Network structure is a mechanism for promoting cooperation in social dilemma games. In the present study, we explore graph surgery, i.e., to slightly perturb the given network, towards a network that better fosters cooperation. To this end, we develop a perturbation theory to assess the change in the propensity of cooperation when we add or remove a single edge to/from the given network. Our perturbation theory is for a previously proposed random-walk-based theory that provides the threshold benefit-to-cost ratio, [Formula: see text], which is the value of the benefit-to-cost ratio in the donation game above which the cooperator is more likely to fixate than in a control case, for any finite networks. We find that [Formula: see text] decreases when we remove a single edge in a majority of cases and that our perturbation theory captures at a reasonable accuracy which edge removal makes [Formula: see text] small to facilitate cooperation. In contrast, [Formula: see text] tends to increase when we add an edge, and the perturbation theory is not good at predicting the edge addition that changes [Formula: see text] by a large amount. Our perturbation theory significantly reduces the computational complexity for calculating the outcome of graph surgery.
