  Cooperation is ubiquitous at every level of living organisms. It is known
that spatial (network) structure is a viable mechanism for cooperation to
evolve. A recently proposed numerical metric, average gradient of selection
(AGoS), a useful tool for interpreting and visualizing evolutionary dynamics on
networks, allows simulation results to be visualized on a one-dimensional phase
space. However, stochastic mutation of strategies was not considered in the
analysis of AGoS. Here we extend AGoS so that it can analyze the evolution of
cooperation where mutation may alter strategies of individuals on networks. We
show that our extended AGoS correctly visualizes the final states of
cooperation with mutation in the individual-based simulations. Our analyses
revealed that mutation always has a negative effect on the evolution of
cooperation regardless of the payoff functions, fraction of cooperators, and
network structures. Moreover, we found that scale-free networks are the most
vulnerable to mutation and thus the dynamics of cooperation are altered from
bistability to coexistence on those networks, undergoing an imperfect pitchfork
bifurcation.
