Ove Frank

Sampling and estimation in large social networks

Abstract An unknown network is modelled by a directed or undirected graph having vertices of different kinds. Partial information is available concerning the vertex labels and the edge occurrences within a simple random sample of vertices. Using this information we find unbiased estimators and variance estimators of such graph parameters which can be given as dyad or triad counts. In particular, we give approximate formulae pertaining to large networks.

Two-Way Traffic on a Single Line of Railway

In this paper a mathematical treatment of two-way traffic on a single line of railway is given. The traffic capacity, the cycle times of the trains, and the number of trains needed to accomplish the transports are studied for certain regular traffic systems.

Cluster Inference by using Transitivity Indices in Empirical Graphs

Abstract A random graph model is introduced for similarities observed between the objects sampled from an unknown cluster structure. We investigate this model and show how some common transitivity indices in empirical graphs can be used for making statistical inferences about cluster structures.

A Survey of Statistical Methods for Graph Analysis

Sociometric problems involving empirical sociograms and more general networks have been one of the major sources of an increasing interest in statistical graph models. The well-known book by Harary, Norman, and Cartwright (1965) has contributed much to the mathematical modeling of social networks, and we now find numerous methodological articles appearing in the literature-for instance, in the Journal of Mathematical Sociology, Social Networks, and other professional journals. Nonetheless, while statistical testing and estimation problems are discussed in some reports, statistical issues are all too often ignored or treated in a very elementary way. Only recently have statistical models and methods been developed that begin to meet the need for proper handling of sampling variation, measurement errors, and other kinds of uncertainty in network data.

Survey sampling in graphs

Abstract The Horvitz-Thompson estimation theory is applied to snowball sampling and some other sampling procedures using a known or unknown graph structure in the survey population. In particular, simple graph-parameter estimators and variance estimators are obtained which are based on various kinds of partial information about the graph.

Integrating individual, relational and structural analysis.

Abstract The first part of this paper presents a statistical model which integrates individual, relational and network data, despite their different units of analysis. The model uses a stepwise approach to find the least number of parameters which adequately fit the data. The second part of this paper uses this model to analyze how the marital status of Torontonians is related to the kinship composition and social density of their intimate networks. It shows that kinship and friendship usually comprise independent social circles within these networks. The larger networks of married respondents tend to contain a higher proportion of kin, and consequently, to be more densely-knit. Yet single respondents tend to have more densely-knit clusters of intimates within their friendship-based networks. This is because marriage rarely joins the intimates of spouses.

Using centrality modeling in network surveys

In a well-known paper [Social Networks 1 (1979) 215] Linton Freeman clarified the importance of the centrality concept in network analysis. There are a variety of centrality measures available, and they are mainly used as descriptive statistics in various network studies. For instance, actor centrality measured by vertex degree captures those aspects of centrality that have an impact on contacts given or received by the actor. The approach taken here is to consider actor centrality as a latent property that manifests itself in generating a particular network structure, and in order to measure centrality we are bound to rely on observable features of this network. By borrowing ideas from recent link-tracing survey methodology, we illustrate how probabilistic network models with centrality parameters can be used to improve on estimators and predictors of various actor attributes related to centrality.

Exploratory statistical analysis of networks

Abstract We review standard multivariate statistical methods useful for exploring network data and discuss various problems related to statistical analysis and modeling. General methods are suggested for three main problem areas, namely whether there is a need for block models, whether there is dependence between dyads, and whether there is dependence between different networks. In particular, we illustrate the use of logit regression analysis in order to fit log-linear models. We comment on various themes in the literature that are important for future research on statistical graph modeling.

Transitivity in stochastic graphs and digraphs

Transitivity is a central concept for many relational structures, e.g. clusterings and partial orderings. Stochastic graph models which are used to describe uncertain relational structures can be tested for transitivity by using indices based on triad counts. The pure random variation of such counts and indices is investigated assuming simple stochastic models. Some earlier results on transitive triads in tournaments are generalized, and a modified version of a randomization model by Holland and Leinhardt which simplifies the required moment calculations is introduced.

MOMENT PROPERTIES OF SUBGRAPH COUNTS IN STOCHASTIC GRAPHS

Statistical analysis of stochastic graph models can be based on subgraph counts. General formulas are given which may be used to derive expected values, variances, and covariances of such graph statistics. Three different stochastic graph models are used as illustrations. Moments are given for certain dyad and triad counts in a graph with stochastically independent edge occurrences. Moments are also given for some subgraph count estimators which are based on a partially erased graph or a sampled subgraph.