In this thesis, we study dynamic (temporal) graphs through simulation and the modeling of static graphs, focusing on graph attributes and community structures.

We first consider dynamic networks that add nodes and edges over time. We study the distribution of maximal clique sizes in graphs generated by the preferential attachment model. Our goal is to preserve average maximal clique size over time. We determine the parameterizations of the preferential attachment model that achieve this goal and we introduce adaptive parameterization as an alternate means of achieving our goal. In addition, we introduce a new algorithm to keep track of maximal cliques in a graph. We show that for sparse graphs, our algorithm performs favourably compared to a widely used maximal clique enumeration algorithm, in terms of memory and time.

We then turn our attention to preserving graph features through the modeling of a static graph. First, we visualize community structures through maximal clique graphs and clique percolation graphs. These community structures, along with other nodal and dyadic covariates, are used to build a logistic regression model with the lasso penalty that explains the existence of edges. Due to clustering of nodes, we consider a quasibinomial family for the model, along with the standard binomial family. The competing models are compared using a goodness of fit statistic based on eigenvalues of the graph Laplacian (Shore and Lubin 2015). We generalize the GOF approach to use any nodal or dyadic covariate to assess how well particular graph structures are preserved. We have used our model successfully to fit a network constructed from mobility data in China and we have illustrated the use of our generalized GOF statistic.

Finally, we generalize the modeling to Bayesian approaches, including empirical Bayes methods and fully Bayesian methods. In the fully Bayesian methods, we use well-known priors that have not been explored much in the context of a network setting. We assess the performance of these models for retaining certain graph structures in graphs simulated from the fitted models and we demonstrate that performance of the various models is robust to the choice of various prior distributions.