assessing influence and selection in network-behavioural co-evolution with an application to smoking...

Assessing influence and selection innetwork-behavioural co-evolution

with an application to smoking and

alcohol consumption among adolescents.

Christian Steglich, Tom Snijders

University of Groningen

Mike Pearson

Napier University Edinburgh

Supported by the Netherlands Organisation for Scientific Research (NWO) under grant # 401-01-550.

Empirical starting point:

“Network autocorrelation” in cross-sectional data:

Friends of smokers are smokers,friends of non-smokers are non-smokers.

Small companies trade with small companies,large companies trade with large companies.

Why that?

Range of theoretical accounts:

influenceselectionRC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam

Influence / contagion paradigm:

Properties of network neighbours are assimilated.

Friends of smokers turn into smokers.

Trade with big companies makes a company big.

Selection paradigm:

Network neighbourhood is chosen to match.

Smokers choose other smokers as friends.

Big companies do not trade with small companies.RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam

How can selection and influence be assessed and separated?

Longitudinal data are a prerequisite, ‘panel density’ sufficiently high:

RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam

Lower actor reciprocates

friendship

Upper actor adapts to

(re-

ciprocal) friend

Upper actor adapts to (per-

ceived) friend

Lower actor reciprocates

friendship

Modelling of network-behavioural co-evolution

Continuous time model

invisibility of to-and-fro changes in panel data poses no problem

evolution can be modelled in micro stepsObserved changes are quite complex – they are interpreted as resulting from a sequence of micro steps.

Actor-driven model

selection and influence conceptually belong to the actor level


Formalization as stochastic process (1)

State spacePair (x,z)(t) contains adjacency matrix x andvector(s) of behaviourals z at time point t.

Transition probabilitiesCo-evolution is modelled by specifying probabilities for simple transitions between states (x,z)(t1) and (x,z)(t2)

•network micro step:(x,z)(t1) and (x,z)(t2) differ in one tie xij only.

•behavioural micro step:(x,z)(t1) and (x,z)(t2) differ in one behavioural

score zi only.RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam

Formalization as stochastic process (2)

Timing of decisions / transitionsWaiting times between decisions are assumed to be exponentially distributed (Markov process); they can depend on state, actor and time.

Actor-driven modellingMicro steps are modelled as outcomes of an actor’s decisions; conditionally independent, given the current state.

Schematic overview of model componentsOccurrence of decisions

Decision rule

Network Network rate function Network decision rule

Behavioural

Behavioural rate function

Behavioural decision rule


Modelling of the actors’ decisions (1)

Network micro step by actor i

• Choice options- change tie variable to one other actor j- change nothing

• Maximize objective function + random disturbancenet net net( , , , , ) ( , , , )i if x z t j x z t j

Random part, i.i.d. over x, z, t, i, j, according to extreme value type I

Deterministic part, depends on network-behavioural neighbourhood of actor i


• Choice probabilities resulting from distribution of are of multinomial logit shape

net net

net net

{1,..., }

,exp ( , , , , )

Pr( | , )exp ( , , , , )

i

ik N

i jf x z t j

x x zf x z t k

x(i,j) is the network obtained from x by changing tie to actor j; x(i,i) formally stands for keeping the network as is


• Objective function f is linear combination of effects, with parameters as effect weights.

Examples:

• reciprocity effect

measures the preference difference of actor i between right and left configuration

• transitivity effect

i i

i i j j

j j

k k

ij jijx x

ij jk ikjkx x x


Modelling of the actors’ decisions (2)

Behavioural micro step by actor i

• Choice options- increase, decrease, or keep score on behavioural

• Maximize objective function + random disturbance

• Choice probabilities analogous to network part

beh beh beh( , , , , ) ( , , , )i if x z t j x z t j

Assume independence also of the network random part

Objective function different from the network objective function


Modelling selection and influence (1)

Influence and selection are based on a measure of behavioural similarity

Friendship similarity of actor i :

Actor i has two ways of increasing friendship similarity:

• by adapting own behaviour to that of friends j,

or

• by choosing friends j who behave the same.

:

i j

ij

z zsim

ij ijjx sim


Modelling selection and influence (2)

• Inclusion of friendship similarityin network objective functionmodels transitions as these:

• Inclusion of friendship similarity in behavioural objective function models transitions as these:

ij ijjx sim

“classical”selection

“classical”influence


Total process model

Transition intensities of Markov process are

Herewaiting times, = change in behavioural, = set of allowed changes in behavioural change, z(i,) = behavioural vector after change.

Together with starting value, process model is fully defined.

net

beh

ˆˆ( , ),( , )

( , ),( , , )( , ),( , , ) ( , )

, ,

, ;

ˆˆPr( | , ) , if and for some , {1,..., }, ;

ˆˆPr( | , ) , if and for some {1,..., }, ( , );

i

i

i

x z x z

x z x z ix z x i j zj i z i

i j i j

i i

x x z x x z z i j N i j

z x z x x z z i N z iq

q q

ˆˆ, if and ;

0 otherwise.

x x z z


Remarks on model estimation:

• The likelihood of an observed data set cannot be calculated in closed form, but can at least be simulated.

‘third generation problem’ of statistical analysis,

simulation-based inference is necessary.

• Currently available:

– Method of Moments estimation (Snijders 2001, 1998)– Maximum likelihood approach (Snijders & Koskinen 2003)

Implementation: program SIENA, part of the StOCNet software package (see link in

the end).RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam

Application to alcohol consumption and smoking behaviour among adolescents

Data three wave panel ’95’96’97,school year group, age 13-16

•alcohol consumption variable ranges from 1 (more than once a week) to 5 (not at all)

•smoking variable ranges from 3 (non-smokers) to 5 (regular smokers)

Method actor-driven modelling, using SIENA

• first run separate analyses per behavioural,

• then analyse them jointly.RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam

Question

Do influence and selection processes based on

(a) smoking behaviour and

(b) drinking behaviour

differ qualitatively?

More precisely:• Is alcohol consumption more “social” and

smoking more “individual”? Is influence stronger on the alcohol

dimension?

• Is alcohol consumption more “accepted” than smoking?

What are the details of the selection mechanisms?RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam

Model components

•covariate effects on both evolution processes- classmate relation (dyadic)- parent smoking, sibling smoking- gender (several effects)

•endogenous effects of network on network evolution

- reciprocity- transitivity (two effects)

•endogenous effects of behaviour on network evolution

- selection based on alcohol consumption- selection based on smoking (three effects each)

•endogenous effects of network on behavioural evolution

- influence from friendsRC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam

Estimation results (excerpts, 1)

• gender-based selection utilities

Based on these estimates,in an artificial choice situationbetween a boy and a girl, ego’schoice probabilities are:

This result is consistent across model specifications.

alter

ego boy girl

boy 0.78 -0.17

girl 0.17 0.77

alter

ego boy girl

boy 72% 28%

girl 35% 65%



• alcohol-based selection utilities

Based on these estimates,in an artificial choice situationbetween a regular drinker and a non-drinker, ego’s choice probabilities are:

This result also is consistent across model specifications.Note that there is a net preference for drinkers as friends!

alter

egonon-

drinkerreg.drink

er

non-drinker 0.31 -0.02

reg.drinker -0.23 0.47

alter

egonon-

drinkerreg.drink

er

non-drinker 58% 42%

reg.drinker 33% 67%


• smoking-based selection utilities

Based on these estimates,in an artificial choice situationbetween a regular smoker and a non-smoker, ego’s choice probabilities are :

This result also is consistent across model specifications.Note that there is a net preference against smokers as friends!

alter

egonon-

smokerreg.smok

er

non-smoker -0.08 -0.55

reg.smoker -0.18 -0.27

alter

egonon-

smokerreg.smok

er

non-smoker 61% 39%

reg.smoker 48% 52%


• smoking-based influence effect:

model without alcohol: controlling for alcohol:

parameter positive, p=0.08 parameter positive, p>0.2

Probabilities shown are for an occasional smoker with 4 friends, depending on the number of regular smokers in his neighbourhood (other friends assumed to be non-smokers)


0%

25%

50%

75%

0 1 2 3 4

increase

decrease

stay

0%

25%

50%

75%

0 1 2 3 4

Weak pos. effect of alcohol consumption on smoking, p=0.08


• alcohol-based influence effect:

model without smoking: controlling for smoking:

parameter positive, p<0.01 parameter positive, p<0.01

Probabilities shown are for an occasional drinker with 4 friends, depending on the number of regular drinkers in his neighbourhood (other friends assumed to be non-drinkers)


0%

25%

50%

75%

0 1 2 3 40%

25%

50%

75%

0 1 2 3 4

increase

decrease

stay

No significant effect of smoking on alcohol consumption, p>0.4

Summary of investigation

• Selection effects occur for both alcohol and smoking.

• Alcohol consumption of a potential friend renders him/her more attractive as friend, while smoking renders him/her less attractive.

• Influence occurs only on the alcohol dimension.

The weak appearance of an influence effect for smoking seems to be due to an effect of alcohol consumption on smoking.


Discussion

• simultaneous statistical modelling of network & behavioural dynamics for longitudinal panel data

• selection and influence effects are disentangled

• many other effects and applications possible

• software SIENA 2.0 beta version available from

http://stat.gamma.rug.nl/stocnet/ (“stable URL”)and via

http://ppswmm.ppsw.rug.nl/steglich/ (current updates)

final version comes soonRC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam

assessing influence and selection in network-behavioural co-evolution with an application to smoking...

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