Analysis of some weighted networks A' Barrat, LPT, Universit ParisSud, France

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Analysis of some weighted networksA. Barrat,
LPT, Université Paris-Sud, France
M. Barthélémy (CEA, France) R. Pastor-Satorras
(Barcelona, Spain) A. Vespignani (LPT, France)
Thanks L. A. N. Amaral, R. Guimerà
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Plan of the talk

• Examples of weighted networks
• Definitions Topological and Weighted quantities
• Data analysis
• Correlations topology/traffic
• Some new definitions
• Perspectives

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A model growing weighted network
S.H. Yook, H. Jeong, A.-L. Barabàsi, Y. Tu P.R.L.
86, 5835 (2001)

• Growing networks with preferential attachment
• Weights on links, driven by network connectivity

• Peaked probability distribution for the weights
• Same universality class as unweighted network

Here analysis of real data
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Examples of weighted networks

• Scientific collaborations
• Actors' collaborations
• Internet
• Emails
• Airports' network
• Finance, economic networks

thanks M. Newman IMDB IATA
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Weighted networks data

• Scientific collaborations cond-mat archive
  N12722 authors, 39967 links
• Actors' collaborations IMDB, N374511 actors
• Airports' network data by IATA N3863 connected
  airports, 18807 links

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Topological quantities

• aij Adjacency matrix
• ki Coordination number of node i P(k)
• Clustering coefficient

ki5 ci0.
i
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Topological quantities

• aij Adjacency matrix
• ki Coordination number of node i P(k)
• Clustering coefficient

ki5 ci0.1
i
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Topological quantities
Assortativity
ki4 knn,i(3447)/44.5
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Topological quantities

• Distances dij
• Centrality
• Efficiency
• Betwenness centrality bci

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Weights (Edge traffic)

• Scientific or actors' collaborations

(M. Newman, P.R.E. 2001)
i, j authors/actors k paper/film nk number
of actors/authors 1 if author/actor i has
contributed to paper/film k

• Internet, emails traffic, number of exchanged
  emails

• Airports number of passengers for the year 2002

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Weighted quantities

• Weights, Edge traffic, P(w)
• Load or Vertex traffic,
  P(L), L(k) (e.g. number
  of papers/films by a given author/actor)
• Weighted distances
• Weighted centrality
• Weighted efficiency

e.g. on a link ij,
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Global data analysis
Number of authors 12722 Maximum
coordination number 97 Average coordination
number 6.28 Maximum weight 21.33 Average
weight 0.57 Clustering coefficient 0.65
Pearson coefficient (assortativity) 0.16
Average shortest path 6.83
Number of actors374511 Maximum coordination
number 3956 Average coordination number
80.2 Maximum weight 37.7 Average weight
0.048 Clustering coefficient 0.67 Pearson
coefficient 0.226
Number of airports 3863 Maximum coordination
number 318 Average coordination number
9.74 Maximum weight 6167177. Average weight
74509. Clustering coefficient 0.53 Pearson
coefficient 0.07 Average shortest path 4.37
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Data analysis P(k), P(L)
knumber of collaborators Lnumber of
papers/films
Airports broad P(k), cf. Guimerà et al. 2003
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Correlations topology/traffic

• Vertex traffic vs. Coordination number
• Vertex traffic vs. Betweenness centrality
• Coordination number vs. Betweenness centrality
• ...

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Vertex traffic vs. Coordination number
L(k) proportional to k
N12722 Largest k 97 Largest L 91
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Vertex traffic vs. Coordination number
L(k) proportional to kb, b1.2
N374511 Largest k 3956 Largest load 645
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Vertex traffic vs. Coordination number
L(k) proportional to kb, b1.5 Randomized
weights b1
N3863 Largest k 318 Largest Load 54 123 800
Correlations between topology and dynamics
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Some new definitions weighted quantities

• Weighted betweenness centrality bci Li
• Weighted assortativity
• Weighted clustering coefficient

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Weighted clustering coefficient
N.B. other definitions are possible, e.g. wjk
(wijwikwjk)/3.
wik
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Clustering vs. weighted clustering coefficient
ci0.1
(e.g. with ltwgt1)
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Clustering vs. weighted clustering coefficient
0.2
ciw0.3 gt ci
ciw0.02 lt ci
ci0.1
(e.g. with ltwgt1)
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Clustering vs. weighted clustering coefficient
k
(wjk)
wik
j
i
wij
Random(ized) weights C Cw C lt Cw more
weights on cliques C gt Cw less weights on
cliques
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Clustering and weighted clustering
Scientific collaborations C 0.65, Cw1.01
C(k) lt Cw(k) at small k, C(k) gt Cw(k) at large k
larger weights on small cliques
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Clustering and weighted clustering
Actor's network C 0.67, Cw 1.44
C(k) lt Cw(k) especially at small k larger
weights on small cliques C(k) close to Cw(k) at
larger k
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Clustering and weighted clustering
Airports' network C 0.53, Cw1.53
C(k) lt Cw(k) larger weights on cliques at all
scales
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Assortativity vs. weighted assortativity
i
k2
k3
ki2, knn,i2.5
ki4 knn,i4.5
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Assortativity vs. weighted assortativity
e.g. ltwgt1
0.5
i
k2
1
2
3
k3
1
10
ki2, knn,iw3.5
ki4 knn,iw22.25
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Assortativity vs. weighted assortativity
e.g. ltwgt1
3
i
k2
1
10
0.5
k3
1
2
ki2, knn,iw18
ki4 knn,iw5.75
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Assortativity vs. weighted assortativity
Comparison of knn(k) and knnw(k)
Informations on the correlations between topology
and dynamics
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Perspectives/ work in progress

• Analysis of other networks internet, emails ?
• Weights vs. connectivities
• More detailed study of new weighted quantities
• Resilience to damage

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Data analysis P(k), P(L)
kcoordination L vertex traffic broad
distributions
cf. also Guimerà et al. 2003
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Note on the exponents
If P(k) and P(L) are power-laws, P(k) dk a k-a
dk P(L) dL a L-g dL and L a kb then b
(g - 1) a - 1
NB -true for the actors' network and for the
airports' network -for randomized weights
b1 and one finds indeed ga
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Betweenness vs. Vertex Traffic

• Correlations
• Large fluctuations

(Red 0.1BC lt bci lt 10 BC)
(Similar plot for the airports' network)
(Scientific collaborations loadnumber of papers)
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Betwenness centrality
Number of shortest paths from j to k
Number of shortest paths from j to k passing
through i
(traffic)
(topology)
larger fluctuations at small values
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How to characterize the centrality?
Coordination number Betwenness centrality Centrali
ty/efficiency Weighted centrality/efficiency Verte
x traffic
All these quantities are statistically correlated
but can give very different rankings !
Cf also www.santafe.edu/mark/collaboration/ M.
Newman, P.R.E. 2001Amaral et al., preprint 2003
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Example various rankings of airports
Coordination
Vertex traffic
BC
Weighted centrality
1 Frankfurt 2 Paris-CDG 3
Munich 4 Amsterdam 5 Atlanta 6
London-Heathrow 7 Chicago (O'Hare) 8
Duesseldorf 9 Vienna 10 Brussels 11 Dallas
Ft. Worth 12 Houston
Atlanta Chicago (O'Hare) London-Heathrow
Tokyo-Haneda Los Angeles Dallas Ft. Worth
Paris-CDG Frankfurt Phoenix Denver Hong
Kong Detroit
Frankfurt Paris-CDG Anchorage Tokyo-Narita Port
Moresby London-Heathrow Los Angeles Singapore Vanc
ouver Bangkok Johannesburg Toronto
London-Heathrow Los Angeles Chicago (O'Hare) New
York(Kennedy) Paris-CDG Tokyo-Narita Singapore San
Francisco Hong Kong Las Vegas Dublin Frankfurt

(cf also Guimerà et al., preprint 2003)