Transitivity testing is considered for graphs and digraphs that are observed in their entirety. 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. Horvitz-Thompson estimators (and variance estimators for those estimators) for the triad counts are given. 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. 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. 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. Two local network attributes that we pay special attention to are the size and the density. A local network for an individual in a friendship network may be regarded as the network between the friends of this person. Throughout this thesis, we focus on the role of local networks.
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. Triads and transitivity are two concepts within the field of social network analysis that are closely related to each other. 1997 (English) Doctoral thesis, comprehensive summary (Other academic) Abstract
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