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

Extended form of the presentation given at the XXV Sunbelt Social Network Conference

Funded by The Netherlands Organization for Scientific Research (NWO) under grant 401-01-550

with an application to the dynamics of music taste, alcohol consumption and friendship

Some notation & clarification

Social networks:

Tie variables

“Behavioural” dimensions:

...can be any changeable dependent actor variable z i :

• overt behaviour • attitudes

• cognitions • ...

1 if the relation exists between actors and

0 otherwiseij

i jx

0 0 1 0 0

0 0 0 0 0

summarised in the 0 1 0 1 1

0 0 0 0 1

0 0 0 1 0

xadjacency matrix

a1

a2 a4

a3

a5

The individual are ij

x

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

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.

Friendship ties inherited from primary schoolgirls yellow boys green

1st wave: August/September 2003

node size indicates strength of delinquency

2nd wave: November/December 2003

3rd wave: February/March 2004

4th wave: May/June 2004

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…

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 (?)

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

Brief sketch of the stochastic modelling framework

- Stochastic process in the space of all possible network-behaviour configurations

(huge!)

- First observation as the process’ starting value.

- 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.

beh

net

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.

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

Timing of decisions Decision rules

Network evolution Network rate function Network decision rule

Behavioural evolution Behavioural rate function Behavioural decision rule

Timing of decisions / transitions

• Waiting times 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

net net net

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

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

net net

i

net net

i

k {1,...,N}

exp f ( , x, z, t, j)Pr(x(i j) | x, z)

exp f ( , 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

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

ij jijx x

ij jk ikjkx x x

• Other possible effects to include in the network objective function:

(from Steglich, Snijders & Pearson 2004)

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

beh beh beh

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

Assume independence also of the network random partObjective function is different from

the network objective function

• Other possible effects to include in the behavioural objective function(s):

(from Steglich, Snijders & Pearson 2004)

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):

i j

ij

z zsim :

ij ijjx sim

assimilation (social influence)

homophily (social selection)

i j

i

i

i

i

i

i

i

j

j

j

j

j j

j

Total process model

• Transition intensities (‘infenitesimal generator’) of Markov process:

Here= waiting times, = change in behavioural, z(i,) = 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.

net

i

beh

i

ˆ ˆ( x,z ),( x,z )

( x,z ),( x( i j ),z ) ( x,z ),( x,z i, )j i { 1,

i, i;

ˆ ˆPr(x(i j) | x, z) , if x x(i j) and z z for some i, j {1, ..., N}, i j;

ˆ ˆPr(z | x, z) , if x x and z z for some i {1, ..., N}, { 1, 1};

qq q

i 1}

ˆ ˆ , if x x and z z;

0 otherwise.

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).

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 12 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?

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

rap

dance

reggae

techno

househiphop

chart

grunge

rave

heavymtl

rock

classica

jazz

indie

sixty70s

folk_trd

scale CLASSICAL

scale ROCK

scale TECHNO

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.

Some descriptives:

0

0.5

1

1.5

2

2.5

3

3.5

wave 1 wave 2 wave 3

techno rock classical alcohol

average dynamics of the four behavioural variables

0

50

100

150

200

250

wave 1 wave 2 wave 3

asymmetric mutual

global dynamics of friendship ties (dyad counts)

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.

Identification of data sourcefiles

Recoding of variables and identification of missing data

Specifying subsets of actors for analyses

Data specification: insert data into the model’s “slots”.

Model specification: select parameters to include for network evolution.

Model specification: select parameters to include for behavioural evolution.

Model specification: some additional features.

Model estimation: stochastic approximation of optimal parameter values.

Network objective function:– intercept:

outdegree

– network-endogenous:reciprocitydistance-2

– covariate-determined:gender homophilygender egogender alter

– behaviour-determined:beh. homophilybeh. egobeh. 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.

parameter s.e. t-scoreoutdegree -1.89 0.29 -6.51reciprocity 2.34 0.12 20.08distance-2 -1.09 0.07 -14.89gender sim 0.80 0.12 6.72

alter -0.21 0.12 -1.73ego 0.24 0.11 2.17

techno sim 0.08 0.33 0.26alter 0.07 0.05 1.30ego -0.10 0.05 -1.93

rock sim 0.11 0.41 0.26alter 0.19 0.07 2.75ego -0.07 0.08 -0.92

classical sim 1.44 0.69 2.07alter 0.15 0.17 0.91ego 0.40 0.17 2.42

alcohol sim 0.83 0.27 3.08alter -0.03 0.04 -0.75ego -0.03 0.03 -0.85

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 -0.62ego girl -0.18 0.41

table gives gender-related contributions to the objective function

There is alcohol homophily:

alter low high

low 0.36 -0.59ego 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

Results: behavioural evolution

par. s.e. par. s.e. par. s.e. par. s.e.intercept -0.30 0.37 0.01 0.25 0.59 0.25 0.67 1.30assimilation 0.94 0.27 0.45 0.18 0.63 0.28 0.42 1.17gender -0.06 0.19 0.25 0.12 0.01 0.19 1.57 0.83techno 0.23 0.16 --- --- -0.25 0.09 -0.46 0.40rock 0.16 0.16 -0.34 0.10 --- --- 0.64 0.39classical -0.59 0.32 -0.13 0.23 -0.34 0.30 --- ---alcohol --- --- 0.07 0.10 -0.11 0.07 -1.03 0.34

alcohol techno rock classical

• 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.

Summary:

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’?

Check out the software at http://stat.gamma.rug.nl/stocnet/