The Statistical Analysis

1
The Statistical Analysis of the Dynamics of
Networks and Behaviour. An Introduction to the
Actor-based Approach. Christian Steglich
and Tom Snijders 2003/04.
2

• Situation investigated
• Given is a group of actors i?1,,N,
• this group is carrier of a
• meaningful social network x , and
• actors in this group show behaviour z .
• Behaviour and network positions
• of actors are interdependent.
• Problem investigated
• How does this interdependence come into
  existence?
• What are the dynamic mechanisms generating
• network ties and behaviour?

3
Two broad types of mechanisms that drive such
co-evolution
Selection mechanisms lead to changes in network
ties
Black actor reciprocates friendship
Influence mechanisms lead to changes in actor
characteristics
White actor adapts to (perceived) friend
4
Both types of mechanisms can occur in the same
process
Selection mechanism followed by influence
mechanism
Black actor reciprocates friendship
White actor adapts to (re- ciprocal) friend
Influence mechanism followed by selection
mechanism
White actor adapts to (per- ceived) friend
Black actor reciprocates friendship
5
Problem Due to sparse data, in many cases the
order of occurrence of these mechanisms cannot
be identified
When working with panel data, dynamics
between measurements are not known.
Black actor reciprocates friendship
White actor adapts to (re- ciprocal) friend
White actor adapts to (per- ceived) friend
Black actor reciprocates friendship
6
Problem but in many cases this order of
occurrence is of focal interest from the theory
perspective.
Black actor reciprocates friendship
White actor adapts to (re- ciprocal) friend
Theory A Relationships are governed by norms
of reciprocity. Adaptive behaviour occurs most
likely within close (reciprocated)
relationships.
White actor adapts to (per- ceived) friend
Black actor reciprocates friendship
7
Problem but in many cases this order of
occurrence is of focal interest from the theory
perspective.
Theory B Influence is strongest in
asymmetrical relationships. Homophily is a
strong deter- minant of starting a new
relationship.
Black actor reciprocates friendship
White actor adapts to (re- ciprocal) friend
White actor adapts to (per- ceived) friend
Black actor reciprocates friendship
8
Theory B Influence is strongest in
asymmetrical relationships. Homophily is a
strong deter- minant of starting a new
relationship.
?
Theory A Relationships are governed by norms
of reciprocity. Adaptive behaviour occurs most
likely within close (reciprocated)
relationships.
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• How to test such theories against each other?
• longitudinal data (we will be studying panel
  data),
• explicit modelling of the mechanisms driving
  co-evolution,
• fit model to data,
• infer relative strength of the different
  mechanisms from parameter estimates,
• draw conclusions about the theories, based on
  evidence for the mechanisms they postulate.

10

• Continuous time Markov process model
• state space consists of all possible
  configurations
• of network ties and behaviourals,
• individual decisions modelled by objective
  functions
• one for behavioural change (in micro steps),
• another one for network change (in micro
  steps)
• timing of individual decisions by rate
  functions
• again one for behavioural decisions,
• and another one for network decisions.

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• State space
• Pair (x,z)(t) contains
• adjacency matrix x and
• vector(s) of behaviourals z
• at time point t.
• Co-evolution is modelled by specifying transition
  probabilities between such states (x,z)(t1) and
  (x,z)(t2).

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16 possible states for a network consisting of
one dyad only. (assuming actor characteristics
and network ties to be dichotomous)
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For the simplest case of dichotomous ties and one
dichotomous actor characteristic, the cardinality
of the state space increases quickly with the
number of actors
Some numbers for illustration
n 2 3 4 5 6 7 8 16 512 64K 32M 64
G 512T 16E
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• Transitions between states
• Not all possible transitions (x,z)(t1) ?(x,z)(t2)
• are modelled, but only micro steps are
• 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.
• Observed transitions are more complex -- they are
  inter-
• preted as resulting from a sequence of such micro
  steps.

15
Possible changes of network ties (diagram
renders possible network micro steps only)
16
Possible changes of behaviourals (diagram
renders possible behavioural micro steps only)
17
All possible micro-transitions for a one-dyad
network
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• Actor based modelling
• The modelled transitions (x,z)(t1) ?(x,z)(t2)
• are results of individual decision making.
• network micro step
• actor i maximises value of his
  network-behavioural neighbourhood by changing
  tie to actor j.
• behavioural micro step
• actor i maximises a similar value of his
  network-behavioural neighbourhood by changing
  his behavioural score.

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• Actor based modelling
• The value of network-behavioural neighbourhood
  is operationalised by satisfaction measures
• satisfaction of actor i from changing the
  network
• tie to actor j
• f deterministic satisfaction measure,
• e random distortion of convenient choice.
• similar (but separate) model for satisfaction
• with behavioural decisions.

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• Actor based modelling
• The deterministic part f of the satisfaction
  measure consists of the following components
• a function measuring utility
• (based only on resulting network configuration),
• a function measuring endowment effects
• (based on current and resulting network),
• a function measuring reinforcement learning
• (also based on current and resulting network).

21

• Actor based modelling
• The probabilistic part e of the satisfaction
  measure is chosen as i.i.d. of extreme value type
  I
• this way, the choice probabilities can be
  expressed as
• (for network decisions,
• behavioural decisions analogous).

22
Changes under control of the upper-left actor
red transitions are behavioural
changes, green transitions are network changes.
23
Changes under control of the lower-right actor
(same colouring) One can see that an
individual actors scope of action is rela-tively
small.
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• Interpretation of parameter estimates
• Rate function parameters indicate the speed of
  the
• respective evolution process.
• positive parameter attached to an effect means
  quicker
• changes in the process when the effect is
  present.
• Objective function parameters indicate the
  actors
• preferences.
• positive parameter attached to an effect means
  a higher preference
• of the actor for a decision in which the effect
  is present.
• Nota bene parameter estimates do NOT indicate
  the
• network-behavioural co-evolution from a macro
  perspective!

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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.