

  10. Size-dependence p >> 1/N => Small-world structure Amaral & Barthélemy Phys Rev Lett 83, 3180 (1999) Newman & Watts,Phys Lett A263, 341 (1999) Barrat & Weigt, Eur Phys J B13, 547 (2000)

  11. Is that all we need ? NO, because... Random graphs, Watts-Strogatz graphs are homogeneous graphs (small fluctuations of the degree k): While.....

  12. Airplane route network

  13. CAIDA AS cross section map

  14. The Internet and the World-Wide-Web • Protein networks • Metabolic networks • Social networks • Food-webs and ecological networks Are Heterogeneous networks P(k) ~ k - • <k>= const • <k2>  Scale-free properties Topological characterization P(k) =probability that a node has k links ( 3) Diverging fluctuations

  15. Exp. vs. Scale-Free Poisson distribution Power-law distribution Exponential Network Scale-free Network

  16. Main Features of complex networks • Many interacting units • Self-organization • Small-world • Scale-free heterogeneity • Dynamical evolution Standard graph theory Random graphs • Static • Ad-hoc topology

  17. Origins SF (1) The number of nodes (N) is NOT fixed. Networks continuously expand by the addition of new nodes Examples: WWW : addition of new documents Citation : publication of new papers (2) The attachment is NOT uniform. A node is linked with higher probability to a node that already has a large number of links. Examples : WWW : new documents link to well known sites (CNN, YAHOO, NewYork Times, etc) Citation : well cited papers are more likely to be cited again Two important observations

  18. BA model Scale-free model P(k) ~k-3 (1)GROWTH: At 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 ki of that node A.-L.Barabási, R. Albert, Science 286, 509 (1999)

  19. Connectivity distribution BA network

  20. More models • Generalized BA model • (Redner et al. 2000) • (Mendes et al. 2000) • (Albert et al. 2000) Non-linear preferential attachment : (k) ~ k Initial attractiveness : (k) ~ A+k • Highly clustered • (Dorogovtsev et al. 2001) • (Eguiluz & Klemm 2002) • Fitness Model • (Bianconi et al. 2001) • Multiplicative noise • (Huberman & Adamic 1999) Rewiring (....)

  21. Tools for characterizing the various models • Connectivity distribution P(k) =>Homogeneous vs. Scale-free • Clustering • Assortativity • ... =>Compare with real-world networks

  22. Topological correlations: clustering aij: Adjacency matrix ki=5 ci=0.1 ki=5 ci=0. i

  23. k=4 k=4 i k=3 k=7 Topological correlations: assortativity ki=4 knn,i=(3+4+4+7)/4=4.5

  24. Assortativity • Assortative behaviour: growing knn(k) Example: social networks Large sites are connected with large sites • Disassortative behaviour: decreasing knn(k) Example: internet Large sites connected with small sites, hierarchical structure

  25. Consequences of the topological heterogeneity • Robustness and vulnerability • Propagation of epidemics

  26. Robustness Robustness 1 node failure S fc 0 1 Fraction of removed nodes, f Complex systems maintain their basic functions even under errors and failures (cell  mutations; Internet  router breakdowns) S: fraction of giant component

  27. Case of Scale-free Networks Random failure fc =1 (2 <g 3) s Attack =progressive failure of the most connected nodes fc<1 fc 1 Internet maps R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)

  28. Robust-SF Failures vs. attacks Failures Topological error tolerance   3 : fc=1 (R. Cohen et al PRL, 2000) fc Attacks 1 S 0 1 f

  29. Other attack strategies • Most connected nodes • Nodes with largest betweenness • Removal of links linked to nodes with large k • Removal of links with largest betweenness • Cascades • ...

  30. bi is large bj is small i j Betweenness • measures the “centrality” of a node i: for each pair of nodes (l,m) in the graph, there are slm shortest paths between l and m silm shortest paths going through i bi is the sum of silm/ slm over all pairs (l,m)

  31. Other attack strategies • Most connected nodes • Nodes with largest betweenness • Removal of links linked to nodes with large k • Removal of links with largest betweenness • Cascades • ... Problem of reinforcement ? P. Holme et al., P.R.E 65 (2002) 056109 A. Motter et al., P.R.E 66 (2002) 065102, 065103 D. Watts, PNAS 99 (2002) 5766

  32. Epidemic spreading on SF networks • Natural computer virus • DNS-cache computer viruses • Routing tables corruption • Data carried viruses • ftp, file exchange, etc. Internet topology • Computer worms • e-mail diffusing • self-replicating E-mail network topology Epidemiology Air travel topology Ebel et al. (2002)

  33. Mathematical models of epidemics • Coarse grained description of individuals and their state • Individuals exist only in few states: • Healthy or Susceptible * Infected * Immune * Dead • Particulars on the infection mechanism on each individual are neglected. • Topology of the system: the pattern of contacts along which infections spread in population is identified by a network • Each node represents an individual • Each link is a connection along which the virus can spread

  34. r Absorbing phase Active phase Finite prevalence Virus death l c l The epidemic threshold is a general result The question of thresholds in epidemics is central (in particular for immunization strategies) SIS model: • Each node is infected with rate n if connected to one or more infected nodes • Infected nodes are recovered (cured) with rate dwithout loss of generality d =1 (sets the time scale) • Definition of an effective spreading rate l=n/d • Non-equilibrium phase transition • epidemic threshold=critical point • prevalence r=order parameter r=prevalence

  35. r Absorbing phase Active phase Finite prevalence Virus death l c l Computer viruses ??? What about computer viruses? • Very long average lifetime (years!) compared to the time scale of the antivirus • Small prevalence in the endemic case • Long lifetime + low prevalence = computer viruses always tuned • infinitesimally closeto the epidemic threshold ???

  36. <k> l c = <k2> SIS model on SF networks SIS= Susceptible – Infected – Susceptible Mean-Field usual approximation: all nodes are “equivalent” (same connectivity) => existence of an epidemic threshold 1/<k> for the order parameter r (density of infected nodes) Scale-free structure => necessary to take into account the strong heterogeneity of connectivities => rk=density of infected nodes of connectivity k =>epidemic threshold

  37. <k> l c = <k2> Epidemic threshold in scale-free networks <k2>   l c  0 Order parameter behavior in an infinite system

  38. Rationalization of computer virus data • Wide range of spreading rate with low prevalence (no tuning) • Lack of healthy phase = standard immunization cannot • drive the system below threshold!!!

  39. Results can be generalized to generic scale-free connectivity distributions P(k)~ k-g • If2 <g 3 we have absence of an epidemic threshold • and no critical behavior. • If 3 <g 4 an epidemic threshold appears, but • it is approached with vanishing slope (no criticality). • If g >4 the usual MF behavior is recovered. • SF networks are equal to random graph.

  40. Main results for epidemics spreading on SF networks • Absence of an epidemic/immunization threshold • The network is prone to infections (endemic state always possible) • Small prevalence for a wide range of spreading rates • Progressive random immunization is totally ineffective • Infinite propagation velocity Very important consequences of the SF topology! (NB: Consequences for immunization strategies) Pastor-Satorras & Vespignani (2001, 2002), Boguna, Pastor-Satorras, Vespignani (2003), Dezso & Barabasi (2001), Havlin et al. (2002), Barthélemy, Barrat, Pastor-Satorras, Vespignani (2004)

  41. Perspectives: Weighted networks • Scientific collaborations • Internet • Emails • Airports' network • Finance, economic networks • ... => are weighted networks !!

  42. Weights: examples • Scientific collaborations: (M. Newman, P.R.E. 2001) i, j: authors; k: paper; nk: number of authors : 1 if author i has contributed to paper k • Internet, emails: traffic, number of exchanged emails • Airports: number of passengers for the year 2002

  43. Weights • Weights: heterogeneous (broad distributions)? • Correlations between topology and traffic ? • Effects of the weights on the dynamics ?

  44. Weights: recent works and perspectives • Empirical studies (airport network; collaboration network: PNAS 2004) • New tools (PNAS 2004) • strength • weighted clustering coefficient (vs. clustering coefficient) • weighted assortativity (vs. assortativity) • New models (PRL 2004) • New effects on dynamics (resilience, epidemics...) on networks (work in progress)

  45. Alain.Barrat@th.u-psud.fr http://www.th.u-psud.fr/

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