weighted networks. how do people find a new job? interviewed 100 people who had changed jobs in the...

Weighted Networks

How do people find a new job?

interviewed 100 people who had changed jobs in the Boston area. More than half found job through personal contacts (at odds with standard economics). Those who found a job, found it more often through “weak ties”.

HYPOTHESIS: The strong ties are within groups, weak ties are between groups.

Network Science: Weighted NetworksGranovetter, 1973

WHY WEIGHTS MATTER: THE STRENGTH OF WEAK TIES

Examples of weighted networks

Extended metrics to weighted networks: Weighted degree and other definitions (C, knn).

Correlations between weights and degrees

Scaling of weights

Betweenness centrality

Granovetter : The strength of weak ties

OUTLINE

Network Science: Weighted Networks

Nodes: airports

Links: direct flights

Weights: Number of seats

2002 IATA database V = 3880 airports E = 18810 direct flights

<k> = 10 kmax = 318 <l> = 4 lmax = 16 <w> = 105 wmax = 107

Nmin = 103 Nmax = 107

Network Science: Weighted Networks

EXAMPLE 1: AIRLINE TRAFFIC

• Nodes: scientists• Links: joint publications• Weights: number of joint pubs.

i, j: authors

k: paper

nk: number of authors

δik=1 if author i contributed

to paper k

Network Science: Weighted NetworksNewman and Girvan, cond-mat/0308217 (2003)

EXAMPLE 2: SCIENCE COLLABORATION NETWORK

EXAMPLE 3: INTERNET

domain2

domain1

domain3

router

• Nodes: routers• Links: physical lines• Link Weights: bandwidth or traffic

Network Science: Weighted Networks

• Nodes: metabolites• Links: reactions• Link Weights: reaction flux

Network Science: Weighted NetworksE. Almaas, B. Kovács, T. Vicsek, Z. N. Oltvai, A.-L. B. Nature, 2004; Goh et al, PRL 2002.

EXAMPLE 4: METABOLIC NETWORK

weak links

strong links

• Nodes: mobile phones• Links: calls• Weights: number of calls or time on the phone

Network Science: Weighted NetworksOnella et al, PNAS 2007

EXAMPLE 5: MOBILE CALL GRAPH

DEFINITIONS:Weighted Degree

Network Science: Degree Correlations

GRAPHOLOGY 2

Unweighted(undirected)

Weighted(undirected)

3

1

4

23

2

14

protein-protein interactions, www Call Graph, metabolic networks

aij and wij

In the literature we often use a double notation: Aij and wij (somewhat redundant)

32

14

Adjacency Matrix (Aij) Weight Matrix (Wij)

Node Strength (weighted degree) s:

Strength distribution P(s): probability that a randomly chosen node has strength s

Weight distribution P(w): probability that a randomly chosen link has weight w

In most real systems P(s) and P(w) are fat tailed.

Network Science: Weighted Networks

EMPIRICAL FINDING 1: P(s) AND P(w) ARE FAT TAILED

If there are no correlations between k and s, we can approximate wij with <w>:

Science collaboration network: Weight appear to be assigned randomly/

Network Science: Weighted NetworksBarrat, Barthelemy, Pastor-Satorras, Vespignani, PNAS 2004

EMPIRICAL 2: RELATIONSHIP BETWEEN STRENGTH AND DEGREE

If there are no correlations between k and s, we can approximate wij with <w>:

Science collaboration network: Weight appear to be assigned randomly/

Barrat, Barthelemy, Pastor-Satorras, Vespignani, PNAS 2004

EMPIRICAL 2: RELATIONSHIP BETWEEN STRENGTH AND DEGREE

Airport network: β=1.5

Randomized weights: s=<w>k: β=1

Correlations between topology and dynamics:

β>1: strength of vertices grows faster than their degree the weight of edges belonging to highly connected vertices have a value higher than the

one corresponding to a random assignment of weights. the larger is an airport’s degree (k), disproportionally more traffic it can handle (s).

In most systems we observe:

EMPIRICAL 2: NODE STRENGHT AND DEGREE

Barrat, Barthelemy, Pastor-Satorras, Vespignani, PNAS 2004

(assuming no

degree correlations)

Weight correlation:

Scaling relationship:

WEIGHT CORRELATIONS(2)

Weight correlation:

Scaling relationship:

Science collaboration:

β=1 Θ=0 (YES!)(confirming the lack of correlations between k and w)

The weight between two authors does not depend on their degree.

Airline network:

β=1.5 Θ=0.5 (YES!)

The traffic between two airports depends on their individual degrees.

Network Science: Weighted NetworksBarrat, Barthelemy, Pastor-Satorras, Vespignani, PNAS 2004

WEIGHT CORRELATIONS(2)

Weight correlation:

Scaling relationship:

E. coli

Network Science: Weighted NetworksMacdonald, Almaas, Barabasi, Europhys Lett 2005

WEIGHT CORRELATIONS(2)

Weight correlation:

Scaling relationship:

Weight-degree:

Airline Collaboration

β=1.5 β=1

Θ=0.5 Θ=0

Network Science: Weighted NetworksMacdonald, Almaas, Barabasi, Europhys Lett 2005

SUMMARY

WEIGHTED CORRELATIONS

(Clustering, Assortativity)

Network Science: Degree Correlations

• If ciw/ci>1: Weights localized on cliques

• If ciw/ci<1: Important links don’t form cliques

• If w and k uncorrelated: ciw=ci

Un-

we

ight

edW

eig

hte

d

• If wij=<w> ci=ciw

aijaihajh=1 only for closed triangles

Barrat, Barthelemy, Pastor-Satorras, Vespignani, PNAS 2004

WEIGHTED CLUSTERING COEFFICIENT

Wei

ght

ed

Collaboration network:C(k) and Cw(k) are comparable

Airline network:C(k) < Cw(k): accumulation of traffic on high degree nodes

Network Science: Weighted Networks

EXAMPLES: WEIGHTED CLUSTERING COEFFICIENT

• If knnw(i)/knn(i) >1: Edges with larger weights point to nodes with larger k

Un-

we

ight

edW

eig

hte

d

• If wij=<w> knn,i=knn,iw

• Measures the affinity to connect with high- or low-degree neighbors according to the magnitude of the interactions.

Network Science: Weighted NetworksBarrat, Barthelemy, Pastor-Satorras, Vespignani, PNAS 2004

WEIGHTED ASSORTATIVITY

Wei

ght

ed

Assortative behavior in agreement with evidence that social networks are usually have a strong assortative character

Assortative only for small degrees. For k>10, knn(k) approaches a constant uncorrelated structure in which vertices with very

different degrees have similar knn

Network Science: Weighted Networks

EXAMPLES: WEIGHTED ASSORTATIVITY

• If Y2(i)» 1/ki 1: No dominant link weights

• If Y2(i)~ 1/ki: A few dominant link weights

Y2(k)» 1/k No dominant connection

Air Traffic:

Network Science: Weighted Networks

DISPARITY

~ k -0.27

Mass flows along linear pathways

E. Almaas, B. Kovács, T. Vicsek, Z. N. Oltvai, A.-L. Barabasi. Nature, 2004Network Science: Weighted Networks

APPLICATION: DISPARITY IN METABOLIC FLUXES

Inhomogeneity in the local flux distribution

~ k -0.27

Mass flows along linear pathways

Mass flows along linear pathways

Network Science: Weighted NetworksE. Almaas, B. Kovács, T. Vicsek, Z. N. Oltvai, A.-L. Barabasi. Nature, 2004

The end

Network Science: Weighted Networks