In this work, we present a fully analytical, non-dimensional model to describe steady-state diffusive flow through a 3-D network. Through a combination of graph theory, linear algebra, and geometry, the model explicitly relates the network’s topology and the morphology of its channels to the material’s effective conductivity. The effective conductivity estimates calculated by the model exhibit good agreement with those from electrochemical fin theory and finite element analysis, but are computed 1.5 and 6 orders of magnitude faster, respectively. In addition, the theory explicitly relates a number of morphological and topological parameters directly to the conductivity, whereby the distributions that characterize the structure are readily available for further analysis, e.g., using network science. We then present an approach for extending the model to account for (1) n interacting flows and (2) for the single-flow case we demonstrate algorithmic approaches for (a) accounting for the influence of reactions at steady state and (b) examining the transient flow properties of the network. The non-dimensional metrics that follow from each model provide a means of comparing different materials and, furthermore, each extension, to a large extent, preserves the relative speed and ease-of-use inherent in the initial steady-state model, thus making them attractive for guiding transport-relevant hypotheses in materials design efforts.