Analysing the co-evolution of social networks

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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
with an application to the dynamics of music
taste, alcohol consumption and friendship
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
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• Some notation clarification
• Social networks
• Tie variables
• Behavioural dimensions
• ...can be any changeable dependent actor variable
  z i
• overt behaviour attitudes
• cognitions ...

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a5
a1
a2
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• 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

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• 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.

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Friendship ties inherited from primary
school girls yellow boys green
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1st wave August/September 2003 node size
indicates strength of delinquency
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2nd wave November/December 2003
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3rd wave February/March 2004
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4th wave May/June 2004
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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
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persistence (?)
beh(tn)
beh(tn1)
selection
influence
net(tn)
net(tn1)
persistence (?)

• 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)

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• 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

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• 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
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• 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).

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• 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
  actors 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
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• 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

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

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
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
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• 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
j
j
j
j
i
i
k
k
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• Other possible effects to include in the network
  objective function
• (from Steglich, Snijders Pearson 2004)

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

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• 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
i
j
homophily (social selection)
i
i
j
j
i
i
j
j
assimilation (social influence)
i
i
j
j
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• 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.

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• 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).

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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?
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Music question 16 items
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 ? 60s/70s ? Heavy
Metal ? House ? Techno ? Grunge
? Folk/Traditional ? Rap ?
Rave ? Hip Hop ? Other
(what?).
Before applying SIENA data reduction to the 3
most informative dimensions
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scale ROCK
scale CLASSICAL
scale TECHNO
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Alcohol question five point scale
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 dont drink (alcohol) ?
5 4 3 2 1
General SIENA requires dichotomous networks
and behavioural variables on an ordinal scale.
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Some descriptives
average dynamics of the four behavioural variables
global dynamics of friendship ties (dyad counts)
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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.
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Recoding of variables and identification of
missing data
Specifying subsets of actors for analyses
Identification of data sourcefiles
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Data specification insert data into the models
slots.
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Model specification select parameters to include
for network evolution.
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Model specification select parameters to include
for behavioural evolution.
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Model specification some additional features.
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Model estimation stochastic approximation of
optimal parameter values.
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Analysis of the music taste data

• 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).

• 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.

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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 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
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
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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.

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• 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/