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Weighted networks: analysis, modeling A. Barrat, LPT, Université Paris-Sud, France

Weighted networks: analysis, modeling A. Barrat, LPT, Université Paris-Sud, France. M. Barthélemy (CEA, France) R. Pastor-Satorras (Barcelona, Spain) A. Vespignani (LPT, France). cond-mat/0311416 PNAS 101 (2004) 3747 cond-mat/0401057 PRL to appear (2004). Plan of the talk.

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Weighted networks: analysis, modeling A. Barrat, LPT, Université Paris-Sud, France

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  1. Weighted networks: analysis, modelingA. Barrat, LPT, Université Paris-Sud, France M. Barthélemy (CEA, France) R. Pastor-Satorras (Barcelona, Spain) A. Vespignani (LPT, France) cond-mat/0311416 PNAS 101 (2004) 3747 cond-mat/0401057 PRL to appear (2004)

  2. Plan of the talk • Complex networks: examples, topology • Topological correlations • The BA model • Weighted networks: examples, analysis • Weighted correlations • A model for weighted networks • Perspectives

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

  4. Airplane route network

  5. CAIDA AS cross section map

  6. Small-world properties Distribution of chemical distances Between two nodes « Six degrees of separation », Milgram 1967 (context: social networks)

  7. Usual random graphs: Erdös-Renyi model (1960) N points, links with proba p: static random graphs Connectivity distribution P(k) = probability that a node has k links BUT...

  8. Main features of complex networks • Many interacting units • Dynamical evolution • Self-organization • Small-world • and...

  9. 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

  10. What does it mean? Poisson distribution Power-law distribution Exponential Network Scale-free Network Strong consequences on the dynamics on the network: • Propagation of epidemics • Robustness • Resilience • ...

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

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

  13. 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

  14. How to generate scale-free graphs: the BA model (Barabàsi and Albert, 1999) Growth : at each time step a new node is added with m links to be connected with previous nodes Preferential attachment: The probability that a new link is connected to a given node is proportional to the number of node’s links. The preferential attachment follows the probability distribution : The generated connectivity distribution is P(k) ~ k -3

  15. Connectivity distribution BA network

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

  17. Weighted networks: examples • Scientific collaborations* • Internet • Emails • Airports' network** • Finance, economic networks • ... *:thanks M. Newman ; **: IATA

  18. Weights • 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

  19. Weighted networks: data • Scientific collaborations: cond-mat archive; N=12722 authors, 39967 links • Airports' network: data by IATA; N=3863 connected airports, 18807 links

  20. Global data analysis Number of authors 12722 Maximum coordination number 97 Average coordination number 6.28 Maximum weight 21.33 Average weight 0.57 Clustering coefficient 0.65 Pearson coefficient (assortativity) 0.16 Average shortest path 6.83 Number of airports 3863 Maximum coordination number 318 Average coordination number 9.74 Maximum weight 6167177. Average weight 74509. Clustering coefficient 0.53 Pearson coefficient 0.07 Average shortest path 4.37

  21. Data analysis: P(k), P(s) Generalization of ki: strength Broad distributions

  22. Correlations topology/traffic Strength vs. Coordination S(k) proportional to k N=12722 Largest k: 97 Largest s: 91

  23. Correlations topology/traffic Strength vs. Coordination S(k) proportional to k=1.5 Randomized weights: =1 N=3863 Largest k: 318 Largest strength: 54 123 800 Correlations between topology and dynamics

  24. Some new definitions: weighted quantities • Weighted clustering coefficient • Weighted assortativity

  25. wij=1 wij=5 Clustering vs. weighted clustering coefficient i i si=8 ciw=0.25 < ci si=16 ciw=0.625 > ci ki=4 ci=0.5

  26. Clustering vs. weighted clustering coefficient k (wjk) wik j i wij Random(ized) weights: C = Cw C < Cw : more weights on cliques C > Cw : less weights on cliques

  27. Clustering and weighted clustering Scientific collaborations: C= 0.65, Cw ~ C C(k) ~ Cw(k) at small k, C(k) < Cw(k) at large k: larger weights on large cliques

  28. Clustering and weighted clustering Airports' network: C= 0.53, Cw=1.1 C C(k) < Cw(k): larger weights on cliques at all scales

  29. 1 5 1 5 5 1 1 5 1 5 Assortativity vs. weighted assortativity i ki=5; knn,i=1.8

  30. 5 5 1 5 5 Assortativity vs. weighted assortativity i ki=5; si=21; knn,i=1.8 ; knn,iw=1.2

  31. 1 1 5 1 1 Assortativity vs. weighted assortativity i ki=5; si=9; knn,i=1.8 ; knn,iw=3.2

  32. Assortativity and weighted assortativity Airports' network knn(k) < knnw(k): larger weights between large nodes

  33. Non-weighted vs. Weighted: Comparison of knn(k) and knnw(k), of C(k) and Cw(k) Informations on the correlations between topology and dynamics

  34. A new model: growing weighted network • Growth: at each time step a new node is added with m links to be connected with previous nodes • Preferential attachment: the probability that a new link is connected to a given node is proportional to the node’s strength The preferential attachment follows the probability distribution : Preferential attachment driven by weights AND...

  35. Redistribution of weights New node: n, attached to i New weight wni=w0=1 Weights between i and its other neighbours: Only parameter si si + w0 + d The new traffic n-i increases the traffic i-j

  36. Evolution equations (mean-field) Also: evolution of weights

  37. Analytical results Power law distributions for k, s and w: P(k) ~ k -g ; P(s)~s-g Correlations topology/weights: wij~ min(ki,kj)a

  38. Numerical results

  39. Numerical results: P(w), P(s)

  40. Numerical results: weights wij~ min(ki,kj)a

  41. Perspectives/ work in progress • Extensions of the model: • fitnesses di ; di depending on ki or si • spatial network • More detailed study of new weighted quantities • Effect of weights on dynamical properties: resilience to damage, propagation of epidemics...

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