Abstract: The self-assembly of individual protein monomers into larger structures is a fundamental biophysical process of considerable functional importance; unfortunately, not all such structures are functional.  Pathological protein aggregation is a critical factor in the etiology of diseases ranging from Alzheimer’s and Lewy Body Dementia to cataract and type-II diabetes, and remains imperfectly understood.  Disease-associated aggregates can vary radically in size and structure, from small annular oligomers to long, highly ordered amyloid fibrils and the disordered structures seen in cataract; aggregation states are also polymorphic, with the same protein sequence leading to different states under different conditions, and different protein sequences leading in some cases to the same states.  From a theoretical standpoint, modeling protein aggregation is further complicated by the large system sizes and timescales involved, with aggregation taking place in systems of large numbers of protein monomers over periods of hours, days, or even years.  This places most such processes out of reach of workhorse techniques such as atomistic molecular dynamics, motivating alternative approaches.  In this talk, I describe the modeling of protein aggregation using Network Hamiltonian Models (NHMs), a broad framework for modeling the formation of complex topological structures.  Formally, the NHMs can be seen as physical interpretations of exponential family random graph models (ERGMs), discrete exponential families of distributions on graph sets that provide a highly general framework for specifying both heterogeneity and dependence among edges.  Originally developed for their inferential properties, the ERGMs enjoy a growing statistical and computational literature that can be exploited to facilitate scalable modeling of physical systems.  As I show, fairly simple families of NHMs can recapitulate the topologies of all so-far identified classes of amyloid fibrils, and kinetic extensions of NHMs can efficiently model fibril formation from free monomers to mature fibrils on commodity hardware.  Simple NHMs can also recapitulate transient aggregates of gamma-D crystallin, as well as disaggregated, gel-like, and oligomeric states; proof-of-principle analyses suggest that NHMs may also be useful for inferring fibril topology from kinetic data (e.g., ThT fluorescence).  Time permitting, I will also discuss current work on terms to capture the effect of unmodeled kinetic degrees of freedom on aggregation states.