The resource cost of cooperative invasion

A contact network can favor cooperation by sheltering a lineage until its descendants compete with defectors as a group. We determine the contact strengths, invasion times and measurement effort needed to realize this advantage at fixed population size. Individuals reproduce according to averaged cooperation payoffs, and offspring replace neighbors through contacts of equal weight in both directions. Contact contrast is the ratio of strongest to weakest contact. Under weak selection on a six-site path, the least contrast needed to favor a single cooperator over the reverse invasion by a defector grows as 656ε-2 for uniformly placed mutants and 4050ε-3 for mutations in newborn offspring, as the benefit-cost ratio b/c=1+ε approaches one. The extra power arises because rare replacement protects a site while suppressing mutation arrival there. Global bounds force two weakly coupled groups and exact mirror symmetry of the least-contrast contacts for sufficiently small ε. Under uniform introduction, additional contrast shortens the leading mean time to extinction or fixation in the absence of selection, from 12/ε toward 6/ε reproduction attempts, and eventually breaks the mirror symmetry of the fastest design. Estimating the uniform-introduction advantage from fixation outcomes of independent completed invasions defines a third optimum. Within the six-site family with mirror-symmetric, strongly separated contacts, we optimize contacts and selection strength with reproduction weighted linearly or exponentially by payoff. At fixed relative precision, with contrast a fixed factor above its minimum, the leading summed trial duration grows as ε-3. Efficient measurement accepts a smaller advantage in exchange for shorter trials. The cheapest, fastest and most efficiently measured networks therefore differ.