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The evolution of a cooperative strategy on multilayer networks is arousing increasing concern. Most of the previous studies assumed that agents can only choose cooperation or defection when interacting with their partners, whereas the actual provisions in real world scenarios might not be discrete, but rather continuous. Furthermore, in evolutionary game, agents often make use of their memory which keeps the most successful strategy in the past, as well as the best current strategy gained by their directed neighbors, to find the best available strategies. Inspired by these observations, we study the impact of the particle swarm optimization (PSO) algorithm on the evolution of cooperation on interdependent networks in the continuous version of spatial prisoner’s dilemma games. Following extensive simulations of this setup, we can observe that the introduction of the PSO mechanism on the interdependent networks can promote cooperation strongly, regardless of the network coupling strength. In addition, we find that the increment of coupling strength is more suitable for the propagation of cooperation. More interestingly, we find that when the coupling strength is relatively large, a spontaneous symmetry breaking phenomenon of cooperation occurs between the interdependent networks. To interpret the symmetry breaking phenomenon, we investigate the asynchronous expansion of heterogeneous strategy couples between different networks. Since this work takes cooperation from a more elaborate perspective, we believe that it may provide a deep understanding of the evolution of cooperation in social networks.

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