Evolution of cooperation with asymmetric social interactions

The paper proves that, on fully directed random k-regular networks (k/2 in- and k/2 out-neighbors), weak selection favors cooperation in the donation game if and only if b/c > k−1 (downstream dispersal) and b/c > k(k−1)/(k−2) (upstream), for sufficiently large N; these are stated explicitly as Eqs. (4)–(5) and derived in the Supporting Information, with exact computations converging to these thresholds as N grows. The candidate model independently derives the same thresholds via the σ-rule and a coalescing-random-walk calculation of σ for the two orientations. The two approaches differ methodologically but agree on the final conditions, up to the stated large-N assumptions. See the paper’s statements of the thresholds and update rule definitions for details .