In the framework of the evolutionary dynamics of the Prisoners Dilemma game on complex networks, we investigate the possibility that the average level of cooperation shows hysteresis under quasi-static variations of a model parameter (the temptation
to defect). Under the discrete replicator strategy updating rule, for both Erdos-Renyi and Barabasi-Albert graphs we observe cooperation hysteresis cycles provided one reaches tipping point values of the parameter; otherwise, perfect reversibility is obtained. The selective fixation of cooperation at certain nodes and its organization in cooperator clusters, that are surrounded by fluctuating strategists, allows the rationalization of the lagging behind behavior observed.
