Abstract: How does the topology of a configuration space behave as we increase the number of points in a configuration? In the case of the ordered configuration space of a non-compact manifold, Church, Ellenberg, and Farb proved that the Betti numbers are eventually polynomial and that the multiplicities of the irreducible symmetric group representations appearing in homology eventually stabilize, though in most cases these stable multiplicities remain unknown. For the ordered configuration space of a graph, the situation is much hairier, as the corresponding stabilizing structure is non-Noetherian. Using ideas coming from twisted algebra and factorization homology to overcome this obstacle, we give a complete asymptotic calculation of the Betti numbers of the ordered configuration space of a graph. Our computations show that asymptotically these groups are torsion free; moreover, they yield the asymptotic multiplicities of the irreducible symmetric group representations appearing in homology. This talk is based on joint work with Ben Knudsen and Louis Hainaut.