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Complex Networks as a tool to studyhuman activity

José Javier Ramasco

Bruno GonçalvesSteven A. Morris

Romualdo Pastor-SatorrasSergei Dorogovtsev

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Social Networks as weighted graphs

Measures and meaning

Social Inertia

Results in Empirical Networks and models

More information, Range

Conclusions

Outline

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Social Networks

Friendster.com

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Social Networks

Some characteristics that make social Some characteristics that make social networksnetworkshard to studyhard to study

Arbitrariness of the definition Arbitrariness of the definition

Variability Variability

Size constrains on the social probes Size constrains on the social probes

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Collaboration Networks

1 2 5

3 4 6

1 2 3 4 5 6

w = 1

2 1

1

1

1

1

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Collaboration Networks

The projected network contains less informationThe projected network contains less informationthan than the original bipartite graph.the original bipartite graph.

A way to avoid that in part is to use a weightedA way to avoid that in part is to use a weightednetwork in the projection.network in the projection.

The weight of a link may be defined asThe weight of a link may be defined as

Note however that there is still a certain loose of Note however that there is still a certain loose of information.information.

M.E.J. Newman, PRE 64, 016132 (2001).

!

wij = "ik" j

k= num. common collab.

k collab

#

!

wij ="ik" j

k

nk #1k collab

$

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Measures

We can define a number of distributions forWe can define a number of distributions forthe weighted networkthe weighted network

Degree distribution Degree distribution

Weight distribution of the edges Weight distribution of the edges

Strength distribution Strength distribution!

Pk(k) k C

k(k)

!

Pw(w) w C

w(w)

!

si = wij

j"# ( i)

$

Ps(s) s Cs(s)

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Measures

Other properties of the projected networkOther properties of the projected networkthat we are interested in that we are interested in

Clustering Clustering

Correlations Correlations

!

ci =2ti

ki(ki "1) C = ci

ciw

=1

si(ki "1)

wij + wim

2j,m#$ (i)

% aijaima jm Cw

= ciw

!

knn,i =1

kik j

j"# ( i)

$ knn (k) = knn,i k

snn (s) = snn,i s

Inn (I) = Inn,i s

A. Barrat, et al., PNAS 101, 3747 (2004).

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Social Inertia

WIJ

Remember that Remember that

that the strength meansthat the strength means

And the degree of a node meansAnd the degree of a node means

Let us define then the Let us define then the Social InertiaSocial Inertia as as

!

wij = num. of works together

!

si = wij

j"# ( i)

$ = total num of partnerships

!

ki= num. of different partners

!

"i=si

ki

J.J. Ramasco and S.A. Morris, PRE 73, 016122 (2006).

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Social Inertia

The extreme values of The extreme values of II are are

In general, represents a measure ofIn general, represents a measure ofthe eagerness of an author for collaborationsthe eagerness of an author for collaborationswith new people.with new people.

This concept could be generalized to other weightedThis concept could be generalized to other weightedgraphs, its physical meaning (?). graphs, its physical meaning (?).

!

Ii "1 if all collaborations happened

with different partners.

Ii " qi if all works where done with the

same team.

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Social Inertia

Warning: the social inertia is averaged overWarning: the social inertia is averaged overtime. It is the same concept as an average time. It is the same concept as an average vvelocity.elocity.

It should be possible to define an instant inertiaIt should be possible to define an instant inertiabut it requires a more detailed knowledge ofbut it requires a more detailed knowledge ofthe empirical databases.the empirical databases.

!

˜ I i="s

i

"ki T

?

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Empirical results

!

Na

= num. agents

Nc = num. collaborations

Nai = num. agents that

work alone

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Empirical results

Main and Inset: movies

Slopes main blue = -3 = 1-δ

!

Pw(w) ~ w

"#

Cw

= P(w')dw' ~ w1"#

w

$

%

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Empirical results

Main: moviesInset: biosensors

Slopes main red = -3

Slopes inset red = -3.8

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Empirical results

Main: moviesInset: superstrings

Slopes main red = 0.7

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Empirical results

Main: moviesInset: infoscience

Slopes main red = 0.4

Slopes inset red = 0.6

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Models

J.J. Ramasco et al., PRE 70, 036106 (2004).R. Guimarà et al., Science 308, 697 (2005).

1 2 3 4 5 6

Each step a new collaboration of size Each step a new collaboration of size nn is added. is added. mm of the agents are new, without experience. of the agents are new, without experience.

The rest The rest m-nm-n are selected from the pool of are selected from the pool of old old agents agents Prob Prob pp →→ one of the previous partners one of the previous partners of an of an ““oldold”” agent is chosen agent is chosen Prob Prob 1-p 1-p →→ an an ““oldold”” agent is chosen with agent is chosen with prob prob proportional to qproportional to q

After After QQcc collaborations, the agents have a collaborations, the agents have a probprob. . 1/1/ττ of becoming inactive.of becoming inactive.

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Simulation results

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Simulation results

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Range, different quality of the connections

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Motivation

• IMDB Actor network, wim = number of times i and m have worked together.

!

Pw(w) ~ w

"#

Cw

= P(w')dw' ~ w1"#

w

$

%

fit # = "4

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Motivation

!

si = wij = wi" 1+

j#" ( i)

$ wi" 2+K+ wi" ki

% s(k) ~ k1/ 2 &%<w> ~ k

'1/ 2

• P(w) has finite second moment.• P(w) has finite second moment.

• <w> does not depend on k

• P(w) has finite second moment.

• <w> does not depend on k

• Are the weight correlated?

(J.J. Ramasco, Eur. Phys. J. ST 143, 47 (2007))

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Models

• We can try to replicate this situation in a toy model

• The simplest case requires P(w,w’), P(w) and P(w’|w)

and also a sets of rules for weight assignment.

• This is not a unique method.

!

P(w) = P(w," w')dw'

P(w' |w) = P(w,w') /P(w)

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Models• We chose three possible functional forms for P(w,w’)

• The reason was that

!

P+(w,w') =X+

(w + w')2+"

PU(w,w') =

XU

(ww')1+"

P#(w,w') =X#

(ww'+1)1+"

!

< w >+ (w0) = (1+" + w

0) /"

< w >U

=" /(" #1)

< w ># (w0) = (" +1/w

0) /(" #1)

!

P(w) ~ w"1"#

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Measures• The goal is to find a measure to estimate the type/intensity of weight

correlations and their intensity.

• P(w) is fixed.

• Compare actual pattern with a random

configuration

!

"w

2(i) = (wij# < w > (i))

2

j

$

!

" =<#w >original

<#w >random

!

Y2(i) =

wij

2

j"

( wij )2

j"

, Y2(k) ~

cons.

1/k

# $ %

!

" =<Y

2>original

<Y2

>random

!

r(i) =wmax (i) " wmin (i)

wmax (i) + wmin (i)

!

" =< r >original

< r >random

M. Barthelemy, Physica A 319, 633 (2003).J.J. Ramasco et al., to appear PRE.

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Measures

• The three magnitudes are able to detect up to some level weightcorrelations

• But there is a resolution problem

Optimal measure?

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Transport• There are many ways to describe network transport properties

• We focus on the so called Superhighways

!

"sphw =Wsph (orig)

Wsph (rand)

Ssphw =Num. nodes sphw (orig)

Num. nodes sphw (rand)

Z. Wu et al., PRL 96, 148702 (2006).

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Transport

• Actor network

ρ ≈ 0.268(1)

Ω ≈ 20(7)

S ≈ 3(1)

• US airport traffic

ρ ≈ 0.983(1)

Ω ≈ 2.6(1)

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ConclusionsWe have studied collaborations networks usingWe have studied collaborations networks usinga weighteda weighted graph representation.graph representation.

This representation allows us to define the This representation allows us to define the Social Inertia in a natural way.Social Inertia in a natural way.

The Inertia (measured in a quantitative way) The Inertia (measured in a quantitative way) grows with the experience and the network is grows with the experience and the network is assortative assortative for the inertia. for the inertia.

The model is able to reproduce some of the The model is able to reproduce some of the quantitative features of the empirical networksquantitative features of the empirical networksbut it is necessary more detail for a better but it is necessary more detail for a better quantitative result.quantitative result.

J.J. Ramasco and S.A. Morris, PRE 73, 016122 (2006).

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Conclusions

Weight correlations appear in real-world networksWeight correlations appear in real-world networks

We have propose a measures to quantify the levelWe have propose a measures to quantify the levelof correlationsof correlations

These correlations have a strong effect on theThese correlations have a strong effect on thetransport properties of the graphtransport properties of the graph

Open questions:Open questions:

–– Origin of the phenomenon in real nets.Origin of the phenomenon in real nets.

–– Effect on disease spreading.Effect on disease spreading.

J.J. Ramasco & B. Gonçalves, to appear in PRE.

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Web Surfing• The database is formed by the weblogs of Emory University from Apr. 1st2005 to Jan. 17th 2006 (41 weeks).

• Each click in a web of the university is registered at the time resolution of 1second.

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Statistics IPs URLs

!

Cx = fx (x ')dx 'x

"

#

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Statistics

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Growth

IPs

URLs

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Growth

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Other aspects

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Other aspects

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Other aspects

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Other aspects

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Conclusions

We have studied an empirical databasegenerating an evolving bipartite graph.

Preferential attachment plays an importantrole in the evolution of the weights but sodoes aging of the connections.

These data allow us to consider also theactivity patterns of the community.

Open questions …

B Gonçalves & JJ Ramasco, in preparation