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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 LNCS 3243 (2004) 56cond-mat/0406238 PRE 70 (2004) 066149physics/0504029
disassortative behaviour typical of growing networksanalytics: 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
• 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 cw and c are close
● old nodes: strong weights so that triangles are more important
Extensions of the model:
i. heterogeneitiesii. non-linearitiesiii. directed modeliv. 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• 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:
i increases with si; saturation effect at s0
Extensions of the model: (ii)-non-linearities
s prop. to k with > 1
N=5000s0=104
Broad P(s) and P(k) with different exponents
Extensions of the model: (iii)-directed network
i
jl nodes i; directed links
Extensions of the model: (iii)- directed network
n i
j (i) Growth
(ii) Strength driven preferential attachment (n: kout=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 sini : approximately proportional to the
total number of in-links i kini , times average weight hwi = 1+
Then: A=1+
sin 2 [2;2+1/m]
Measure of A
prediction of
Numerical simulations
Approx of
Numerical simulations
NB: broad P(sout) even if kout=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
•knn: disassortative behaviour, as usual in growing networksmodels, 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/0408343
(i) choose at random a weighted edge i-j, with probability / wij
(ii) reinforcement wij ! wij + (iii) attach a new node to the extremities of i-j
broad P(s), P(k), P(w)large clusteringlinear correlations between s and k
“BUSY GETS BUSIER”
G. Bianconi“Emergence of weight-topology correlations
in complex scale-free networks ”cond-mat/0412399
(i) new nodes use preferential attachment driven byconnectivity to establish m links(ii) random selection of m’ weighted edges i-j, with probability / wij
(iii) reinforcement of these edges wij ! wij+w0
=>broad distributions of k,s,w=>non-linear correlations s / k > 1 iff m’ > m
“BUSY GETS BUSIER”
Summary/ Perspectives
•Empirical analysis of weighted networksweights heterogeneitiescorrelations weights/topologynew metrics to quantify these correlations
•New mechanism for growing network which couples topology and weightsbroad distributions of weights, strengths, connectivitiesextensions of the model
randomness, non linearities, directed networkspatial network: physics/0504029
Perspectives:
Influence of weights on the dynamics on the networks
COevolution and Self-organization In dynamical Networkshttp://www.cosin.org
http://delis.upb.de
http://www.th.u-psud.fr/page_perso/Barrat/
•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) 1079.
•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, 2003