Largest Laplacian eigenvalue predicts the emergence of costly punishment in the evolutionary ultimatum game on networks

In recent years, there has been a growing interest in studying the role of costly punishment in promoting altruistic behaviors among selfish individuals. Rejections in ultimatum bargaining as a metaphor exemplify costly punishment, where the division of a sum of resources proposed by one side may be rejected by the other side, and both sides get nothing. Under a setting of the network of contacts among players, we find that the largest Laplacian eigenvalue of the network determines the critical division of players' proposals, below which pure punishers who never accept any offers will emerge as a phase transition in the system. The critical division of offers that predicts the emergence of costly punishment is termed as the selfishness tolerance of a network within evolutionary ultimatum game, and extensive numerical simulations on the data of the science collaboration network, and computer-generated small-world/scale-free networks support the analytical findings.