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The theoretical core of Network Science is the study of network models. The space of network models, and the properties of individual models therein, are explored in this dissertation. The projects presented are unified by the themes of entropy and dynamics. in the context of random graphs. In the first project, we study maximum-entropy models of unlabeled networks, comparing them to their labeled analogues. We find that stark differences sometimes arise between the two, and that simply removing labels after sampling from a labeled network model can induce significant statistical biases. In the second project, we scan through the parameter-spaces of various popular network models and examine graph distances between pairs of i.i.d.'ly sampled graphs therefrom. Depending on the choice of graph distance metric, different ensemble characteristics and structural transitions are identifiable. This procedure allows the use of graph distance metrics to study network ensembles, and vice versa. In the third project, we introduce a class of dynamic network models governed by dynamic nodewise variables, directly tuning the rate of change of network structure and rate of change of node-intrinsic variables. We find that depending on the pace of hidden-variable dynamics and link-dynamics, substantial deviations of network snapshots can arise in comparison with the associated static hidden-variables models. In the fourth project, we develop canonical ensembles of temporal network models, imposing constraints on combinations of structural and dynamical variables. We establish mappings to time-inhomogeneous binary Markovian stochastic processes, deriving the associated transition probabilities and evaluating the implied constraint consistency conditions. We derive dynamic analogues of the Erdős-Rényi model and of the soft configuration model, as well as hyperparametric variants thereof. Altogether, the work in this dissertation takes substantial steps forward in the ongoing endeavor of exploration and elucidation of the space of network models of various kinds and across choices of sample space.