An inverse voter model for co-evolutionary networks: Stationary efficiency and phase transitions

In some co-evolutionary networks, the nodes always flip their states between two opposite ones, changing the types of the links to others correspondingly. Meanwhile, the link-rewiring and state-flipping processes feed back each other, and only the links between the nodes in the opposite states are productive in generating flow for the network. We propose an inverse voter model to depict the basic features of them. New phase transitions from full efficiency to deficiency state are found by both the analysis and simulations starting from the random graphs and small-world networks. We suggest a new way to measure the efficiency of networks.