Akademisk avhandling

som för avläggande av filosofie
doktorsexamen

vid Stockholms universitet

offentligen försvaras i

hörsal 4, hus B, Södra huset,
Frescati

tisdagen den 27 maj 1997 kl 10.00

av

**Martin Karlberg**

fil lic

Statistiska institutionen

Stockholms universitet

**ABSTRACT**

Triads and transitivity
are two concepts within the field of social network analysis that are closely
related to each other. We study some estimation and testing problems related
to those concepts, using the tools of graph theory; the results obtained
could be applied to graphs representing other kinds of data than social
relations.

Throughout this
thesis, we focus on the role of local networks. A local network for an
individual in a friendship network may be regarded as the network between
the friends of this person. Two local network attributes that we pay special
attention to are the size and the density. The local size of a vertex is
defined so that it counts the number of transitive relationships in which
that vertex is involved; this definition is not undisputable in the digraph
situation, since not all edges in the local network are counted using that
definition. We define the local density of a vertex in such a way, that
its expected value is equal to the expected overall density of the network
under some commonly used simple random graph and random digraph models.

When dealing with
triad count estimation, we consider the situation when we have observed
information about a probability sample of vertices in a graph or digraph;
we let the amount of information observed for each vertex range from the
vertex degree to the entire local network of that vertex. Horvitz-Thompson
estimators (and variance estimators for those estimators) for the triad
counts are given. A main result is that when local networks without information
on the identities of the vertices in that network are observed, the triad
counts may be expressed as sums of vertex attributes; this greatly facilitates
the estimation based on vertex sampling designs more complex than simple
random sampling.

Transitivity testing
is considered for graphs and digraphs that are observed in their entirety.
We study two different kinds of transitivity tests; tests based on the
counts of transitive and intransitive triads and triples, and tests based
on the mean local density over all vertices. The null hypothesis used is
that the graphs and digraphs observed have been generated according to
some conditional uniform random graph model that does not imply a high
degree of transitivity. The powers of the tests against random graph distributions
that generate highly transitive graphs are examined in simulation studies.
In other simulation studies, the tests are applied to a large set of school
class sociograms. When (undirected) graphs are considered, the test based
on the proportion of transitive triads out of the non-vacuously transitive
ones is found to be best at detecting transitivity; for digraphs, the test
based on the difference between the mean local density and the overall
network density is the best transitivity detector.

ISBN 91-7153-595-0

*Last update: 990916/CE*