Architecture of Complex Weighted Networks

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

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 • 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)

5. Beyond Topology: Weighted Networks w ij j i w 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. Motivation Why study weighted networks ? • )The weights can modify the behavior predicted • by topology: • Resilience • Epidemics • …

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

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

10. Topological Characterization • Diameter: d» logN) ‘small-world’ • d» N1/D ) ‘large world’ • Clustering coeff.: CÀ CRG» 1/N • C(k)» k-) Hierarchy • Assortativity: knn versus k ? • Betweenness centrality, modularity, …

11. Topological Characterization: P(k) • Connectivity k (kÀ 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 Power-law distribution Exponential Network Scale-free Network

13. Weighted Networks )New measurement tools needed !

14. Weighted networks characterization Generalization of ki: strength • For wij=w0: • For wij and ki independent:

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

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

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

18. Weighted networks characterization

19. Weighted networks characterization • « Disparity »: • If Y2(i)» 1/ki¿ 1: No dominant connections • If Y2(i)À 1/ki: A few dominant connections

20. Weighted networks characterization • Disparity:

21. Case study: Transportation networks Different studies at different scales: • 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)

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

26. Correlations topology-traffic • <wij>» (kikj) ¼ 0.5

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

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

29. Disparity: Airport Y2(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 km2 • - 1,600,000 inhabitants

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

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

34. Case study: Inter-cities movements-Topology • Clustering: <C>¼ 0.26' CRG¼ 0.24

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

36. Case study: Inter-cities movements-Traffic • <w>¼ 23, wmax¼ 14.000 (!) P(w)» w-w w¼ 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 • Y2(k) » k-, ' 0.4 ) Traffic jams !

41. Transportation networks: Summary

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)/ si

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 -g ; P(s)~s-g 2 <  < 3: • Strong coupling ! 2 • Weak coupling ! 3

47. Analytical results • Power law distributions for w: • P(w)» w- • Correlations topology/weights: si ' (2+1)ki  <w> ki

48. Nonlinear correlations ? • Correlations topology/weights: si ' (2+1)ki  <w> ki) = 1 )How can we obtain   1 ? • Inclusion of space • …

49. Nonlinear correlations ? • Growing network: addition of nodes + distance • Proba(n! i)/ sif(dni) With: f(d)» e-d/d0 • d0/LÀ 1 ) = 1 • d0/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 ?