Title: The Statistical Analysis
1The 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?
3Two 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
4Both 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
5Problem 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
6Problem 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
7Problem 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.
9- 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.
11- 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).
1216 possible states for a network consisting of
one dyad only. (assuming actor characteristics
and network ties to be dichotomous)
13For 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
14- 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.
15Possible changes of network ties (diagram
renders possible network micro steps only)
16Possible changes of behaviourals (diagram
renders possible behavioural micro steps only)
17All possible micro-transitions for a one-dyad
network
18- 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.
19- 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.
20- 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).
22Changes under control of the upper-left actor
red transitions are behavioural
changes, green transitions are network changes.
23Changes under control of the lower-right actor
(same colouring) One can see that an
individual actors scope of action is rela-tively
small.
24- 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!
25- 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.