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 LNCS 3243 (2004) 56 cond-mat/0406238 PRE 70 (2004) 066149 physics/0504029

●

Complex networks:

examples, models, topological correlations

●

Weighted networks:

●examples, empirical analysis

●new metrics: weighted correlations

●models of 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

• <k²>  

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

What does it mean?

Poisson distribution

Exponential Network

Power-law distribution

Scale-free Network

Strong consequences on the dynamics on the network:

● Propagation of epidemics

● Robustness

● Resilience

● ...

Topological correlations:

clustering

i

k_i=5 c_i=0.

k_i=5 c_i=0.1

a_ij: Adjacency matrix

Topological correlations:

assortativity

k_i=4

k_nn,i=(3+4+4+7)/4=4.5

i

k=7 k=3

k=4 k=4

Assortativity

● Assortative behaviour: growing k_nn(k)

Example: social networks

Large sites are connected with large sites

● Disassortative behaviour: decreasing k_nn(k)

Example: internet

Large sites connected with small sites, hierarchical structure

Models for growing scale-free graphs

Barabási and Albert, 1999: growth + preferential attachment

P(k) ~ k

Generalizations and variations:

Non-linear preferential attachment : (k) ~ k^ Initial attractiveness : (k) ~ A+k^

Highly clustered networks Fitness model: (k) ~_ik_i 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

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) : SCN

● World-wide Airports' network*: WAN

● ...

*: data from IATA www.iata.org

are weighted heterogeneous networks, with broad distributions of weights

Weights

● Scientific collaborations:

i, j: authors; k: paper; n_k: 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

●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

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

Generalization of k

: 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_ij ~ (k_ik_j)^s_i =  w_ij ; s(k) ~ k^

WAN: no degree correlations =>  = 1 +  SCN: 

Some new definitions:

weighted metrics

●

Weighted clustering coefficient

●

Weighted assortativity

●

Disparity

Clustering vs. weighted clustering coefficient

s_i=16

c_i^w=0.625 > c_i k_i=4 c_i=0.5

s_i=8

c_i^w=0.25 < c_i

w_ij⁼¹ w_ij⁼⁵

i i

Clustering vs. weighted clustering coefficient

Random(ized) weights: C = C_w C < C_w : more weights on cliques

C > C_w : less weights on cliques

i j

k (w_jk)

w_ij w_ik

Clustering and weighted clustering

Scientific collaborations: C= 0.65, C_w ~ C

C(k) ~ C_w(k) at small k, C(k) < C_w(k) at large k: larger weights on large cliques

Clustering and weighted clustering

Airports' network: C= 0.53, C_w=1.1 C

C(k) < C_w(k): larger weights on cliques at all scales, especially for the hubs

Another definition for the weighted clustering

J.-P. Onnela, J. Saramäki, J. Kertész, K. Kaski, cond-mat/0408629

uses a global normalization and the weights of the three edges of the triangle, while:

uses a local normalization and focuses on node i

Assortativity vs. weighted assortativity

k_i=5; k_nn,i=1.8

5 1 1

1

1

1 5 5

5

5

i

Assortativity vs. weighted assortativity

k_i=5; s_i=21; k_nn,i=1.8 ; k_nn,i^w=1.2:

k

> k

1 5 5

5

5

i

Assortativity vs. weighted assortativity

k_i=5; s_i=9; k_nn,i=1.8 ; k_nn,i^w=3.2:

k

< k

5 1 1

1

1

i

Assortativity and weighted assortativity

Airports' network

k_nn(k) < k_nn^w(k): larger weights towards large nodes

Assortativity and weighted assortativity

Scientific collaborations

k_nn(k) < k_nn^w(k): larger weights between large nodes

Non-weighted vs. Weighted:

Comparison of k_nn(k) and k_nn^w(k), of C(k) and C_w(k)

Informations on the correlations between topology and dynamics

Disparity

weights of the same order => y

» 1/k

small number of dominant edges => y

» O(1)

identification of local heterogeneities between weighted links,

existence of dominant pathways...

Models of weighted networks:

static weights

S.H. Yook et al., P.R.L. 86, 5835 (2001); Zheng et al. P.R.E 67, 040102 (2003):

● growing network with preferential attachment

● weights driven by nodes degree

● static weights

More recently, studies of weighted models:

W. Jezewski, Physica A 337, 336 (2004); K. Park et al., P. R. E 70, 026109 (2004); E.

Almaas et al, P.R.E 71, 036124 (2005); T. Antal and P.L. Krapivsky, P.R.E 71, 026103 (2005)

in all cases:

no dynamical evolution of weights nor feedback mechanism between topology and weights

A new (simple) mechanism for growing weighted networks

• 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:

feedback mechanism

New node: n, attached to i New weight w_ni=w₀=1

Weights between i and its other neighbours:

s

s

+ w

+ 

parameter

n i

j

Redistribution of weights:

feedback mechanism

The new traffic n-i increases the traffic i-j and the strength/attractivity of i

=> feedback mechanism

n i

j

“Busy gets busier”

Evolution equations (mean-field)

s

changes because

• a new node connects to i

• a new node connects to a neighbour j of i

Evolution equations (mean-field)

changes because

• a new node connects to i

• a new node connects to j

Evolution equations (mean-field)

•m new links

•global increase of strengths: 2m(1+) each new node:

Analytical results

Correlations topology/weights:

power law growth of s_i (i introduced at time t_i=i)

Analytical results:

Probability distributions

t

uniform 2 [1;t]

P(s) ds » s

ds

= 1+1/a

Analytical results:

degree, strength, weight distributions

Power law distributions for k, s and w:

P(k) ~ k

; P(s)~s

Numerical results

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

(N=10⁵)

Numerical results: weights

w_ij ~ min(k_i,k_j)^a

Numerical results:

assortativity

disassortative behaviour typical of growing networks analytics: k_nn / k^-3

(Barrat and Pastor-Satorras, Phys. Rev. E 71 (2005) )

Numerical results:

assortativity

Weighted k_nn^w much larger than k_nn : larger weights contribute to the links towards

vertices with larger degree

Disassortativity

during the construction of the network: new nodes attach to nodes with large strength

=>hierarchy among the nodes:

-new vertices have small k and large degree neighbours -old vertices have large k and many small k neighbours reinforcement: edges between “old” nodes get reinforced

=>larger k_nnw , especially at large k

Numerical results:

clustering

•  increases => clustering increases

• clustering hierarchy emerges

• analytics: C(k) proportional to k^-3

(Barrat and Pastor-Satorras, Phys. Rev. E 71 (2005) )

Numerical results:

clustering

Weighted clustering much larger than unweighted one, especially at large degrees

Clustering

● as  increases: larger probability to build

triangles, with typically one new node and 2 old nodes => larger increase at small k

● new nodes: small weights so that c^w and c are close

● old nodes: strong weights so that triangles are more important

Extensions of the model:

i.

heterogeneities

ii.

non-linearities

iii.

directed model

iv.

other similar mechanisms

Extensions of the model:

(i)-heterogeneities

Random redistribution parameter _i (

i.i.d. with

 self-consistent analytical solution

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

Results

• s_i(t) grows as t^a(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 i

j

New node: n, attached to i New weight w_ni=w₀=1

Weights between i and its other neighbours:

_i increases with s_i; saturation effect at s₀

Extensions of the model:

(ii)-non-linearities

s prop. to k^ with > 1 N=5000

s₀=10⁴



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

Extensions of the model:

(iii)-directed network

i

l j

Extensions of the model:

(iii)- directed network

n i

j

(ii) Strength driven

preferential attachment (n: k^out=m outlinks)

AND...

“Busy gets busier”

Weights reinforcement mechanism

i

j n

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

“Busy gets busier”

Evolution equations

(Continuous approximation)

Coupling term

Resolution

Ansatz

supported by numerics:

Results

Approximation

Total in-weight _i sⁱⁿ_i : approximately proportional to the

total number of in-links _i kⁱⁿ_i , times average weight hwi = 1+

Then: A=1+

^s_in 2 [2;2+1/m]

Measure of A prediction of 

Numerical simulations

Approx of 

Numerical simulations

NB: broad P(s^out) even if k^out=m

Clustering spectrum

•  increases => clustering increases

• New pages: point to various well-known pages, often connected together => large clustering for small nodes

• Old, popular nodes with large k: many in-links from many less popular nodes which are not connected together

=> smaller clustering for large nodes

Clustering and weighted clustering

Weighted Clustering larger than topological clustering:

triangles carry a large part of the traffic

Assortativity

Average connectivity of nearest neighbours of i

Assortativity

•k_nn: disassortative behaviour, as usual in growing networks models, and typical in technological networks

•lack of correlations in popularity as measured by the in-degree

S.N. Dorogovtsev and J.F.F. Mendes

“Minimal models of

weighted scale-free networks ”

cond-mat/0408343

(i) choose at random a weighted edge i-j, with probability / w_ij

(ii) reinforcement w_ij ! w_ij + 

(iii) attach a new node to the extremities of i-j

broad P(s), P(k), P(w)

large clustering

linear correlations between s and k

“BUSY GETS BUSIER”

G. Bianconi

“Emergence of weight-topology correlations in complex scale-free networks ”

cond-mat/0412399

(i) new nodes use preferential attachment driven by connectivity to establish m links

(ii) random selection of m’ weighted edges i-j, with probability / w_ij

(iii) reinforcement of these edges w_ij ! w_ij+w₀

=>broad distributions of k,s,w

=>non-linear correlations

s / k

 > 1 iff m’ > m

“BUSY GETS BUSIER”

Summary/ Perspectives

•Empirical analysis of weighted networks

weights heterogeneities

correlations weights/topology

new metrics to quantify these correlations

•New mechanism for growing network which couples topology and weights

broad distributions of weights, strengths, connectivities

extensions of the model

randomness, non linearities, directed network

spatial network: physics/0504029

Perspectives:

Influence of weights on the dynamics on the networks

COevolution and Self-organization In dynamical Networks

http://www.cosin.org

http://delis.upb.de

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

•R. Albert, A.-L. Barabási, “Statistical mechanics of complex networks”, Review of Modern Physics 74 (2002) 47.

•S.N. Dorogovtsev, J.F.F. Mendes, “Evolution of networks”, Advances in

•Physics 51 (2002) 1079.

•S.N. Dorogovtsev, J.F.F. Mendes, “Evolution of networks: From biological nets to the Internet and WWW”, Oxford University Press, Oxford, 2003

•R. Pastor-Satorras, A. Vespignani, “Evolution and structure of the Internet:

A statistical physics approach”, Cambridge University Press, Cambridge, 2003 +other books/reviews to appear soon....

Some useful reviews/books