This article describes families of finite multigraphs with labeled or unlabeled edges and vertices. It shows how size and complexity vary for different types of equivalence classes of graphs defined by ignoring only edge labels or ignoring both edge and vertex labels. Complexity is quantified by the distribution of edge multiplicities, and different complexity measures are discussed. Basic occupancy models for multigraphs are used to illustrate different graph distributions on isomorphism and complexity. The loss of information caused by ignoring edge and vertex labels is quantified by entropy and joint information that provide tools for studying properties of and relations between different graph families.

This thesis is concerned with multigraphs and their complexity which is defined and quantified by the distribution of edge multiplicities. Two random multigraph models are considered.  The first model is random stub matching (RSM) where the edges are formed by randomly coupling pairs of stubs according to a fixed stub multiplicity sequence. The second model is obtained by independent edge assignments (IEA) according to a common probability distribution over the edge sites. Two different methods for obtaining an approximate IEA model from an RSM model are also presented.

In Paper I, multigraphs are analyzed with respect to structure and complexity by using entropy and joint information. The main results include formulae for numbers of graphs of different kinds and their complexity. The local and global structure of multigraphs under RSM are analyzed in Paper II. The distribution of multigraphs under RSM is shown to depend on a single complexity statistic. The distributions under RSM and IEA are used for calculations of moments and entropies, and for comparisons by information divergence. The main results include new formulae for local edge probabilities and probability approximation for simplicity of an RSM multigraph. In Paper III, statistical tests of a simple or composite IEA hypothesis are performed using goodness-of-fit measures. The results indicate that even for very small number of edges, the null distributions of the test statistics under IEA have distributions that are  well approximated by their asymptotic χ²-distributions. Paper IV contains the multigraph algorithms that are used for numerical calculations in Papers I-III.

Snowball sampling, where existing study subjects recruit further subjects from amongtheir acquaintances, is a popular approach when sampling from hidden populations.Since people with many in-links are more likely to be selected, there will be a selectionbias in the samples obtained. In order to eliminate this bias, the sample data must beweighted. However, the exact selection probabilities are unknown for snowball samplesand need to be approximated in an appropriate way. This paper proposes differentways of approximating the selection probabilities and develops weighting techniquesusing the inverse of the selection probabilities. Some numerical examples for smallgraphs and simulations on larger networks are provided to compare the efficiencyof the weighting techniques. The simulation results indicate that the suggested re-weighted estimators should be preferred to traditional estimators with equal sampleweights for the initial snowball sampling waves.

A commonly used sampling design in economic valuation studies is on-sitesampling. If this sampling design is used, the sampling inclusion probabil-ities may be correlated with respondents’ valuations, invalidating welfaremeasures derived from estimates of the probit model. This problem is re-ferred to a length-bias, a problem discovered in other fields of applicationof statistics.The first paper in this thesis outlines different application fields thathave length-bias problems and the suggested model solutions in the litera-ture are presented.The second paper of this thesis proposes a model based on the bivariateordinal probit, a model that can be used to analyze binary choice CV datagathered by on-site sampling. The models is presented, the log-likelihoodis derived, and the properties of the MLE’s are illustrated using a smallsimulation study. The simulation results show the proposed estimator tobe an interesting alternative.