Impact of density and interconnectedness of influential players on social welfare

•Influential players may promote cooperation in evolutionary social dilemmas.•Influential players must be rare and weakly interconnected for cooperation to thrive.•Influential players enhance network reciprocity.

We show that in defection prone environments influential players must be rare and weakly interconnected to optimally promote cooperation in the prisoner's dilemma game. Conversely, low temptations to defect warrant a high level of social welfare even if influential players are common, yet still demand the latter be weakly interconnected for cooperation to thrive.