  Population structure and spatial heterogeneity are integral components of
evolutionary dynamics, in general, and of evolution of cooperation, in
particular. Structure can promote the emergence of cooperation in some
populations and suppress it in others. Here, we provide results for weak
selection to favor cooperation on regular graphs for any configuration, meaning
any arrangement of cooperators and defectors. Our results extend previous work
on fixation probabilities of single, randomly placed mutants. We find that for
any configuration cooperation is never favored for birth-death (BD) updating.
In contrast, for death-birth (DB) updating, we derive a simple, computationally
tractable formula for weak selection to favor cooperation when starting from
any configuration containing any number of cooperators and defectors. This
formula elucidates two important features: (i) the takeover of cooperation can
be enhanced by the strategic placement of cooperators and (ii) adding more
cooperators to a configuration can sometimes suppress the evolution of
cooperation. These findings give a formal account for how selection acts on all
transient states that appear in evolutionary trajectories. They also inform the
strategic design of initial states in social networks to maximally promote
cooperation. We also derive general results that characterize the interaction
of any two strategies, not only cooperation and defection.
