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 scalefree 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. A cooperator is someone who pays a cost, c, for another individual to receive a benefit, b. A defector pays no cost and does not distribute any benefits. In evolutionary biology, cost and benefit are measured in terms of fitness. Reproduction can be genetic or cultural. In the latter case, the strategy of someone who does well is imitated by others. In an unstructured population, where all individuals are equally likely to interact with each other, defectors have a higher average payoff than unconditional cooperators. Therefore, natural selection increases the relative abundance of defectors and drives cooperators to extinction. These evolutionary dynamics hold for the deterministic setting of the replicator equation 27, 28 and for stochastic game dynamics of finite populations 29 . In our model, the players of an evolutionary game occupy the vertices of a graph. The edges denote links between individuals in terms of game dynamical interaction and biological reproduction. We assume that the graph is fixed for the duration of the evolutionary dynamics. Consider a population of N individuals consisting of cooperators and defectors. A cooperator helps all individuals to whom it is connected. If a cooperator is connected to k other individuals and i of those are cooperators, then its payoff is bi 2 ck. A defector does not provide any help, and therefore has no costs, but it can receive the benefit from neighbouring cooperators. If a defector is connected to j cooperators, then its payoff is bj. The fitness of an individual is given by a constant term, denoting the baseline fitness, plus the payoff that arises from the game. Strong selection means that the payoff is large compared to the baseline fitness; weak selection means the payoff is small compared to the baseline fitness. The idea behind weak selection is that many different factors contribute to the overall fitness of an individual, and the game under consideration is just one of those factors. At first, we will study the following update rule for evolutionary dynamics ( Fig. 1) : in each time step, a random individual is chosen to die, and the neighbors compete for the empty site proportional to their fitness. We call this mechanism 'death-birth' updating, because it involves a death event followed by a birth. Later we will investigate other update mechanisms. Let us explore whether natural selection can favour cooperation on certain graphs. To do this, we need to calculate the probability that a single cooperator starting in a random position turns the whole population from defection to cooperation. If selection neither favours nor opposes cooperation, then this probability is 1/N, which is the fixation probability of a neutral mutant. If the fixation LETTERS Figure 1 | The rules of the game. Each individual occupies the vertex of a graph and derives a payoff, P, from interactions with adjacent individuals. A cooperator (blue) pays a cost, c, for each neighbour to receive a benefit, b.
Ohtsuki et al. REF generalised this result showing that natural selection favours cooperation if the benefit of the altruistic act divided by the cost exceeds the average number of neighbours on the network.
A simple rule for the evolution of cooperation on graphs and social networks
Nature
Nature
Biology Medicine