A' Barrat, M' Barthelemy,

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The architecture of complex weighted networks

• A. Barrat, M. Barthelemy,
• R. Pastor-Satorras, and A. Vespignani

Çaglayan Dicle
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Main Issue

• Correlate the model with topology
• Past studies
• Links are represented as binary states
  (present/absent)
• This study
• Links are represented with weights proportional
  to intensity of capacity of the connection.

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Many Networks

• Internet
• WWW
• Transport networks
• Power grids
• Protein interaction networks
• Food webs
• Metabolic networks
• Social networks

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Many are small-world

• the network has an average topological distance
  between the various nodes increasing very slowly
  with the number of nodes (logarithmically or even
  slower)

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Many have scale-free properties
P(k) probability that a node has k links
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• OK
• Threat analysis
• Policy decisions
• BUT
• The Internet and the World-Wide-Web
• Protein networks
• Metabolic networks
• Social networks
• Food-webs and ecological networks
• !!! Are HETEROGENOUS Networks

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Weighted Networks Data

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

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WAN
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SCN
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• Generalization of ki strength

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P(s), P(k) similaritySCN case
S(k) proportional to k Proof?
N12722 Largest k 97 Largest s 91
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P(s), P(k) similarityWAN case
S(k) proportional to k????1.5 Randomized
weights ?1
N3863 Largest k 318 Largest strength 54 123 800
!!!Strong correlations between topology and
dynamics
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Correlations Weights vs. Coordination
!!!Lack of correlation between weightsvertex
degrees
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Clustering vs. weighted clustering coefficient
si16 ciw0.625 gt ci
ki4 ci0.5
si8 ciw0.25 lt ci
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Clustering vs. weighted clustering coefficient
What does figure imply?
C(k) lt Cw(k) clustering in large weighted edges,
major affect on network C(k) gt Cw(k) clustering
in low weighted edges, minor affect on network
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Assortativity vs. weighted assortativity
ki5 knn,i1.8
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Assortativity vs. weighted assortativity
ki5 si21 knn,i1.8 knn,iw1.2 knn,i gt
knn,iw
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Assortativity vs. weighted assortativity(WAN)
knn(k) lt knnw(k) larger weights towards large
nodes
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Assortativity vs. weighted assortativity(SCN)
knn(k) lt knnw(k) larger weights between large
nodes
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Conclussion

• Weighted quantities bring a complementary
  perspective on the structural organization of the
  networks.