Architecture of Complex Weighted Networks

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Architecture of Complex Weighted Networks
Marc Barthélemy CEA, France
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Collaborators

• A. Barrat (LPT-Orsay, France)
• R. Pastor-Satorras (Politechnica Univ. Catalunya)
• A. Vespignani (Indiana Univ., USA)
• A. Chessa (Univ. Cagliari, Italy)
• A. de Montis (Univ. Cagliary, Italy)

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Outline

• Weighted Complex networks
• Motivations
• Characterization Measurement tools

• II. Case-studies Transportation networks
• Inter-cities network Sardinia
• Global network World Airport Network

• III. Modeling
• Necessity of topology-traffic coupling Simple
  model

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Complex Networks

• Recent studies on topological properties showed

- broad distribution of connectivities -
impact on different processes (eg. Resilience,
epidemics)
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Beyond Topology Weighted Networks
w
ij
j
i
w
ji
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Beyond Topology Weighted Networks

• Internet, Web, Emails importance of traffic
• Ecosystems prey-predator interaction
• Airport network number of passengers
• Scientific collaboration number of papers
• Metabolic networks fluxes heterogeneous

Are - Weighted networks - With broad
distributions of weights
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Motivation
Why study weighted networks ?

• ) The weights can modify the behavior predicted
• by topology
• Resilience
• Epidemics

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Motivation Epidemics

• ) Epidemics spread on a contact network
• Social networks (STDs on sexual contact network)
• Transportation network (Airlines, railways,
  highways)
• WWW and Internet (e-viruses)

) The weights will affect the propagation of the
disease ) Immunization strategies ?
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Topological Characterization of Large Networks
All these networks are

• Complex

• Very large

• Statistical tools needed !
• Statistical mechanics of large networks

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Topological Characterization

• Diameter d logN) small-world
• d N1/D ) large world

• Clustering coeff. CÀ CRG 1/N
• C(k) k-? ) Hierarchy

• Assortativity knn versus k ?

• Betweenness centrality, modularity,

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Topological Characterization P(k)

• Connectivity k (kÀ 1 Hubs)
• Connectivity distribution P(k)
• probability that a node has k links
• Usual random graphs Erdös-Renyi model (1960)

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Classes of networks
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Weighted Networks
) New measurement tools needed !
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Weighted networks characterization
Generalization of ki strength

• For wijw0

• For wij and ki independent

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Weighted networks characterization

• In general

• If ? gt 1 or if ?1 and A?ltwgt

) Existence of strong correlations !
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Weighted networks characterization

• Weighted clustering coefficient

• If ciw/cigt1 Weights localized on clicques

• If ciw/cilt1 Important links dont form clicques

• If w and k uncorrelated ) ciwci

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Weighted networks characterization

• Weighted assortativity

• If knnw(i)/knn(i) gt1 Edges with larger weights
• point to nodes
  with larger k

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Weighted networks characterization
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Weighted networks characterization

• Disparity

• If Y2(i) 1/ki 1 No dominant connections
• If Y2(i)À 1/ki A few dominant connections

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Weighted networks characterization

• Disparity

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Case study Transportation networks
Different studies at different scales

• Intra-urban flows (Eubank et al, PRE 2003,
  Nature 2004)

• Inter-cities flows (with A. Chessa and A. de
  Montis)

• Global flows Word Airport network (PNAS, 2004)

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Airplane route network
Nodes airports Links direct flight
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Case study Global Air Travel
Number of airports 3863 18807 links
Topology Maximum coordination number 318 Average
coordination number 9.74 Average clustering
coefficient 0.53 Average shortest path 4.37
Weights Maximum weight 6167177 (seats/year,
2002) Average weight 74509
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Case study Airport network

• Broad distribution connectivity and weights

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Correlations topology-traffic Airports
s(k) proportional to k????1.5 (Randomized
weights sltwgtk ?1)
Strong correlations between topology and dynamics
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Correlations topology-traffic

• ltwijgt (kikj)?

?¼ 0.5
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Weighted clustering coefficient Airport
Cw(k) gt C(k) larger weights on cliques at all
scales
(esp. for large k)
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Weighted assortativity Airport
knn(k) lt knnw(k) larger weights between large
nodes
For large k ) Large traffic between hubs
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Disparity Airport
Y2(k) 1/k ) No dominant connection
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Airport Summary

• Topology Scale-free network
• Rich traffic structure
• Strong correlations traffic-topology

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Case study Inter-cities movements

• Sardinia
• - Italian island 24,000 km2
• - 1,600,000 inhabitants

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Case study Inter-cities movements

• Sardinian network
• Nodes 375 Cities
• Link wjiwij
• of individuals
• going from i
• to j (daily and by any means)

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Case study Inter-cities movements-Topology

• N375, E16,248 ) ltkgt43, kmax279

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Case study Inter-cities movements-Topology

• Clustering ltCgt¼ 0.26' CRG¼ 0.24

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Case study Inter-cities movements-Topology

• Slightly disassortative network

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Case study Inter-cities movements-Traffic

• ltwgt¼ 23, wmax¼ 14.000 (!)

P(w) w-?w ?w¼ 2.2
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Case study Inter-cities movements-Traffic

• Correlations s k?, ?' 1.9

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Case study Inter-cities movements-Traffic

• Weighted clustering Hubs form large w-clicques

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Case study Inter-cities movements-Traffic

• Weighted assortativity Large w between hubs

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Case study Inter-cities movements-Traffic

• Y2(k) k-?, ?' 0.4 ) Traffic jams !

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Transportation networks Summary
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Summary Weighted networks

• Broad strength distributions ) weights are
  relevant !
• (independently from topology)

• Topology-weight correlations important

• ) Model for networks with heterogeneous and
  correlated connectivities and weights ?

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Weighted networks Model

• Growing network addition of nodes
• Proba(n! i)/ si

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Weighted networks Model

• Rearrangement of weights

• ? 1 No effect (?0 BA model)

• ?À 1 Traffic stimulation

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Evolution equations (mean-field)
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Analytical results

• Power law distributions for k and s
• P(k) k -g P(s)s-g

2 lt ? lt 3

• Strong coupling ?! 2
• Weak coupling ?! 3

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Analytical results

• Power law distributions for w
• P(w) w-?

• Correlations topology/weights

si ' (2?1)ki ? ltwgt ki
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Nonlinear correlations ?
Correlations topology/weights
si ' (2?1)ki ? ltwgt ki) ? 1
) How can we obtain ? ? 1 ?

• Inclusion of space

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Nonlinear correlations ?

• Growing network addition of nodes distance
• Proba(n! i)/ si f(dni)

With f(d) e-d/d0

• d0/LÀ 1 ) ? 1

• d0/L 1 ) ? gt 1 !

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Summary Perspectives

• Weighted networks Complexity not only
  topological !
• Very rich traffic structure
• Correlations between weights and topology
• Model for weighted networks topology-traffic
  coupling (variants)

• Perspectives
• Effect of weights heterogeneity on dynamical
  processes (epidemics)
• Getting more data common features ?

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References

• A. Barrat, MB, R. Pastor-Satorras, A.
  Vespignani, PNAS 101, 3747 (2004)
• A. Barrat, MB, A. Vespignani, PRL 92, 228701
  (2004)
• A. Barrat, MB, A. Vespignani, LNCS 3243, 56
  (2004)
• A. Barrat, MB, A. Vespignani, PRE 70, 066149
  (2004)
• MB, A. Barrat, R. Pastor-Satorras, A.
  Vespignani, Physica A 346, 34 (2005)
• A.de Montis, MB, A. Chessa, A. Vespignani (in
  preparation)
• A. Barrat, MB, A. Vespignani (in preparation)

marc.barthelemy_at_th.u-psud.fr
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Numerical results clustering
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Numerical results assortativity
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Numerical results
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Numerical results P(w), P(s)
(N105)
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Numerical results weights
wij min(ki,kj)a
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Numerical results assortativity
analytics knn proportional to k(g-3)
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Numerical results clustering
analytics C(k) proportional to k(g-3)
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Extensions of the model (i)-heterogeneities

• Random redistribution parameter di (i.i.d. with
  r(d) )
• self-consistent analytical solution
• (in the spirit of the fitness model, cf. Bianconi
  and Barabási 2001)
• Results
• si(t) grows as ta(di)
• s and k proportional
• broad distributions of k and s
• same kind of correlations

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Extensions of the model (i)-heterogeneities
late-comers can grow faster
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Extensions of the model (i)-heterogeneities
Uniform distributions of d
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Extensions of the model (i)-heterogeneities
Uniform distributions of d
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Extensions of the model (ii)-non-linearities
New node n, attached to i New weight
wniw01 Weights between i and its other
neighbours
Dwij f(wij,si,ki)
Example Dwij d (wij/si)(s0 tanh(si/s0))a di
increases with si saturation effect at s0
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Extensions of the model (ii)-non-linearities
Dwij d (wij/si)(s0 tanh(si/s0))a
N5000 s0104 d0.2
s prop. to kb with b gt 1
Broad P(s) and P(k) with different exponents
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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) Ak? Highly clustered
networks Fitness model ?(k) hiki Inclusion of
space
P(k) k -g
(....) gt many available models
Redner et al. 2000, Mendes et al. 2000, Albert et
al. 2000, Dorogovtsev et al. 2001, Bianconi et
al. 2001, Barthélemy 2003, etc...
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Topological correlations clustering
ki5 ci0.
ki5 ci0.1
i

• aij Adjacency matrix

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General Motivation Ubiquity of Networks

• Economical and technological realms
• Internet, WWW (sites, hyperlinks)
• Power grids (power plants, electric lines)
• Transportation networks (airports, direct
  flights)

• Social realm
• Actors network (actors, in the same movie)
• Collaboration network (scientists, common paper)
• Citation network (scientists, cited ref.)
• Acquaintances (people, social relation)

• Biological realm
• Neural networks (neurons, axons)
• Ecosystems Food-webs (species, who eats who)
• Metabolic networks (metabolites, chem. Reaction)

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Topological Characterization Diameter
Diameter maxi,j2 G d(i,j) (1) or ltd(i,j)gt (
2)
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Topological Characterization Diameter

• Stanley Milgram (1967) Average distance in
• North-America d ¼ 6
• Six degrees of separation

• Usually d log N ( N1/dim)
• ) Small-World

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Topological Characterization Clustering

• Random graph CRN 1/N 1
• Observed
• - C À CRN
• - Hierarchy C(k) k-? ?¼ 1

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Topological Characterization Clustering
Do your friends know each other ?
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Topological correlations assortativity
ki4 knn,i(3447)/44.5
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Topological Characterization Assortativity
Are your friends similar to you ?
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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

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Topological CharacterizationBetweenness
Centrality
k
i
ij large centrality
j
jk small centrality

• ?st of shortest paths from s to t
• ?st(ij) of shortest paths from s to t via (ij)

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Topological Characterization Modularity

• Real networks are fragmented into group or
  modules

• Society Granovetter, M. S. (1973) Girvan,
  M., Newman, M.E.J. (2001) Watts, D. J., Dodds,
  P. S., Newman, M. E. J. (2002).
• WWW Flake, G. W., Lawrence, S., Giles. C.
  L. (2000).
• Biology Hartwell, L.-H., Hopfield, J. J.,
  Leibler, S., Murray, A. W. (1999).
• Internet Vasquez, Pastor-Satorras,
  Vespignani(2001).

Modularity vs. Fonctionality ?
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Weights

• Airports number of available seats for the year
  2002

• Scientific collaborations

i, j authors k paper nk number of
authors ???? 1 if author i has contributed to
paper k
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Case study Collaboration network

• (1) Broad distribution connectivity and weights

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Global data analysis Collaboration network

• Number of authors 12722 39967 links
• Topology
• Maximum coordination number 97
• Average coordination number 6.28
• Clustering coefficient 0.65
• Pearson coefficient (assortativity) 0.16
• Average shortest path 6.83
• Weight
• Maximum weight 21.33
• Average weight 0.57

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Weighted assortativity Collab.
) High-degree nodes publish together many papers !
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Weighted clustering coefficient Collab.
) For high-degree nodes most papers done in
well-connected groups
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Weighted clustering coefficient Airports
C(k) lt Cw(k) larger weights on cliques at all
scales
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Weighted clustering coefficient Airport
) Rich-club phenomenon
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Case study Inter-urban movements-Traffic

• Weighted assortativity Large w between hubs

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Correlations topology-weight Collab.
S(k) proportional to k