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/ LNCS 3243 (2004) 56 cond-mat/ PRE 70 (2004) physics/

● Complex networks: examples, models, topological correlations ● Weighted networks: ● examples, empirical analysis ● new metrics: weighted correlations ● models of 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

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

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 - 

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) : SCN ● World-wide Airports' network*: WAN ●... *: 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 ● 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

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 ● Disparity

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, especially for the hubs

Another definition for the weighted clustering J.-P. Onnela, J. Saramäki, J. Kertész, K. Kaski, cond-mat/ uses a global normalization and the weights of the three edges of the triangle, while: uses a local normalization and focuses on node i

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 towards 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

Disparity weights of the same order => y 2 » 1/k i small number of dominant edges => y 2 » O(1) identification of local heterogeneities 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, (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, (2004); E. Almaas et al, P.R.E 71, (2005); T. Antal and P.L. Krapivsky, P.R.E 71, (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 w ni =w 0 =1 Weights between i and its other neighbours: s i s i + w 0 +  Only parameter ni j

Redistribution of weights: feedback mechanism The new traffic n-i increases the traffic i-j and the strength/attractivity of i => feedback mechanism ni j “Busy gets busier”

Evolution equations (mean-field) s i 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 Correlations topology/weights: power law growth of s i (i introduced at time t i =i)

Analytical results: Probability distributions t i 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  ; P(s)~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 disassortative behaviour typical of growing networks analytics: k nn / k -3 (Barrat and Pastor-Satorras, Phys. Rev. E 71 (2005) )

Numerical results: assortativity Weighted k nn w much larger than k nn : 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 k nn w, especially at large k

Numerical results: clustering  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

Clustering ● as  increases: larger probability to build triangles, with typically one new node and 2 old nodes => larger increase at small k ● new nodes: small weights so that c w and c are close ● old nodes: strong weights so that triangles are more important

Extensions of the model: i.heterogeneities ii.non-linearities iii.directed model iv.other similar mechanisms

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:  i increases with s i ; saturation effect at s 0

Extensions of the model: (ii)-non-linearities s prop. to k  with  > 1 N=5000 s 0 =10 4  Broad P(s) and P(k) with different exponents

Extensions of the model: (iii)-directed network i j l nodes i; directed links

Extensions of the model: (iii)- directed network n i j (i) Growth (ii) Strength driven preferential attachment (n: k out =m outlinks) AND... “Busy gets busier”

Weights reinforcement mechanism i j n The new traffic n-i increases the traffic i-j “Busy gets busier”

Evolution equations (Continuous approximation) Coupling term

Resolution Ansatz supported by numerics:

Results

Approximation Total in-weight  i s in i : approximately proportional to the total number of in-links  i k in i, times average weight h w i = 1+ Then: A=1+  s in 2 [2;2+1/m]

Measure of A prediction of  Numerical simulations Approx of 

Numerical simulations NB: broad P(s out ) even if k out =m

Clustering spectrum  increases => clustering increases New pages: point to various well-known pages, often connected together => large clustering for small nodes Old, popular nodes with large k: many in-links from many less popular nodes which are not connected together => smaller clustering for large nodes

Clustering and weighted clustering Weighted Clustering larger than topological clustering: triangles carry a large part of the traffic

Assortativity Average connectivity of nearest neighbours of i

Assortativity k nn : disassortative behaviour, as usual in growing networks models, and typical in technological networks lack of correlations in popularity as measured by the in-degree

S.N. Dorogovtsev and J.F.F. Mendes “Minimal models of weighted scale-free networks ” cond-mat/ (i)choose at random a weighted edge i-j, with probability / w ij (ii) reinforcement w ij ! w ij +  (iii) attach a new node to the extremities of i-j  broad P(s), P(k), P(w)  large clustering  linear correlations between s and k “BUSY GETS BUSIER”

G. Bianconi “Emergence of weight-topology correlations in complex scale-free networks ” cond-mat/ (i) new nodes use preferential attachment driven by connectivity to establish m links (ii) random selection of m’ weighted edges i-j, with probability / w ij (iii) reinforcement of these edges w ij ! w ij +w 0 =>broad distributions of k,s,w =>non-linear correlations s / k   > 1 iff m’ > m “BUSY GETS BUSIER”

Summary/ Perspectives Empirical analysis of weighted networks  weights heterogeneities  correlations weights/topology  new metrics to quantify these correlations New mechanism for growing network which couples topology and weights  broad distributions of weights, strengths, connectivities  extensions of the model  randomness, non linearities, directed network  spatial network: physics/ Perspectives: Influence of weights on the dynamics on the networks

COevolution and Self-organization In dynamical Networks

R. Albert, A.-L. Barabási, “Statistical mechanics of complex networks”, Review of Modern Physics 74 (2002) 47. S.N. Dorogovtsev, J.F.F. Mendes, “Evolution of networks”, Advances in Physics 51 (2002) S.N. Dorogovtsev, J.F.F. Mendes, “Evolution of networks: From biological nets to the Internet and WWW”, Oxford University Press, Oxford, 2003 R. Pastor-Satorras, A. Vespignani, “Evolution and structure of the Internet: A statistical physics approach”, Cambridge University Press, Cambridge, other books/reviews to appear soon.... Some useful reviews/books