1. Marc Barthélemy CEA, France Architecture of Complex Weighted Networks

2. Collaborators  A. Barrat (LPT-Orsay, France)  R. Pastor-Satorras (Politechnica Univ. Catalunya)  A. Vespignani (Indiana Univ., USA)  A. Chessa (Univ. Cagliari, Italy)  A. de Montis (Univ. Cagliary, Italy)

3. Outline I.Weighted Complex networks Motivations Characterization: Measurement tools II. Case-studies: Transportation networks Inter-cities network: Sardinia Global network: World Airport Network III. Modeling Necessity of topology-traffic coupling: Simple model

4. Complex Networks  Recent studies on topological properties showed: - broad distribution of connectivities - impact on different processes (eg. Resilience, epidemics) ji

5. Beyond Topology: Weighted Networks i j w w ij ji

6. Beyond Topology: Weighted Networks  Internet, Web, Emails: importance of traffic  Ecosystems: prey-predator interaction  Airport network: number of passengers  Scientific collaboration: number of papers  Metabolic networks: fluxes heterogeneous Are: - Weighted networks - With broad distributions of weights

7. Why study weighted networks ? Motivation ) The weights can modify the behavior predicted by topology: Resilience Epidemics …

8. Motivation: Epidemics ) The weights will affect the propagation of the disease ) Immunization strategies ? ) Epidemics spread on a ‘contact network’ Social networks (STDs on sexual contact network) Transportation network (Airlines, railways, highways) WWW and Internet (e-viruses)

9. Topological Characterization of Large Networks Complex All these networks are: Very large  Statistical tools needed !  Statistical mechanics of large networks

10. Topological Characterization Diameter: d » logN ) ‘small-world’ d » N 1/D ) ‘large world’ Clustering coeff.: C À C RG » 1/N C(k) » k -   ) Hierarchy Assortativity: k nn versus k ? Betweenness centrality, modularity, …

11. Topological Characterization: P(k) Connectivity k (k À 1: Hubs) Connectivity distribution P(k) : probability that a node has k links Usual random graphs: Erdös-Renyi model (1960)

12. Classes of networks Poisson distribution Exponential Network Power-law distribution Scale-free Network

13. Weighted Networks ) New measurement tools needed !

14. Weighted networks characterization Generalization of k i : strength For w ij =w 0 : For w ij and k i independent:

15. Weighted networks characterization In general: If  > 1 or if  =1 and A  ) Existence of strong correlations !

16. Weighted networks characterization Weighted clustering coefficient: If c i w /c i >1: Weights localized on clicques If c i w /c i <1: Important links don’t form clicques If w and k uncorrelated ) c i w =c i

17. Weighted networks characterization Weighted assortativity: If k nn w (i)/k nn (i) >1: Edges with larger weights point to nodes with larger k

18. Weighted networks characterization

19. « Disparity »: If Y 2 (i) » 1/k i ¿ 1: No dominant connections If Y 2 (i) À 1/k i : A few dominant connections

20. Weighted networks characterization Disparity:

21. Case study: Transportation networks Intra-urban flows (Eubank et al, PRE 2003, Nature 2004) Inter-cities flows (with A. Chessa and A. de Montis) Global flows: Word Airport network (PNAS, 2004) Different studies at different scales:

22. Airplane route network Nodes: airports Links: direct flight

23. Case study: Global Air Travel Number of airports 3863; 18807 links Topology: Maximum coordination number 318 Average coordination number 9.74 Average clustering coefficient 0.53 Average shortest path 4.37 Weights: Maximum weight 6167177 (seats/year, 2002) Average weight 74509

24. Case study: Airport network Broad distribution: connectivity and weights

25. s(k) proportional to k     =1.5 (Randomized weights: s= k:  =1) Strong correlations between topology and dynamics Correlations topology-traffic: Airports

26. Correlations topology-traffic » (k i k j )   ¼ 0.5

27. Weighted clustering coefficient: Airport (esp. for large k) C w (k) > C(k): larger weights on cliques at all scales

28. Weighted assortativity: Airport For large k ) Large traffic between hubs k nn (k) < k nn w (k): larger weights between large nodes

29. Disparity: Airport Y 2 (k) » 1/k ) No dominant connection

30. Airport: Summary Topology: Scale-free network Rich traffic structure Strong correlations traffic-topology

31. Case study: Inter-cities movements Sardinia: - Italian island 24,000 km 2 - 1,600,000 inhabitants

32. Case study: Inter-cities movements Sardinian network: -Nodes: 375 Cities - Link w ji =w ij : # of individuals going from i to j (daily and by any means)

33. Case study: Inter-cities movements-Topology N=375, E=16,248 ) =43, k max =279

34. Case study: Inter-cities movements-Topology Clustering: ¼ 0.26 ' C RG ¼ 0.24

35. Case study: Inter-cities movements-Topology Slightly disassortative network

36. Case study: Inter-cities movements-Traffic ¼ 23, w max ¼ 14.000 (!) P(w) » w -  w  w ¼ 2.2

37. Case study: Inter-cities movements-Traffic Correlations: s » k ,  ' 1.9

38. Case study: Inter-cities movements-Traffic Weighted clustering: Hubs form large w-clicques

39. Case study: Inter-cities movements-Traffic Weighted assortativity: Large w between hubs

40. Case study: Inter-cities movements-Traffic Y 2 (k) » k - ,  ' 0.4 ) Traffic jams !

41. Transportation networks: Summary NetworkP(k)P(s)s » k  Y 2 (k) » k - Clustering Assort. Global (WAN) Heavy tail ( ' 2.0) Scale-free Broad  ' 1.5 ' 1.0 (good) C w /C>1 k w /k>1 Inter-citiesFast tail ( ' 3.5) Random Graph Broad ( s ' 2.0)  ' 1.9 ' 0.4 (bad) C w /C>1 k w /k>1 Intra-urban (Eubank) Heavy tail ( ' 2.4) Scale-free Broad ( s ' 2.7)  ' 1.0 --C(k) » 1/k --

42. Summary: Weighted networks ● Broad strength distributions ) weights are relevant ! (independently from topology) ● Topology-weight correlations important ) Model for networks with heterogeneous and correlated connectivities and weights ?

43. Weighted networks: Model Growing network: addition of nodes Proba(n ! i) / s i

44. Weighted networks: Model Rearrangement of weights  ¿ 1: No effect (  =0: BA model)  À 1: Traffic stimulation

45. Evolution equations (mean-field)

46. Analytical results Power law distributions for k and s: P(k) ~ k  ; P(s)~s  2 <  < 3:  Strong coupling  ! 2  Weak coupling  ! 3

47. Analytical results Power law distributions for w: P(w) » w -  Correlations topology/weights: s i ' (2  +1)k i  k i

48. Nonlinear correlations ? Correlations topology/weights: s i ' (2  +1)k i  k i )  = 1 ) How can we obtain  1 ? - Inclusion of space - …

49. Nonlinear correlations ? Growing network: addition of nodes + distance Proba(n ! i) / s i f(d ni ) With: f(d)» e -d/d 0 d 0 /LÀ 1 )  = 1 d 0 /L¿ 1 )  > 1 !

50. Summary & Perspectives Weighted networks: Complexity not only topological !  Very rich traffic structure  Correlations between weights and topology  Model for weighted networks: topology-traffic coupling (variants…) Perspectives:  Effect of weights heterogeneity on dynamical processes (epidemics)  Getting more data: common features ?

51. References A. Barrat, MB, R. Pastor-Satorras, A. Vespignani, PNAS 101, 3747 (2004) A. Barrat, MB, A. Vespignani, PRL 92, 228701 (2004) A. Barrat, MB, A. Vespignani, LNCS 3243, 56 (2004) A. Barrat, MB, A. Vespignani, PRE 70, 066149 (2004) MB, A. Barrat, R. Pastor-Satorras, A. Vespignani, Physica A 346, 34 (2005) A.de Montis, MB, A. Chessa, A. Vespignani (in preparation) A. Barrat, MB, A. Vespignani (in preparation) marc.barthelemy@th.u-psud.fr

52. Numerical results: clustering

53. Numerical results: assortativity

54. Numerical results

55. Numerical results: P(w), P(s) (N=10 5 )

56. Numerical results: weights w ij ~ min(k i,k j ) a

57. Numerical results: assortativity analytics: k nn proportional to k ( 

58. Numerical results: clustering analytics: C(k) proportional to k ( 

59. Extensions of the model: (i)-heterogeneities Random redistribution parameter  i ( i.i.d. with  )  self-consistent analytical solution (in the spirit of the fitness model, cf. Bianconi and Barabási 2001) Results s i (t) grows as t a( i ) s and k proportional broad distributions of k and s same kind of correlations

60. Extensions of the model: (i)-heterogeneities late-comers can grow faster

61. Extensions of the model: (i)-heterogeneities Uniform distributions of 

62. Extensions of the model: (i)-heterogeneities Uniform distributions of 

63. Extensions of the model: (ii)-non-linearities ni j New node: n, attached to i New weight w ni =w 0 =1 Weights between i and its other neighbours: Example  w ij =  (w ij /s i )(s 0 tanh(s i /s 0 )) a  i increases with s i ; saturation effect at s 0  w ij = f(w ij,s i,k i )

64. Extensions of the model: (ii)-non-linearities s prop. to k  with  > 1 N=5000 s 0 =10 4   w ij =  (w ij /s i )(s 0 tanh(s i /s 0 )) a Broad P(s) and P(k) with different exponents

65. Models for growing scale-free graphs Barabási and Albert, 1999: growth + preferential attachment P(k) ~ k -3 Generalizations and variations: Non-linear preferential attachment :  (k) ~ k  Initial attractiveness :  (k) ~ A+k  Highly clustered networks Fitness model:  (k) ~  i k i Inclusion of space Redner et al. 2000, Mendes et al. 2000, Albert et al. 2000, Dorogovtsev et al. 2001, Bianconi et al. 2001, Barthélemy 2003, etc... (....) => many available models P(k) ~ k - 

66. Topological correlations: clustering i k i =5 c i =0. k i =5 c i =0.1 a ij : Adjacency matrix

67. Economical and technological realms Internet, WWW (sites, hyperlinks) Power grids (power plants, electric lines) Transportation networks (airports, direct flights) General Motivation: Ubiquity of Networks Social realm Actors network (actors, in the same movie) Collaboration network (scientists, common paper) Citation network (scientists, cited ref.) Acquaintances (people, ‘social relation’) Biological realm Neural networks (neurons, axons) Ecosystems: Food-webs (species, who eats who) Metabolic networks (metabolites, chem. Reaction)

68. Topological Characterization: Diameter Diameter = max i,j 2 G d(i,j)(1) or:= (2)

69. Topological Characterization: Diameter Stanley Milgram (1967): Average distance in North-America: d ¼ 6 « Six degrees of separation » Usually d » log N ( ¿ N 1/dim ) ) ‘Small-World’

70. Topological Characterization: Clustering Random graph: C RN » 1/N ¿ 1 Observed: - C À C RN - Hierarchy: C(k) » k -  ¼ 1

71. Topological Characterization: Clustering # of links between neighbors C = k(k-1)/2 Do your friends know each other ?

72. Topological correlations: assortativity k i =4 k nn,i =(3+4+4+7)/4=4.5 i k=3 k=7 k=4

73. Topological Characterization: Assortativity Are your friends similar to you ?

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

75. Topological Characterization: Betweenness Centrality  st = # of shortest paths from s to t  st (ij)= # of shortest paths from s to t via (ij) i j k ij: large centrality jk: small centrality

76.  Real networks are fragmented into group or modules  Society: Granovetter, M. S. (1973) ; Girvan, M., & Newman, M.E.J. (2001); Watts, D. J., Dodds, P. S., & Newman, M. E. J. (2002).  WWW: Flake, G. W., Lawrence, S., & Giles. C. L. (2000).  Biology: Hartwell, L.-H., Hopfield, J. J., Leibler, S., & Murray, A. W. (1999).  Internet: Vasquez, Pastor-Satorras, Vespignani(2001). Topological Characterization: Modularity Modularity vs. Fonctionality ?

77. Weights i, j: authors; k: paper; n k : number of authors  : 1 if author i has contributed to paper k ● Airports: number of available seats for the year 2002 ● Scientific collaborations:

78. Case study: Collaboration network (1) Broad distribution: connectivity and weights

79. Global data analysis: Collaboration network Number of authors 12722; 39967 links Topology: Maximum coordination number 97 Average coordination number 6.28 Clustering coefficient 0.65 Pearson coefficient (assortativity) 0.16 Average shortest path 6.83 Weight: Maximum weight 21.33 Average weight 0.57

80. Weighted assortativity: Collab. ) High-degree nodes publish together many papers !

81. Weighted clustering coefficient: Collab. ) For high-degree nodes: most papers done in well-connected groups

82. Weighted clustering coefficient: Airports C(k) < C w (k): larger weights on cliques at all scales

83. Weighted clustering coefficient: Airport ) Rich-club phenomenon

84. Case study: Inter-urban movements-Traffic Weighted assortativity: Large w between hubs

85. Correlations topology-weight: Collab. S(k) proportional to k