  A growing body of empirical evidence indicates that social and cooperative
behavior can be affected by cognitive and neurological factors, suggesting the
existence of state-based decision-making mechanisms that may have emerged by
evolution. Motivated by these observations, we propose a simple mechanism of
anonymous network interactions identified as a form of generalized reciprocity
- a concept organized around the premise "help anyone if helped by someone",
and study its dynamics on random graphs. In the presence of such mechanism, the
evolution of cooperation is related to the dynamics of the levels of
investments (i.e. probabilities of cooperation) of the individual nodes
engaging in interactions. We demonstrate that the propensity for cooperation is
determined by a network centrality measure here referred to as neighborhood
importance index and discuss relevant implications to natural and artificial
systems. To address the robustness of the state-based strategies to an invasion
of defectors, we additionally provide an analysis which redefines the results
for the case when a fraction of the nodes behave as unconditional defectors.
