= Network reciprocity
<Network reciprocity> is the promotion of <cooperation> by the pattern of interactions in a <biological population>. When cooperators interact preferentially with other cooperators, their help can sustain local clusters. A <graph> makes the mechanism precise: individuals occupy <graph vertices>, and <graph edges> specify who interacts with whom. Recognition of a previous helper is not required.
In the <donation game>, let each cooperator pay a cost $c$ per neighbour and supply benefit $b$ to that neighbour. On a <regular graph> with four neighbours per individual, a cooperator surrounded by four cooperators receives payoff $4b-4c$; an isolated cooperator surrounded by nonhelpers receives $-4c$. Thus the same behaviour has different consequences in different local arrangements.
<Hisashi Ohtsuki> and colleagues derived the approximate condition $b/c>k$ for <natural selection> to favour <cooperation> on large <regular graphs> of <degree of a vertex> $k$, under <weak selection>. Their updating rule first removed a random individual, then let its neighbours compete for the vacancy according to <biological fitness>. For $k=4$, $b=5$ and $c=1$ satisfy the asymptotic condition. The rule concerns <fixation probability> under stated assumptions; changing the order of death and reproduction can change the result.<A simple rule for the evolution of cooperation on graphs and social networks by Hisashi Ohtsuki and others>[Nature 441, pp. 502–504; large-population weak-selection approximation, Figure 2 and comparison of update mechanisms]{ref}
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