  We present a technique for calculating free-energy profiles for the
nucleation of multicomponent structures that contain as many species as
building blocks. We find that a key factor is the topology of the graph
describing the connectivity of the target assembly. By considering the designed
interactions separately from weaker, incidental interactions, our approach
yields predictions for the equilibrium yield and nucleation barriers. These
predictions are in good agreement with corresponding Monte Carlo simulations.
We show that a few fundamental properties of the connectivity graph determine
the most prominent features of the assembly thermodynamics. Surprisingly, we
find that polydispersity in the strengths of the designed interactions
stabilizes intermediate structures and can be used to sculpt the free-energy
landscape for self-assembly. Finally, we demonstrate that weak incidental
interactions can preclude assembly at equilibrium due to the combinatorial
possibilities for incorrect association.
