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

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?

Black actor reciprocates friendship White actor adapts to (perceived) friend Selection mechanisms lead to changes in network ties: Influence mechanisms lead to changes in actor characteristics: Two broad types of mechanisms that drive such co-evolution:

Black actor reciprocates friendship Selection mechanism followed by influence mechanism: Influence mechanism followed by selection mechanism: White actor adapts to (per- ceived) friend White actor adapts to (re- ciprocal) friend Black actor reciprocates friendship Both types of mechanisms can occur in the same process:

Black actor reciprocates friendship White actor adapts to (per- ceived) friend White actor adapts to (re- ciprocal) friend Black actor reciprocates friendship 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 (per- ceived) friend White actor adapts to (re- ciprocal) friend Black actor reciprocates friendship Problem: …but in many cases this order of occurrence is of focal interest from the theory perspective. Theory A: Relationships are governed by norms of reciprocity. Adaptive behaviour occurs most likely within close (reciprocated) relationships.

Black actor reciprocates friendship White actor adapts to (per- ceived) friend White actor adapts to (re- ciprocal) friend Black actor reciprocates friendship 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.

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. ?

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.

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.

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)(t 1 ) and (x,z)(t 2 ).

16 possible states for a network consisting of one dyad only. (assuming actor characteristics and network ties to be dichotomous)

n # K32M64G512T16E 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:

Transitions between states: Not all possible transitions (x,z)(t 1 )  (x,z)(t 2 ) are modelled, but only “micro steps” are: – network micro step: (x,z)(t 1 ) and (x,z)(t 2 ) differ in one tie x ij only. –behavioural micro step: (x,z)(t 1 ) and (x,z)(t 2 ) differ in one behavioural score z i only. Observed transitions are more complex -- they are inter- preted as resulting from a sequence of such micro steps.

Possible changes of network ties: (diagram renders possible network micro steps only)

Possible changes of behaviourals: (diagram renders possible behavioural micro steps only)

All possible micro- transitions for a one-dyad network:

Actor based modelling: The modelled transitions (x,z)(t 1 )  (x,z)(t 2 ) 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.

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,  random distortion of convenient choice. – similar (but separate) model for satisfaction with behavioural decisions.

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).

Actor based modelling: The probabilistic part  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).

Changes under control of the upper-left actor: red transitions are behavioural changes, green transitions are network changes.

Changes under control of the lower-right actor: (same colouring) One can see that an individual actor’s scope of action is rela- tively small.

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 actor’s 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!

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.