In this paper we study the evolution of cooperation in the one-shot prisoner's dilemma in heterogeneous populations.
Heterogeneity is defined by a contact network, which gives interaction partners and possibilities for
strategy spreading in the population. We find that the high levels of cooperation previously reported for scalefree
networks (SFNWs)<sup>2-4</sup> are unstable, when small probabilities for unmotivated strategy changes (mutations)
are included in the evolution dynamics. Investigating the role of the network structure for the persistence and
stability of cooperation, we use an optimization technique to generate small networks that allow a large fraction
of cooperators to prosper. Analyzing these networks three key characteristics are identified: (i) degree heterogeneity,
(ii) a set of hubs characterized by many connections to very low degree nodes, and (iii) a relatively large
link density between these hub nodes. These findings motivate the introduction of a class of periphery-core networks,
on which complete cooperation can evolve over the entire range of game parameters. On these networks
cooperation also overwhelmingly dominates, even when relatively large mutation probabilities are included.