  How cooperation emerges in human societies is both an evolutionary enigma,
and a practical problem with tangible implications for societal health.
Population structure has long been recognized as a catalyst for cooperation
because local interactions enable reciprocity. Analysis of this phenomenon
typically assumes bi-directional social interactions, even though real-world
interactions are often uni-directional. Uni-directional interactions -- where
one individual has the opportunity to contribute altruistically to another, but
not conversely -- arise in real-world populations as the result of
organizational hierarchies, social stratification, popularity effects, and
endogenous mechanisms of network growth. Here we expand the theory of
cooperation in structured populations to account for both uni- and
bi-directional social interactions. Even though directed interactions remove
the opportunity for reciprocity, we find that cooperation can nonetheless be
favored in directed social networks and that cooperation is provably maximized
for networks with an intermediate proportion of directed interactions, as
observed in many empirical settings. We also identify two simple structural
motifs that allow efficient modification of interaction directionality to
promote cooperation by orders of magnitude. We discuss how our results relate
to the concepts of generalized and indirect reciprocity.
