Slide1: Analysing the co-evolution of social networks
and “behavioural” dimensions with SIENA
Christian Steglich University of Groningen
Tom Snijders University of Groningen
Patrick West University of Glasgow
Andrea Knecht Utrecht University SIENA workshop Groningen, 8-11 January,2007
Funded by The Netherlands Organization for Scientific Research (NWO) under grant 401-01-550
Slide2: Some notation & clarification
Social networks:
Tie variables
“Behavioural” dimensions:
...can be any changeable dependent actor variable z i :
overt behaviour • attitudes
cognitions • ... a1 a2 a4 a3 a5
Slide3: Social network dynamics often depend on actors’ characteristics…
– patterns of homophily:
interaction with similar others can be more rewarding than interaction with dissimilar others
– patterns of exchange:
selection of partners such that they complement own abilities
…but also actors’ characteristics can depend on the social network:
– patterns of assimilation:
spread of innovations in a professional community
pupils copying ‘chic’ behaviour of friends at school
traders on a market copying (allegedly) successful behaviour of competitors
– patterns of differentiation:
division of tasks in a work team
Slide4: Example 1: (Andrea Knecht, 2003/04)
Data on the co-evolution of petty delinquency (graffiti, fighting, stealing,
breaking something, buying illegal copies) and friendship among first-grade pupils at Dutch secondary schools (“bridge class”).
125 school classes
4 measurement points
Questions to be addressed:
Is petty crime a dimension that plays a role in friendship formation?
Is petty crime a habit that is acquired by peer influence?
The following slides show how the type of data “look like”
that we are analysing.
Note: we analyse panel data – in principle,
continuous-time data is easier to
analyse – but the methods are not
yet implemented.
Slide5: Friendship ties inherited from primary school
girls yellow boys green
Slide6: 1st wave: August/September 2003
node size indicates strength of delinquency
Slide7: 2nd wave: November/December 2003
Slide8: 3rd wave: February/March 2004
Slide9: 4th wave: May/June 2004
Slide10: What do these pictures tell us?
– there is segregation of the friendship network according to gender (although not as strong as in other classes)
– delinquency is stronger among boys than among girls
Questions unanswered:
– to what degree can social influence and social selection processes account for the observed dynamics?
More general…
Slide11: How to analyse this?
structure of complete networks is complicated to model
additional complication due to the interdependence with behavior
and on top of that often incomplete observation (panel data) beh(tn) beh(tn+1) net(tn) net(tn+1) persistence (?) selection influence persistence (?)
Slide12: Agenda for this talk:
Presentation of the stochastic modelling framework
An illustrative research question (Example 2)
Data for Example 2
Software
Analysis
Interpretation of results
Summary
Slide13: Brief sketch of the stochastic modelling framework (1)
Stochastic process in the space of all possible network-behaviour configurations
(huge!)
First observation of the network as the process’ starting value. beh net For the simplest case of dichotomous ties and one dichotomous actor characteristic, the cardinality of the state space increases at a squared exponential rate with the number of actors:
Slide14: 16 possible states for a network consisting of one dyad only.
(assuming actor characteristics to be dichotomous)
Slide15: Brief sketch of the stochastic modelling framework (2)
Change is modelled as occurring in continuous time.
Network actors drive the process: individual decisions.
two domains of decisions*:
decisions about network neighbours (selection, deselection),
decisions about own behaviour.
per decision domain two submodels:
When can actor i make a decision? (rate function)
Which decision does actor i make? (objective function)
Technically: Continuous time Markov process.
Beware: model-based inference!
* assumption: conditional independence, given the current state of the process.
Slide16: How does the model look like?
State space
Pair (x,z)(t) contains adjacency matrix x and vector(s)
of behavioural variables z at time point t.
Stochastic process
Co-evolution is modelled by specifying transition probabilities
between such states (x,z)(t1) and (x,z)(t2).
Continuous time model
invisibility of to-and-fro changes in panel data poses no problem,
evolution can be modelled in smaller units (‘micro steps’).
Observed changes are quite complex – they are interpreted as resulting from a sequence of micro steps.
Slide17: Micro steps that are modelled explicitly
network micro steps:
(x,z)(t1) and (x,z)(t2) differ in one tie variable xij only.
behavioural micro steps:
(x,z)(t1) and (x,z)(t2) differ (by one) in one behavioural score variable zi only.
Actor-driven model
Micro steps are modelled as outcomes of an actor’s decisions;
these decisions are conditionally independent, given the current state of the process.
Schematic overview of model components
Slide18: 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.
Network micro step / network decision by actor i
Choice options
change tie variable to one other actor j
change nothing
Maximize objective function + random disturbance 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 Choice probabilities resulting from distribution of e are of multinomial logit shape 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
Slide19: Network micro step / network decision by actor i
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
Slide20: Other possible effects to include in the network objective function:
(from Steglich, Snijders & Pearson 2004)
Slide21: Possible changes of network ties
in the dyad case:
(diagram renders possible network micro steps only)
Slide22: Behavioural micro step by actor i
Choice options
increase, decrease, or keep score on behavioural variable
Maximize objective function + random disturbance
Choice probabilities analogous to network part Assume independence also of the network random part Objective function is different from the network objective function
Slide23: Possible effects to include in the behavioural objective function(s):
(from Steglich, Snijders & Pearson 2004)
Slide24: Possible changes of behaviour in the dyad case:
(diagram renders possible behavioural micro steps only)
Slide25: Total process model
Transition intensities (‘infenitesimal generator’) of Markov process:
Here l = waiting times, d = change in behavioural,
z(i,d) = behavioural vector after change.
Together with starting value, process model is fully defined.
Parametrisation of process implies equilibrium distribution, process is a ‘drift’ from 1st observation towards regions of high probability under this equilibrium.
Slide26: Modelling selection and influence
Influence and selection are based
on a measure of behavioural similarity
Similarity of actor i to network neighbours :
Actor i has two ways of increasing friendship similarity:
by choosing friends j who behave the same (network effect):
by adapting own behaviour to that of friends j (behaviour effect):
assimilation
(social influence) homophily
(social selection) i j i i i i i i i j j j j j j j
Slide27: 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).
Slide28: Example (2):
A set of illustrative research questions:
To what degree is music taste acquired via friendship ties?
Does music taste (co-)determine the selection of friends?
Data: social network subsample of the West of Scotland 11-16 Study
(West & Sweeting 1996)
three waves, 129 pupils (13-15 year old) at one school
pupils named up to 6 friends
Take into account previous results on same data (Steglich, Snijders & Pearson 2004):
What is the role played by alcohol consumption in both friendship formation and the dynamics of music taste?
Slide29: 43. Which of the following types of music do you like listening to? Tick one or more boxes.
Rock Indie
Chart music Jazz
Reggae Classical
Dance 60’s/70’s
Heavy Metal House
Techno Grunge
Folk/Traditional Rap
Rave Hip Hop
Other (what?)…………………………………. Music question: 16 items Before applying SIENA: data reduction to
the 3 most informative dimensions
Slide30: scale CLASSICAL scale ROCK scale TECHNO
Slide31: 32. How often do you drink alcohol?
Tick one box only.
More than once a week
About once a week
About once a month
Once or twice a year
I don’t drink (alcohol) 5
4
3
2
1 Alcohol question: five point scale General: SIENA requires dichotomous networks
and behavioural variables on an ordinal scale.
Slide32: Some descriptives: average dynamics of the four behavioural variables global dynamics of friendship ties (dyad counts)
Slide33: Software:
The models briefly sketched above are instantiated in the SIENA program. Optionally, evolution models can be estimated from given data, or evolution processes can be simulated, given a model parametrisation and starting values for the process.
SIENA is implemented in the StOCNET program package, available at http://stat.gamma.rug.nl/stocnet (release 14-feb-05).
Currently, it allows for analysing the co-evolution of one social network (directed or undirected) and multiple behavioural variables.
Slide34: Identification of data sourcefiles Recoding of variables and identification of missing data Specifying subsets of actors for analyses
Slide36: Data specification: insert data into the model’s “slots”.
Slide37: Model specification: select parameters to include for network evolution.
Slide38: Model specification: select parameters to include for behavioural evolution.
Slide39: Model specification: some additional features.
Slide40: Model estimation: stochastic approximation of optimal parameter values.
Slide41: Network objective function:
intercept:
outdegree
network-endogenous:
reciprocity
distance-2
covariate-determined:
gender homophily
gender ego
gender alter
behaviour-determined:
beh. homophily
beh. ego
beh. alter
Rate functions were kept as simple as possible (periodwise constant). Analysis of the music taste data: Behaviour objective function(s):
intercept:
tendency
network-determined:
assimilation to neighbours
covariate-determined:
gender main effect
behaviour-determined:
behaviour main effect
“behaviour” stands shorthand for the three music taste dimensions and alcohol consumption.
Slide42: Results: network evolution
Ties to just anyone are but costly. Reciprocated ties are valuable (overcompensating the costs). There is a tendency towards transitive closure. There is gender homophily:
alter
boy girl
boy 0.38 -0.62
ego
girl -0.18 0.41
table gives gender-related contributions to the objective function There is alcohol homophily:
alter
low high
low 0.36 -0.59
ego
high -0.59 0.13
table shows contributions to the objective function for highest / lowest possible scores There is no general homophily according to music taste:
alter
techno rock classical
techno -0.06 0.25 -1.39
ego rock -0.15 0.54 -1.31
classical 0.02 0.50 1.73
table renders contributions to the objective function for highest possible scores & mutually exclusive music tastes
Slide43: Results: behavioural evolution
Assimilation to friends occurs:
on the alcohol dimension,
on the techno dimension,
on the rock dimension. There is evidence for mutual exclusiveness of:
listening to techno and listening to rock,
listening to classical and drinking alcohol. The classical listeners tend to be girls.
Slide44: Summary (1)
Does music taste (co-)determine the selection of friends?
Somewhat.
There is no music taste homophily
(possible exception: classical music).
Listening to rock music seems to coincide with popularity,
listening to classical music with unpopularity.
To what degree is music taste acquired via friendship ties?
It depends on the specific music taste:
Listening to techno or rock music is ‘learnt’ from peers,
listening to classical music is not – maybe a ‘parent thing’?
Slide45: Summary (2)
What is the role played by alcohol consumption in friendship formation?
There is homophilous selection going on:
Friends select each other based on similarity in alcohol consumption.
What is the role played by alcohol consumption in the dynamics
of music taste?
Only for the classical scale, an effect was found:
Drinking alcohol reduces the chances of listening to classical music (and vice versa).
Literature:
Christian Steglich, Tom Snijders, and Patrick West, 2006.
Applying SIENA: An illustrative analysis of the co-evolution of adolescents' friendship networks, taste in music, and alcohol consumption.
Methodology 2(1), 48-56.