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/ PNAS 101 (2004) 3747 cond-mat/ PRL 92 (2004) cs.NI/

● Complex networks: examples, models, topological correlations ● Weighted networks: ● examples, empirical analysis ● new metrics: weighted correlations ● a model for weighted networks ● Perspectives Plan of the talk

Examples of complex networks ● Internet ● WWW ● Transport networks ● Power grids ● Protein interaction networks ● Food webs ● Metabolic networks ● Social networks ●...

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

Airplane route network

CAIDA AS cross section map

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

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 - 

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

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

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

Beyond topology: Weighted networks ● Internet ● s ● Social networks ● Finance, economic networks (Garlaschelli et al. 2003) ● Metabolic networks (Almaas et al. 2004) ● Scientific collaborations (Newman 2001) ● Airports' network* ●... *: data from IATA are weighted heterogeneous networks, with broad distributions of weights

Weights ● Scientific collaborations: i, j: authors; k: paper; n k : number of authors  : 1 if author i has contributed to paper k (Newman, P.R.E. 2001) ● Internet, s: traffic, number of exchanged s ● Airports: number of passengers for the year 2002 ● Metabolic networks: fluxes ● Financial networks: shares

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

Global data analysis Number of authors Maximum coordination number 97 Average coordination number 6.28 Maximum weight 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 Average weight Clustering coefficient 0.53 Pearson coefficient 0.07 Average shortest path 4.37

Data analysis: P(k), P(s) Generalization of k i : strength Broad distributions

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

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

Correlations topology/traffic Weights vs. Coordination See also Macdonald et al., cond-mat/ w ij ~ (k i k j )   s i =  w ij ; s(k) ~ k  WAN: no degree correlations =>  = 1 +  SCN: 

Some new definitions: weighted metrics ● Weighted clustering coefficient ● Weighted assortativity

Clustering vs. weighted clustering coefficient s i =16 c i w =0.625 > c i k i =4 c i =0.5 s i =8 c i w =0.25 < c i w ij =1 w ij =5 i i

Clustering vs. weighted clustering coefficient Random(ized) weights: C = C w C < C w : more weights on cliques C > C w : less weights on cliques i j k (w jk ) w ij w ik

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

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

Assortativity vs. weighted assortativity k i =5; k nn,i = i

Assortativity vs. weighted assortativity k i =5; s i =21; k nn,i =1.8 ; k nn,i w =1.2: k nn,i > k nn,i w i

Assortativity vs. weighted assortativity k i =5; s i =9; k nn,i =1.8 ; k nn,i w =3.2: k nn,i < k nn,i w i

Assortativity and weighted assortativity Airports' network k nn (k) < k nn w (k): larger weights between large nodes

Assortativity and weighted assortativity Scientific collaborations k nn (k) < k nn w (k): larger weights between large nodes

Non-weighted vs. Weighted: Comparison of k nn (k) and k nn w (k), of C(k) and C w (k) Informations on the correlations between topology and dynamics

A model of growing weighted network S.H. Yook, H. Jeong, A.-L. Barabási, Y. Tu, P.R.L. 86, 5835 (2001) ● Peaked probability distribution for the weights ● Same universality class as unweighted network ● Growing networks with preferential attachment ● Weights on links, driven by network connectivity ● Static weights See also Zheng et al. Phys. Rev. E (2003)

A new model of 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...

Redistribution of weights New node: n, attached to i New weight w ni =w 0 =1 Weights between i and its other neighbours: s i s i + w 0 +  The new traffic n-i increases the traffic i-j Only parameter ni j

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

Analytical results Power law distributions for k, s and w: P(k) ~ k  ; P(s)~s  Correlations topology/weights: w ij ~ min(k i,k j ) a, a=2  /(2  +1) power law growth of s k proportional to s

Numerical results

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

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

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

Numerical results: assortativity

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

Numerical results: clustering

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

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

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

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

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 )

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

Summary/ Perspectives/ Work in progress Empirical analysis of weighted networks  weights heterogeneities  correlations weights/topology  new metrics to quantify these correlations New model of growing network which couples topology and weights  analytical+numerical study  broad distributions of weights, strengths, connectivities  extensions of the model  randomness, non linearities  spatial network: work in progress  other ? Influence of weights on the dynamics on the networks: work in progress