(A) The critical b/c ratio for public goods production to be favored for various graph structures, plotted against the diffusion rate λ. These results are derived from Equation 2 and the expressions for ϕ0 in Table 1. For a well-mixed population (complete graph), cooperation is favored if and only if b/c > 1 + λ; for other graph structures, the critical b/c ratio is a increasing, convex function of λ. In general, the conditions for cooperation become increasingly stringent with both the degree and the dimensionality of the graph. (B) Our results are confirmed by simulations on a 15 × 15 periodic triangular lattice with uniform edge weights and cost c = 5%. The critical b/c threshold from Equation 2 is plotted in black. A plus (+) indicates that the frequency of cooperator fixation exceeded the frequency of defector fixation (ρC > ρD), while a minus (−) indicates the opposite. In all cases the results were statistically significant (two-proportion pooled z-test, p<0.05). (C) Adding decay of rate d effectively reduces both λ and b by the factor 1/(1 + d), reflecting greater locality in sharing but reduced overall concentration of public good. On a graph of b/c versus λ, this moves each point (b/c, λ) along a straight line toward the origin. Since the increase in the critical b/c ratio with λ is in all cases sublinear, this change always hinders cooperation. The critical b/c ratio for a planar triangular lattice is plotted in black. Adding a decay rate equal to the utilization rate (d = 1) changes favorable (b/c, λ) combinations (marked by circles) to unfavorable ones (arrowheads).