  Since Nowak & May's (1992) influential paper, network reciprocity--the fact
that individuals' interactions repeated within a local neighborhood support the
evolution of cooperation--has been confirmed in several theoretical models.
Essentially, local interactions allow cooperators to stay protected from
exploiters by assorting into clusters, and the heterogeneity of the network of
contacts--the co-presence of low- and high-connected nodes--has been shown to
further favor cooperation. The few available large-scale experiments on humans
have however missed these effects. The reason is that, while models assume that
individuals update strategy by imitating better performing neighbors,
experiments showed that humans are more prone to reciprocate cooperation than
to compare payoffs. Inspired by the empirical results, we rethink network
reciprocity as a rational form of direct reciprocity on networks--networked
rational reciprocity--indeed made possible by the locality of interactions. We
show that reciprocal altruism in a networked prisoner's dilemma can invade and
fixate in any network of rational agents, profit-maximizing over an horizon of
future interactions. We find that networked rational reciprocity works better
at low average connectivity and we unveil the role of network heterogeneity.
Only if cooperating hubs invest in the initial cost of exploitation, the
invasion of cooperation is boosted; it is otherwise hindered. Although humans
might not be as rational as here assumed, our results could help the design and
interpretation of new experiments in social and economic networks
