A fundamental aspect of all biological systems is cooperation. Cooperative interactions are required for many levels of biological organization ranging from single cells to groups of animals^{1, 2, 3, 4}. Human society is based to a large extent on mechanisms that promote cooperation^{5, 6, 7}. It is well known that in unstructured populations, natural selection favours defectors over cooperators. There is much current interest, however, in studying evolutionary games in structured populations and on graphs^{8, 9, 10, 11, 12, 13, 14, 15, 16, 17}. These efforts recognize the fact that who-meets-whom is not random, but determined by spatial relationships or social networks^{18, 19, 20, 21, 22, 23, 24}. Here we describe a surprisingly simple rule that is a good approximation for all graphs that we have analysed, including cycles, spatial lattices, random regular graphs, random graphs and scale-free networks^{25, 26}: natural selection favours cooperation, if the benefit of the altruistic act, b, divided by the cost, c, exceeds the average number of neighbours, k, which means b/c > k. In this case, cooperation can evolve as a consequence of 'social viscosity' even in the absence of reputation effects or strategic complexity.

1. Department of Biology, Kyushu University, Fukuoka 812-8581, Japan
2. Program for Evolutionary Dynamics, Department of Organismic and Evolutionary Biology, Department of Mathematics,
3. Department of Applied Mathematics, Harvard University, Cambridge, Massachusetts 02138, USA

Correspondence to: Martin A. Nowak² Correspondence and requests for materials should be addressed to M.A.N. (Email: martin_nowak@harvard.edu).

Received 7 December 2005 | Accepted 26 January 2006

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