  Evolutionary graph theory is a well established framework for modelling the
evolution of social behaviours in structured populations. An emerging consensus
in this field is that graphs that exhibit heterogeneity in the number of
connections between individuals are more conducive to the spread of cooperative
behaviours. In this article we show that such a conclusion largely depends on
the individual-level interactions that take place. In particular, averaging
payoffs garnered through game interactions rather than accumulating the payoffs
can altogether remove the cooperative advantage of heterogeneous graphs while
such a difference does not affect the outcome on homogeneous structures. In
addition, the rate at which game interactions occur can alter the evolutionary
outcome. Less interactions allow heterogeneous graphs to support more
cooperation than homogeneous graphs, while higher rates of interactions make
homogeneous and heterogeneous graphs virtually indistinguishable in their
ability to support cooperation. Most importantly, we show that common measures
of evolutionary advantage used in homogeneous populations, such as a comparison
of the fixation probability of a rare mutant to that of the resident type, are
no longer valid in heterogeneous populations. Heterogeneity causes a bias in
where mutations occur in the population which affects the mutant's fixation
probability. We derive the appropriate measures for heterogeneous populations
that account for this bias.
