Equilibrium analysis of edge-heterogeneous binary network games

Recent studies on the equilibrium of binary network games have primarily focused on scenarios characterized by agent heterogeneity, where agents exhibit unique attributes but their interactions with different neighbors remain uniform. In this paper, we investigate the edge-heterogeneous binary network game, a more general framework that incorporates heterogeneity into agent interactions. We establish two sufficient equilibrium conditions under asynchronous best-response dynamics from different perspectives. The first condition requires underlying symmetry in interactions between neighboring agents, integrating and generalizing three classical convergence situations in binary network games. The second condition focuses on network balance, positing that equilibrium is achievable if the coordination value network of a game is structurally balanced. Additionally, for games meeting this condition, we develop a method to predict the final state based on initial state information. These results reveal factors that steer edge-heterogeneous binary network games towards equilibrium, providing valuable insights for controlling such highly nonlinear systems. Lastly, we extend the analysis to higher-order network games and propose an equilibrium condition for edge-heterogeneous 2-order network games.