Protected domains and the cost of cooperation on weighted networks

Abstract: A costly inherited trait can spread when its descendants create a favorable local environment. We ask which finite contact structures sustain such a collective competitor and what their advantage costs. Individuals play a donation game, conferring benefit $b$ at cost $c$. Reciprocal weighted contacts determine averaged payoffs and copying recipients, while sources are chosen globally according to payoff. Cooperation is favored when a uniformly introduced cooperator takes over more often than a defector in the complementary experiment. Exact fluctuation response and elimination of fast configurations identify the formation and competition of protected domains as the controlling processes. The sharp weak-selection threshold infima are $b/c=3$ on the four-cycle, $7$ on the five-site path, and $1$ on every fixed path with at least six sites. On the six-site path, a global bound forces near-barrier designs into this domain hierarchy. Polynomial relations among the contact ratios then control unrestricted optimization and imply eventual exact reflection symmetry. Reaching threshold $1+ε$ requires minimum contact contrast $656ε<sup>{-2}[1+O(ε)]$ and neutral absorption time of order $ε<sup>{-1}$. We determine their tradeoff and the associated design tolerance. Finite selection consumes the same margin. Within the controlled small-selection window, the maximal fixation advantage is of order $ε<sup>{5/2}$ when the contrast exceeds its exact minimum by order $ε<sup>{-1}$. Thus an improving invasion threshold can require increasing contact heterogeneity, precision and observation time.