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This theoretical study examines how cooperation can evolve in populations structured as weighted networks, where individuals interact with neighbors through contacts of varying strength. The researchers demonstrate that on networks with at least six nodes, asymmetric contact weights can create "protected domains" that favor cooperative behavior over defection at any benefit-to-cost ratio greater than one, by converting individual reproduction into collective competition between aligned groups. The advantage depends critically on both the network structure and whether mutants are placed uniformly or arise in newborn offspring.
Why it matters
This work provides mathematical foundations for understanding how spatial structure and interaction patterns enable cooperation to persist in biological and social systems, with potential applications to evolutionary biology, social network dynamics, and the design of systems where cooperative behavior is desirable but individually costly.
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Abstract: Reciprocal contacts can favor cooperation by protecting a lineage until its descendants compete as a group. An individual is chosen to reproduce from the whole population, with a probability increasing with its averaged cooperation payoff. Its offspring replaces a neighbor through contacts weighted equally in both directions. We compare a single cooperator taking over defectors with the reverse invasion, using the same distribution of introduction sites. When payoffs weakly affect reproduction, every network with at least six sites and a path visiting them all admits weights favoring cooperation at any benefit-cost ratio $b/c>1$. This holds for mutants placed uniformly or arising in newborn offspring. Unequal contacts sustain aligned domains around two protected sites, converting individual reproduction into collective competition. Cooperation cannot have an advantage when its benefit does not exceed its cost, at any positive selection strength with positive reproduction rates. Where mutants arise changes which contact patterns can use this mechanism. On a five-site path, the weak-selection threshold has the sharp lower bound seven under uniform placement; newborn introduction never favors cooperation at this order. We classify the contact patterns with at most five sites that can favor cooperation under either rule; newborn introduction requires a four- or five-site cycle. Tracing copies backward to their ancestors separates domain competition from the probability of reaching it. Exact polynomial inequalities prove sharp threshold bounds for paths and the four- and five-site cycles. The same ancestry reconstructs evolution without selection on weighted paths and cycles and fixes the slowest relaxation rate on every connected reciprocal network. Cooperation thus depends on both the collective competitors created by contacts and the route by which a new lineage enters them.
Source: Protected domains and cooperative invasion on weighted networks