• altruism;
  • inclusive fitness;
  • network;
  • relatedness;
  • structured populations


A recent model studies the evolution of cooperation on a network, and concludes with a result connecting the benefits and costs of interactions and the number of neighbours. Here, an inclusive fitness analysis is conducted of the only case solved analytically, of a cycle, and the identical result is obtained. This brings the result within a biologically familiar framework. It is notable that the benefits and costs in the inclusive fitness framework need to be derived, and are not the benefits and costs that are the parameters in the original model. The relatedness is a quadratic function of position in a cycle of size N: an individual is related by 1 to itself, by (N − 5)/(N + 1) to an immediate neighbour, and by very close to −1/2 to the most distant individuals. The inclusive fitness analysis explains hitherto puzzling features of the results.