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 to appear (2004)

● Complex networks: examples, topology ● Topological correlations ● The BA model ● Weighted networks: examples, analysis ● 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 ● Social networks ●...

Airplane route network

CAIDA AS cross section map

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

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

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

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

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

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 -  How to generate scale-free graphs: the BA model (Barabàsi and Albert, 1999)

BA network Connectivity distribution

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  Rewiring Highly clustered (Eguiluz & Klemm 2002) Fitness Model (Bianconi et al. 2001) Multiplicative noise (Huberman & Adamic 1999)

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

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

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: Correlations between topology and dynamics Correlations topology/traffic Strength vs. Coordination

Some new definitions: weighted quantities ● 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 = i

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

Assortativity and weighted assortativity Airports' network 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 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...

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

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

Numerical results

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

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

Perspectives/ work in progress ●Extensions of the model: ●fitnesses  i ;  i depending on k i or s i ●spatial network ●More detailed study of new weighted quantities ●Effect of weights on dynamical properties: resilience to damage, propagation of epidemics...