Assessing influence and selection in
network-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:
influence
selection
RC33 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:
Lower actor
reciprocates
Upper actor
adapts to (re-
friendship
ciprocal) friend
Upper actor
adapts to (per-
Lower actor
reciprocates
ceived) friend
friendship
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
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 steps
Observed 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
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
Formalization as stochastic process (1)
State space
Pair (x,z)(t) contains adjacency matrix x
andvector(s) of behaviourals z at time point t.
Transition probabilities
Co-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 / transitions
Waiting times l between decisions are assumed to be
exponentially distributed (Markov process);
they can depend on state, actor and time.
Actor-driven modelling
Micro steps are modelled as outcomes of an actor’s
decisions; conditionally independent, given the current state.
Schematic overview of model components
Occurrence of decisions Decision rule
Network
Network rate function
Network decision rule
Behavioural
Behavioural rate function
Behavioural decision rule
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
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 disturbance
finet (net , x, z, t, j )  inet ( x, z, t, j )
Deterministic part, depends
on network-behavioural
neighbourhood of actor i
Random part, i.i.d. over
x, z, t, i, j, according to
extreme value type I
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
• Choice probabilities resulting from
distribution of  are of multinomial logit shape
Pr( x
i, j
| x, z ) 
exp  finet (  net , x , z, t , j ) 

exp  finet (  net , x , z, t , k ) 
k{1,..., N }
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
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
• Objective function f is linear combination of
effects, with parameters as effect weights.
Examples:
• reciprocity effect
xx
ij
j
ji
i
j
i
j
measures the preference difference of actor i
between right and left configuration
• transitivity effect

j
i
k
jk
x ij x jk x ik
j
i
k
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
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
f
beh
i
( , x, z, t , j )   ( x, z, t , j )
beh
Objective function different
from the network objective
function
beh
i
Assume independence
also of the network
random part
• Choice probabilities analogous to network part
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
Modelling selection and influence (1)
Influence and selection are based on a measure of
behavioural similarity
simij :
Friendship similarity of actor i :
  zi  zj

 x sim
j
ij
ij
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.
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
Modelling selection and influence (2)
• Inclusion of friendship similarity
in network objective function
models transitions as these:
 x sim
j
ij
ij
“classical”
selection
• Inclusion of friendship similarity in behavioural
objective function models transitions as these:
“classical”
influence
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
Total process model
Transition intensities of Markov process are
q( x ,z ),( xˆ ,zˆ )
ˆ
ˆ
l net
i Pr(x i , j | x , z ) , if x  x i , j and z  z for some i , j  {1, ..., N}, i  j;

l ibeh Pr(z i , d | x , z ) , if xˆ  x and zˆ  z i ; d for some i  {1, ..., N}, d  G( z, i );


     j  i q( x , z ),( x i , j , z )   dG( z , i ) q( x , z ),( x , z i , d )  , if xˆ  x and zˆ  z;

 0 otherwise.
i
Here l  waiting times, d = change in behavioural, G = set of allowed
changes in behavioural change, z(i,d) = behavioural vector after change.
Together with starting value, process model is fully defined.
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
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 friends
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
Estimation results (excerpts, 1)
• gender-based selection utilities
alter
ego
Based on these estimates,
in an artificial choice situation
between a boy and a girl, ego’s
choice probabilities are:
boy
girl
boy
0.78
-0.17
girl
0.17
0.77
alter
ego
boy
girl
boy
72%
28%
girl
35%
65%
This result is consistent across model specifications.
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
Estimation results (excerpts, 2)
• alcohol-based selection utilities
ego
alter
non-drinker reg.drinker
non-drinker
0.31
-0.02
reg.drinker
-0.23
0.47
Based on these estimates,
in an artificial choice situation
between a regular drinker and
a non-drinker, ego’s choice
probabilities are:
ego
alter
non-drinker reg.drinker
non-drinker
58%
42%
reg.drinker
33%
67%
This result also is consistent across model specifications.
Note that there is a net preference for drinkers as friends!
Estimation results (excerpts, 3)
• smoking-based selection utilities
ego
alter
non-smoker reg.smoker
non-smoker
-0.08
-0.55
reg.smoker
-0.18
-0.27
Based on these estimates,
in an artificial choice situation
between a regular smoker and
a non-smoker, ego’s choice
probabilities are :
ego
alter
non-smoker reg.smoker
non-smoker
61%
39%
reg.smoker
48%
52%
This result also is consistent across model specifications.
Note that there is a net preference against smokers as friends!
Estimation results (excerpts, 4)
Weak pos. effect of
alcohol consumption
on smoking, p=0.08
• smoking-based influence effect:
model without alcohol:
controlling for alcohol:
75%
75%
50%
50%
25%
25%
0%
0%
0
1
2
3
4
parameter positive, p=0.08
increase
stay
decrease
0
1
2
3
4
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)
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
Estimation results (excerpts, 5)
No significant effect of
smoking on alcohol
consumption, p>0.4
• alcohol-based influence effect:
model without smoking:
controlling for smoking:
75%
75%
50%
50%
25%
25%
0%
0%
0
1
2
3
4
parameter positive, p<0.01
increase
stay
decrease
0
1
2
3
4
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)
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
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.
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam
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/
and via
http://ppswmm.ppsw.rug.nl/steglich/
(“stable URL”)
(current updates)
final version comes soon
RC33 Sixth International Conference on Social Science Methodology, 17-20 August 2004, Amsterdam