Akademisk avhandling

som för avläggande av filosofie
doktorsexamen

vid Stockholms universitet

offentligen försvaras i

hörsal 3, hus B, Södra huset,
Frescati

fredagen den 31 oktober 1997 kl 13.00

av

**Ingegerd Jansson**

fil lic

Statistiska institutionen

Stockholms universitet

**ABSTRACT**

A social network
is formed by a collection of individuals and the contacts or relations
between them. The focus is here on relations defined by sociometric choices,
implying that the individuals in the network have been required to choose
among the other individuals in the group according to some criteria.

Some statistical models
of the sociometric choice structure in a social network are presented.
A recurrent assumption in the models is the assumption that there exists
latent or underlying structures in a network, structures that can not be
directly observed. The models also have in common the assumption of independence
or conditional independence between entities of the network, assumptions
not uncommon in statistical network models. These assumptions are often
crucial for the possibility to find simple ways of estimating models. Here
an attempt is made to find probabilistic models where independence is an
ingredient but which also allow for structure.

The models are
illustrated and evaluated by application to data where the sociometric
choices are based on friendship or cooperation. The large number of networks
available makes it possible to find and evaluate empirical distributions
of the test statistics.

The simplest model
presented assumes dyad independence. Possible simplifications by choice
or edge independence are also discussed. It appears that dyad independence
can not be ruled out, but that choice independence is inappropriate.

In order to model the
transitive structure of a network, a model is introduced that is based
on a random choice structure and an unknown underlying clique structure.
Two different approaches for estimating the clique structure are discussed,
the maximum likelihood approach and the Bayesian approach. The empirical
results show no large deviation between model and observations.

A model of popularity
structure is introduced where popularity is viewed as a latent attribute
of the individuals in the network. The group of individuals is assumed
to have a latent popularity structure, composed of individuals from three
popularity groups. It is shown how the popularity structure can be estimated
and how latent popularity can be considered in combination with manifest
individual attributes.

A model is also
presented for the situation when two sociometric relations are measured
on one set of individuals. The model assumes that there exists a latent
network structure which can not be observed directly. When the relations
are measured on the network, deviations from the underlying structure might
occur with some probability. Approximate estimates of model parameters
are given. Empirical findings suggest that the birelational latent model
is sufficiently accurate for the data, but that there appear to be convergence
problems, possibly due to the relatively large number of parameters in
the model.

ISBN 91-7153-665-5

*Last update: 990916/CE*