Weighted networks: analysis, modeling

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 cs.NI/0405070

http://www.th.u-psud.fr/page_perso/Barrat

● Complex networks:

examples, models, topological correlations

● Weighted networks:

● examples, empirical analysis

● new metrics: weighted correlations

● a model for 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

• <k

>  

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

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) ~ 

k

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 ^-

Topological correlations:

clustering

i

k

=5 c

=0.

k

=5 c

=0.1

a

: Adjacency matrix

Topological correlations:

assortativity

k

=4

k

=(3+4+4+7)/4=4.5

i

k=7 k=3

k=4 k=4

Assortativity

● Assortative behaviour: growing k

(k)

Example: social networks

Large sites are connected with large sites

● Disassortative behaviour: decreasing k

(k)

Example: internet

Large sites connected with small sites, hierarchical structure

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)

● Airports' network*

● ...

*: data from IATA www.iata.org

are weighted heterogeneous networks,

with broad distributions of weights

Weights

●

Scientific collaborations:

i, j: authors; k: paper; n

: 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 for the year 2002

●

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

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 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

Data analysis: P(k), P(s)

Generalization of k _i : strength

Broad distributions

Correlations topology/traffic Strength vs. Coordination

S(k) proportional to k

N=12722

Largest k: 97

Largest s: 91

S(k) proportional to k



=1.5 Randomized weights: =1

N=3863

Largest k: 318

Largest 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

w

~ (k

k

)

s

=  w

; s(k) ~ k

WAN: no degree correlations =>  = 1 + 

SCN: 

Some new definitions:

weighted metrics

●

Weighted clustering coefficient

●

Weighted assortativity

Clustering vs. weighted clustering coefficient

s

=16

c

=0.625 > c

k

=4 c

=0.5

s

=8

c

=0.25 < c

w

⁼¹ w

⁼⁵

i i

Clustering vs. weighted clustering coefficient

Random(ized) weights: C = C

C < C

: more weights on cliques

C > C

: less weights on cliques

i j

k (w

)

w

w

Clustering and weighted clustering

Scientific collaborations: C= 0.65, C

~ C

C(k) ~ C

(k) at small k, C(k) < C

(k) at large k: larger weights on large cliques

Clustering and weighted clustering

Airports' network: C= 0.53, C

=1.1 C

C(k) < C

(k): larger weights on cliques at all scales

Assortativity vs. weighted assortativity

k

=5; k

=1.8

5 1 1

1

1

1 5 5

5

5

i

Assortativity vs. weighted assortativity

k

=5; s

=21; k

=1.8 ; k

=1.2: k _nn,i > k _nn,i ^w

1 5 5

5

5

i

Assortativity vs. weighted assortativity

k

=5; s

=9; k

=1.8 ; k

=3.2: k _nn,i < k _nn,i ^w

5 1 1

1

1

i

Assortativity and weighted assortativity

Airports' network

k

(k) < k

(k): larger weights between large nodes

Assortativity and weighted assortativity

Scientific collaborations

k

(k) < k

(k): larger weights between large nodes

Non-weighted vs. Weighted:

Comparison of k

(k) and k

(k), of C(k) and C

(k)

Informations on the correlations between topology and dynamics

A model of growing weighted network

S.H. Yook, H. Jeong, A.-L. Barabási, Y. Tu, P.R.L. 86, 5835 (2001)

●

Peaked probability distribution for the weights

●

Same universality class as unweighted network

●

Growing networks with preferential attachment

●

Weights on links, driven by network connectivity

●

Static weights

See also Zheng et al. Phys. Rev. E (2003)

A new model of

growing weighted network

• 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

New node: n, attached to i New weight w

=w

=1

Weights between i and its other neighbours:

s _i s _i + w ₀ + 

The new traffic n-i increases the traffic i-j

Only

parameter

n i

j

Evolution equations (mean-field)

Also: evolution of weights

Analytical results

Power law distributions for k, s and w:

P(k) ~ k ^ ; P(s)~s ^

Correlations topology/weights:

w

~ min(k

,k

)

, a=2/(2+1)

•power law growth of s

•k proportional to s

Numerical results

Numerical results: P(w), P(s)

(N=10

)

Numerical results: weights

w

~ min(k

,k

)

Numerical results:

assortativity

analytics: k

proportional to k

Numerical results:

assortativity

Numerical results:

clustering

analytics: C(k) proportional to k

Numerical results:

clustering

Extensions of the model:

(i)-heterogeneities

Random redistribution parameter 

( i.i.d. with  )

 self-consistent analytical solution

(in the spirit of the fitness model, cf. Bianconi and Barabási 2001)

Results

•

s

(t) grows as t

•

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 i New weight w

=w

=1

Weights between i and its other neighbours:

Examplew _ij =  (w _ij /s _i )(s ₀ tanh(s _i /s ₀ )) ^a



increases with s

; saturation effect at s

w _ij = f(w _ij ,s _i ,k _i )

Extensions of the model:

(ii)-non-linearities

s prop. to k

with > 1 N=5000

s

=10



w _ij =  (w _ij /s _i )(s ₀ tanh(s _i /s ₀ )) ^a

Broad P(s) and P(k) with different exponents

Summary/ Perspectives/

Work in progress

• Empirical analysis of weighted networks

weights heterogeneities

correlations weights/topology

new metrics to quantify these correlations

• New model of growing network which couples topology and weights

analytical+numerical study

broad distributions of weights, strengths, connectivities

extensions of the model

randomness, non linearities

spatial network: work in progress

other ?

• Influence of weights on the dynamics on the networks: work in progress