Amyloid fibrils are locally ordered protein aggregates that self-assemble under a variety of physiological and in vitro conditions. Their formation is of fundamental interest as a physical chemistry problem and plays a central role in Alzheimer's disease, Type II diabetes, and other human diseases. As the number of known amyloid fibril structures has grown, the need has arisen for a nomenclature for describing and classifying fibril types, as well as a theoretical description of the physics that gives rise to the self-assembly of these structures. Here, we introduce a systematic nomenclature and coarse-graining methodology for describing the topology of fibrils and other protein aggregates, along with a computational methodology for simulating protein aggregation. Both have mathematical underpinnings in graph theory and statistical mechanics and are consistent with available experimental data on the fibril structure and aggregation kinetics. Our graph representation of the fibril topology enables us to define a network Hamiltonian based on connectivity patterns among monomers rather than detailed intermolecular interactions, greatly speeding up the simulation of large ensembles. Our simulation strategy is capable of recapitulating the formation of all currently known amyloid fibril topologies found in the Protein Data Bank, as well as the formation kinetics of fibrils and oligomers.