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Weighted networks: analysis, modeling A. Barrat, LPT, UniversitÃ© Paris-Sud, France. M. BarthÃ©lemy (CEA, France) R. Pastor-Satorras (Barcelona, Spain) A. Vespignani (LPT, France). cond-mat/0311416 PNAS 101 (2004) 3747 cond-mat/0401057 PRL 92 (2004) 228701
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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 92 (2004) 228701 cs.NI/0405070 LNCS 3243 (2004) 56 cond-mat/0406238 PRE 70 (2004) 066149 physics/0504029
Plan of the talk • Complex networks: • examples, models, topological correlations • Weighted networks: • examples, empirical analysis • new metrics: weighted correlations • models of weighted networks • Perspectives
Examples of complex networks • Internet • WWW • Transport networks • Power grids • Protein interaction networks • Food webs • Metabolic networks • Social networks • ...
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...
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
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 • ...
Topological correlations: clustering aij: Adjacency matrix ki=5 ci=0.1 ki=5 ci=0. i
k=4 k=4 i k=3 k=7 Topological correlations: assortativity ki=4 knn,i=(3+4+4+7)/4=4.5
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
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) ~hiki • Inclusion of space P(k) ~ k-g (....) => many available models Redner et al. 2000, Mendes et al. 2000, Albert et al. 2000, Dorogovtsev et al. 2001, Bianconi et al. 2001, Barthélemy 2003, etc...
Beyond topology: Weighted networks • Internet • Emails • Social networks • Finance, economic networks (Garlaschelli et al. 2003) • Metabolic networks (Almaas et al. 2004) • Scientific collaborations (Newman 2001) : SCN • World-wide Airports' network*: WAN • ... are weighted heterogeneous networks, with broad distributions of weights *: data from IATA www.iata.org
Weights • Scientific collaborations: (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 • Metabolic networks: fluxes • Financial networks: shares
Weighted networks: data • Scientific collaborations: cond-mat archive; N=12722 authors, 39967 links • Airports' network: data by IATA; N=3863 connected airports, 18807 links
Data analysis: P(k), P(s) Generalization of ki: strength Broad distributions
Correlations topology/traffic Strength vs. Coordination S(k) proportional to k N=12722 Largest k: 97 Largest s: 91
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 Strong correlations between topology and dynamics
Correlations topology/traffic Weights vs. Coordination wij ~ (kikj)q ; si = Swij ; s(k) ~ kb WAN: no degree correlations => b = 1 + q SCN: q~0 => b=1 See also Macdonald et al., cond-mat/0405688
Some new definitions: weighted metrics • Weighted clustering coefficient • Weighted assortativity • Disparity
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
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
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
Clustering and weighted clustering Airports' network: C= 0.53, Cw=1.1 C C(k) < Cw(k): larger weights on cliques at all scales, especially for the hubs
Another definition for theweighted clustering J.-P. Onnela, J. Saramäki, J. Kertész, K. Kaski, cond-mat/0408629 uses a global normalization and the weights of the three edges of the triangle, while: uses a local normalization and focuses on node i
1 5 1 5 5 1 1 5 1 5 Assortativity vs. weighted assortativity i ki=5; knn,i=1.8
5 5 1 5 5 Assortativity vs. weighted assortativity i ki=5; si=21; knn,i=1.8 ; knn,iw=1.2: knn,i > knn,iw
1 1 5 1 1 Assortativity vs. weighted assortativity i ki=5; si=9; knn,i=1.8 ; knn,iw=3.2: knn,i < knn,iw
Assortativity and weighted assortativity Airports' network knn(k) < knnw(k): larger weights towards large nodes
Assortativity and weighted assortativity Scientific collaborations knn(k) < knnw(k): larger weights between large nodes
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
Disparity weights of the same order => y2» 1/ki small number of dominant edges => y2» O(1) identification of localheterogeneities between weighted links, existence of dominant pathways...
Models of weighted networks:static weights S.H. Yook et al., P.R.L. 86, 5835 (2001); Zheng et al. P.R.E 67, 040102 (2003): • growing network with preferential attachment • weights driven by nodes degree • static weights More recently, studies of weighted models: W. Jezewski, Physica A 337, 336 (2004); K. Park et al., P. R. E 70, 026109 (2004); E. Almaas et al, P.R.E 71, 036124 (2005); T. Antal and P.L. Krapivsky, P.R.E 71, 026103 (2005) in all cases: no dynamical evolution of weights nor feedback mechanism between topology and weights
A new (simple) mechanism for growing weighted networks • 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:feedback mechanism New node: n, attached to i New weight wni=w0=1 Weights between i and its other neighbours: n i j sisi + w0 + d Only parameter
Redistribution of weights:feedback mechanism n i j The new traffic n-i increases the traffic i-j and the strength/attractivity of i => feedback mechanism “Busy gets busier”
Evolution equations (mean-field) • si changes because • a new node connects to i • a new node connects to a neighbour j of i
Evolution equations (mean-field) • changes because • a new node connects to i • a new node connects to j
Evolution equations (mean-field) • m new links • global increase of strengths: 2m(1+) each new node:
Analytical results power law growth of si (i introduced at time ti=i) Correlations topology/weights:
Analytical results:Probability distributions ti uniform 2 [1;t] P(s) ds » s- ds = 1+1/a
Analytical results:degree, strength, weight distributions Power law distributions for k, s and w: P(k) ~ k -g ; P(s)~s-g
Numerical results: P(w), P(s) (N=105)
Numerical results: weights wij~ min(ki,kj)a
Numerical results: assortativity disassortative behaviour typical of growing networks analytics: knn/ k-3 (Barrat and Pastor-Satorras, Phys. Rev. E 71 (2005) )
Numerical results: assortativity Weighted knnw much larger than knn : larger weights contribute to the links towards vertices with larger degree
Disassortativity during the construction of the network: new nodes attach to nodes with large strength =>hierarchy among the nodes: -new vertices have small k and large degree neighbours -old vertices have large k and many small k neighbours reinforcement: edges between “old” nodes get reinforced =>larger knnw , especially at large k
Numerical results: clustering • d increases => clustering increases • clustering hierarchy emerges • analytics: C(k) proportional to k-3 • (Barrat and Pastor-Satorras, Phys. Rev. E 71 (2005) )
Numerical results: clustering Weighted clustering much larger than unweighted one, especially at large degrees