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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) 3747cond-mat/0401057 PRL 92 (2004) 228701cs.NI/0405070
●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)
• <k>= const• <k2>
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 networksFitness model: (k) ~ iki
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
ki=5ci=0.ki=5ci=0.1
aij: Adjacency matrix
Topological correlations: assortativity
ki=4knn,i=(3+4+4+7)/4=4.5
i
k=3k=7
k=4k=4
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
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)● Airports' network*● ...
*: data from IATA www.iata.org
are weighted heterogeneous networks,
with broad distributions of weights
Weights
● Scientific collaborations:
i, j: authors; k: paper; nk: number of authors
: 1 if author i has contributed to paper k
(Newman, P.R.E. 2001)
●Internet, emails: traffic, number of exchanged emails●Airports: number of passengers for the year 2002●Metabolic networks: fluxes●Financial networks: shares
●Airports' network: data by IATA; N=3863 connected airports, 18807 links
Global data analysis
Number of authors 12722 Maximum coordination number 97Average coordination number 6.28Maximum weight 21.33Average weight 0.57 Clustering coefficient 0.65 Pearson coefficient (assortativity) 0.16 Average shortest path 6.83
Number of airports 3863Maximum coordination number 318Average coordination number 9.74Maximum weight 6167177.Average weight 74509.Clustering coefficient 0.53Pearson coefficient 0.07Average shortest path 4.37
Data analysis: P(k), P(s)
Generalization of ki: strength
Broad distributions
Correlations topology/traffic Strength vs. Coordination
S(k) proportional to k
N=12722Largest k: 97Largest s: 91
S(k) proportional to k=1.5
Randomized weights: =1
N=3863Largest k: 318Largest strength: 54 123 800
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/0405688
wij ~ (kikj)si = wij ; s(k) ~ k
WAN: no degree correlations => = 1 + SCN:
Some new definitions: weighted metrics
● Weighted clustering coefficient
● Weighted assortativity
Clustering vs. weighted clustering coefficient
si=16ci
w=0.625 > ci
ki=4ci=0.5
si=8ci
w=0.25 < ci
wij=1
wij=5
i i
Clustering vs. weighted clustering coefficient
Random(ized) weights: C = Cw
C < Cw : more weights on cliques
C > Cw : less weights on cliques
ij
k(wjk)
wij
wik
Clustering and weighted clusteringScientific 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
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• si(t) grows as ta(
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
n i
j
New node: n, attached to iNew weight wni=w0=1Weights between i and its other neighbours:
Examplewij = (wij/si)(s0 tanh(si/s0))a
i increases with si; saturation effect at s0
wij = f(wij,si,ki)
Extensions of the model: (ii)-non-linearities
s prop. to k with > 1
N=5000s0=104
wij = (wij/si)(s0 tanh(si/s0))a
Broad P(s) and P(k) with different exponents
Summary/ Perspectives/ Work in progress
•Empirical analysis of weighted networksweights heterogeneitiescorrelations weights/topologynew metrics to quantify these correlations
•New model of growing network which couples topology and weightsanalytical+numerical studybroad distributions of weights, strengths, connectivitiesextensions of the model
randomness, non linearitiesspatial network: work in progressother ?
•Influence of weights on the dynamics on the networks: work in progress