A cooperation-competition evolutionary dynamic model over signed networks

Signed networks are widely applied to describe cooperative and competitive interaction relationships arising from various social, biological, and physical systems. In this article, a cooperation-competition evolutionary dynamic model is offered to explain the gradual evolution of the interaction relationship between neighboring nodes in signed networks. The core feature of our model is to portray the interaction relationship between neighboring nodes as a nonlinear function of the state difference, i.e., neighboring nodes cooperate with each other only if their state difference is within a limited range (called cooperation range); otherwise, they antagonize with each other. We investigate the leader-follower dynamics, and present the lower bound of the convergence rate by using the infinity norm of products of substochastic matrices. For the leaderless dynamics, we establish algebraic conditions with respect to the state differences and the cooperation range for realizing consensus and polarization behaviors. Finally, numerical examples are provided to illustrate the evolution dynamics.