Tom A.B. Snijders
Department of Statistics and Measurement Theory / ICS
University of Groningen
Grote Kruisstraat 2/1, NL-9712 TS Groningen
Social networks are structures of relations between individuals.
The most common representation of a social network is a directed
graph, in which the arcs indicate for each of the ordered pairs
of individuals whether the relation in question (e.g., friendship)
is present or not.
Repeated measures on social networks represent a complicated
data structure, and few probability models and statistical methods
have been proposed for such data. Computer simulation offers
fruitful possibilities here, because it greatly expands the scope
of modeling beyond the models for which likelihood and other
functions can be analytically calculated. Continuous-time models
are more appropriate for modeling longitudinal social network data
than discrete-time models because of the endogenous feedback
processes involved in network evolution.
The probability models for the evolution of social networks
proposed here are based on the idea of actor-oriented modeling:
the vertices in the network represent actors who change their
relations in a process of optimizing their "utility function".
This function includes a random component to represent unexplained
change. The resulting model constitutes a continuous-time Markov
chain, and can be simulated in a straightforward manner. It can be
applied also when the actor-oriented interpretation is not so
obvious. The change in the network is modeled as the stochastic
result of network effects (reciprocity, transitivity, etc.) and
effects of covariates.
The main parameters of this model are weights in the utility
function, representing these various effects. The parameters
can be estimated using a stochastic version of the method of
moments, implemented by a Robbins-Monro-type algorithm.
An example is given of the evolution of the friendship network
in a group of university freshmen students.
Key words: simulation, evolution