weighted networks: analysis, modeling a. barrat, lpt, université paris-sud, france
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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) 3747 cond-mat/0401057 PRL 92 (2004) 228701 - PowerPoint PPT PresentationTRANSCRIPT
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
●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)
• <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
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
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
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 -
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● ...
*: 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●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=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
● Disparity
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 clustering
Scientific collaborations: C= 0.65, Cw
~ C
C(k) ~ Cw(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, 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
Assortativity vs. weighted assortativity
ki=5; knn,i=1.8
5
11
1
1
1
55
5
5i
Assortativity vs. weighted assortativity
ki=5; si=21; k
nn,i=1.8 ; knn,i
w=1.2: knn,i > knn,iw
1
55
5
5
i
Assortativity vs. weighted assortativity
ki=5; si=9; k
nn,i=1.8 ; knn,i
w=3.2: knn,i < knn,iw
511
1
1
i
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 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, 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 iNew weight wni=w0=1Weights between i and its other neighbours:
si si + w0 + Onlyparameter
n i
j
Redistribution of weights:feedback mechanism
The new traffic n-i increases the traffic i-j
and the strength/attractivity of i
=> feedback mechanism
n i
j
“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
Correlations topology/weights:
power law growth of si
(i introduced at time ti=i)
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 ; P(s)~s
Numerical results
Numerical results: P(w), P(s)
(N=105)
Numerical results: weights
wij ~ min(ki,kj)a
Numerical results: assortativity
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
+other books/reviews to appear soon....
Some useful reviews/books