Architecture of Complex
Weighted Networks
Marc Barthélemy
CEA, France
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)
Outline
I. 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
Complex Networks
 Recent studies on topological
properties showed:
- broad distribution of connectivities
- impact on different processes (eg. Resilience,
epidemics)
i
j
Beyond Topology: Weighted Networks
w ij
j
i
wji
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
Motivation
Why study weighted networks ?
) The weights can modify the behavior predicted
by topology:
• Resilience
• Epidemics
• …
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 ?
Topological Characterization of Large Networks
All these networks are:
• Complex
• Very large
Statistical tools needed !
Statistical mechanics of large networks
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, …
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)
Classes of networks
Poisson distribution
Exponential Network
Power-law distribution
Scale-free Network
Weighted Networks
) New measurement tools needed !
Weighted networks characterization
Generalization of ki: strength
• For wij=w0:
• For wij and ki independent:
Weighted networks characterization
• In general:
• If  > 1 or if =1 and A<w>
) Existence of strong correlations !
Weighted networks characterization
• Weighted clustering coefficient:
• If ciw/ci>1: Weights localized on clicques
• If ciw/ci<1: Important links don’t form clicques
• If w and k uncorrelated ) ciw=ci
Weighted networks characterization
• Weighted assortativity:
• If knnw(i)/knn(i) >1: Edges with larger weights
point to nodes with larger k
Weighted networks characterization
Weighted networks characterization
• « Disparity »:
• If Y2(i)» 1/ki ¿ 1: No dominant connections
• If Y2(i)À 1/ki: A few dominant connections
Weighted networks characterization
• Disparity:
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)
Airplane route network
Nodes: airports
Links: direct flight
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
Case study: Airport network
• Broad distribution: connectivity and weights
Correlations topology-traffic: Airports
s(k) proportional to k =1.5 (Randomized weights: s=<w>k: =1)
Strong correlations between topology and dynamics
Correlations topology-traffic
• <wij>» (kikj)
¼ 0.5
Weighted clustering coefficient: Airport
Cw(k) > C(k): larger weights on cliques at all scales
(esp. for large k)
Weighted assortativity: Airport
knn(k) < knnw(k): larger weights between large nodes
For large k ) Large traffic between hubs
Disparity: Airport
Y2(k)» 1/k ) No dominant connection
Airport: Summary
• Topology: Scale-free network
• Rich traffic structure
• Strong correlations traffic-topology
Case study: Inter-cities movements
• Sardinia:
- Italian island 24,000 km2
- 1,600,000 inhabitants
Case study: Inter-cities movements
• Sardinian network:
-Nodes: 375 Cities
- Link wji=wij:
# of individuals
going from i
to j (daily and by
any means)
Case study: Inter-cities movements-Topology
• N=375, E=16,248 ) <k>=43, kmax=279
Case study: Inter-cities movements-Topology
• Clustering: <C>¼ 0.26' CRG¼ 0.24
Case study: Inter-cities movements-Topology
• Slightly disassortative network
Case study: Inter-cities movements-Traffic
• <w>¼ 23, wmax¼ 14.000 (!)
P(w)» w-w
w¼ 2.2
Case study: Inter-cities movements-Traffic
• Correlations: s» k, ' 1.9
Case study: Inter-cities movements-Traffic
•Weighted clustering: Hubs form large w-clicques
Case study: Inter-cities movements-Traffic
•Weighted assortativity: Large w between hubs
Case study: Inter-cities movements-Traffic
• Y2(k) » k-, ' 0.4 ) Traffic jams !
Transportation networks: Summary
Network
P(k)
P(s)
s» k
Y2(k)» k-
Clustering
Assort.
Global
(WAN)
Heavy tail
('2.0)
Scale-free
Broad
 ' 1.5
' 1.0
(good)
Cw/C>1
kw/k>1
Inter-cities
Fast tail
(' 3.5)
Random
Graph
Broad
(s'2.0)
 ' 1.9
' 0.4
(bad)
Cw/C>1
kw/k>1
Intra-urban
(Eubank)
Heavy tail
('2.4)
Scale-free
Broad
(s'2.7)
 ' 1.0
--
C(k)» 1/k
--
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 ?
Weighted networks: Model
• Growing network: addition of nodes
Proba(n! i)/ si
Weighted networks: Model
• Rearrangement of weights
• ¿ 1: No effect (=0: BA model)
• À 1: Traffic stimulation
Evolution equations (mean-field)
Analytical results
• Power law distributions for k and s:
P(k) ~ k - ; P(s)~s-
2 <  < 3:
 Strong coupling ! 2

Weak coupling ! 3
Analytical results
• Power law distributions for w:
P(w)» w-
• Correlations topology/weights:
si ' (2+1)ki  <w> ki
Nonlinear correlations ?
Correlations topology/weights:
si ' (2+1)ki  <w> ki)  = 1
) How can we obtain   1 ?
- Inclusion of space
-…
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 )  > 1 !
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 ?
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@th.u-psud.fr
Numerical results: clustering
Numerical results: assortativity
Numerical results
Numerical results: P(w), P(s)
(N=105)
Numerical results: weights
wij ~ min(ki,kj)a
Numerical results:
assortativity
analytics: knn proportional to k(-3)
Numerical results:
clustering
analytics: C(k) proportional to k(-3)
Extensions of the model:
(i)-heterogeneities
Random redistribution parameter i (i.i.d.
with r() )
 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
New node: n, attached to i
New weight wni=w0=1
Weights between i and its other neighbours:
i
Dwij = f(wij,si,ki)
Example: Dwij =  (wij/si)(s0 tanh(si/s0))a
i increases with si; saturation effect at s0
j
Extensions of the model:
(ii)-non-linearities
Dwij =  (wij/si)(s0 tanh(si/s0))a
N=5000
s0=104
=0.2
s prop. to k with  > 1
Broad P(s) and P(k) with different exponents
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 networks
Fitness model: (k) ~ hiki
Inclusion of space
P(k) ~ k -
(....) => 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...
Topological correlations: clustering
ki=5
ci=0.1
=0.
i
aij: Adjacency
matrix
General Motivation: Ubiquity of Networks

Economical
Social
Biological
realm
realm
and technological realms
 Neural
Actors
Internet,
network
networks
WWW(actors,
(neurons,
(sites, hyperlinks)
in the
axons)
same movie)
 Ecosystems:
Power grids (power
Collaboration
Food-webs
network
plants,
(scientists,
(species,
electric
who
common
lines)
eats who)
paper)
 Metabolic
Citation
Transportation
network
networks
networks
(scientists,
(metabolites,
(airports,
cited chem.
ref.)
direct
Reaction)
flights)
 Acquaintances (people, ‘social relation’)
Topological Characterization: Diameter
Diameter = maxi,j2 G d(i,j)
or:
=<d(i,j)>
(1)
(2)
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’
Topological Characterization: Clustering
• Random graph: CRN» 1/N ¿ 1
• Observed:
- C À CRN
- Hierarchy: C(k) » k-
¼ 1
Topological Characterization: Clustering
Do your friends know each other ?
C=
# of links between neighbors
k(k-1)/2
Topological correlations: assortativity
k=4
k=4
ki=4
knn,i=(3+4+4+7)/4=4.5
i
k=7
k=3
Topological Characterization: Assortativity
Are your friends similar to you ?
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
Topological Characterization:
Betweenness Centrality
k
i
j
ij: large centrality
jk: small centrality
st = # of shortest paths from s to t
st(ij)= # of shortest paths from s to t via (ij)
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 ?
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
Case study: Collaboration network
• (1) Broad distribution: connectivity and weights
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
Weighted assortativity: Collab.
) High-degree nodes publish together many papers !
Weighted clustering coefficient: Collab.
) For high-degree nodes: most papers done
in well-connected groups
Weighted clustering coefficient: Airports
C(k) < Cw(k): larger weights on cliques at all scales
Weighted clustering coefficient: Airport
) Rich-club phenomenon
Case study: Inter-urban movements-Traffic
•Weighted assortativity: Large w between hubs
Correlations topology-weight: Collab.
S(k) proportional to k