Supplemental Information from Proximity inheritance explains the evolution of cooperation under natural selection and mutation

In this paper, a mechanism called proximity inheritance is introduced in the birth–death process of a networked population involving in Prisoner's Dilemma game. Different from the traditional birth–death process, in the proposed model, players are distributed in a spatial space and offspring is distributed in the neighbourhood of its parents. That is, offspring inherits not only the strategy but also the proximity of its parents. In this coevolutionary game model, a cooperative neighbourhood gives more neighbouring cooperative offspring and a defective neighbourhood gives more neighbouring defective offspring, leading to positive feedback among cooperative interactions. It is shown that with the help of proximity inheritance, natural selection will favour cooperation over defection under various conditions, even in the presence of mutation. Furthermore, the coevolutionary dynamics could lead to self-organized substantial network clustering, which promotes an assortment of cooperative interactions. This study provides a new insight into the evolutionary mechanism of cooperation in the absence of social attributions such as reputation and punishment.