  We discuss a model for evolutionary game dynamics in a growing,
network-structured population. In our model, new players can either make
connections to random preexisting players or preferentially attach to those
that have been successful in the past. The latter depends on the dynamics of
strategies in the game, which we implement following the so-called Fermi rule
such that the limits of weak and strong strategy selection can be explored. Our
framework allows to address general evolutionary games. With only two
parameters describing the preferential attachment and the intensity of
selection, we describe a wide range of network structures and evolutionary
scenarios. Our results show that even for moderate payoff preferential
attachment, over represented hubs arise. Interestingly, we find that while the
networks are growing, high levels of cooperation are attained, but the same
network structure does not promote cooperation as a static network. Therefore,
the mechanism of payoff preferential attachment is different to those usually
invoked to explain the promotion of cooperation in static, already-grown
networks.
