This paper presents a new, systematic approach to analyzing network topologies. We first introduce a series of probability distributions specifying all degree correlations within d-sized subgraphs of a given graph G. Using this series, we can quantitatively evaluate how close synthetic topologies are to G, construct graphs that accurately reproduce the values of commonly-used graphmetrics of G, and provide a rigorous basis for capturing any future metrics that may be of interest. The d = 0 and d = 1 cases reduce to the known classical (Erdõs-Rényi) random graphs and random graphs with prescribed degree distributions respectively. However, recent research shows that simply reproducing a graph's degree distribution is insufficient for capturing important properties of network topologies. Using our approach, we construct graphs for d = 0, 1, 2, 3 and demonstrate that these graphs reproduce, with increasing accuracy, important properties of measured and modeled Internet topologies. We find that the d = 2 case is sufficient for most practical purposes, while d = 3 essentially reconstructs the Internet AS- and router-level topologies exactly. Overall, the availability of a systematic method to analyze and synthesize topologies offers a significant improvement to the set of tools available to network topology and protocol researchers.