  In order to accommodate the empirical fact that population structures are
rarely simple, modern studies of evolutionary dynamics allow for complicated
and highly-heterogeneous spatial structures. As a result, one of the most
difficult obstacles lies in making analytical deductions, either qualitative or
quantitative, about the long-term outcomes of evolution. The
"structure-coefficient" theorem is a well-known approach to this problem for
mutation-selection processes under weak selection, but a general method of
evaluating the terms it comprises is lacking. Here, we provide such a method
for populations of fixed (but arbitrary) size and structure, using easily
interpretable demographic measures. This method encompasses a large family of
evolutionary update mechanisms and extends the theorem to allow for asymmetric
contests to provide a better understanding of the mutation-selection balance
under more realistic circumstances. We apply the method to study social goods
produced and distributed among individuals in spatially-heterogeneous
populations, where asymmetric interactions emerge naturally and the outcome of
selection varies dramatically depending on the nature of the social good, the
spatial topology, and frequency with which mutations arise.
