This paper proposes a co-evolutionary model of directed graphs and three opinions, i.e., conservative$(+)$, neutral$(\odot)$ and liberal$(-)$. Agents update both opinions and social relationships with bias. We find that an emergent game suffices to predict the stability of bi-polarization under a rare opinion updating limit and a large system size limit. The bi-polarization is stable if and only if the emergent game has an internal Nash equilibrium. The necessary and sufficient condition is explained by both risk dominance and evolutionary stability. This game approach facilitates us to reveal the stability of bi-polarization in empirical systems. Our work fosters the understanding of opinion formation for controversial topics, and shows a deep connection between opinion dynamics and evolutionary game theory.

In this paper, we apply the coordinate increment discrete gradient (CIDG) method to solve the Lorentz force system which can be written as a non-canonical Hamiltonian system. Then we can obtain a new energy-preserving CIDG-I method for the system. The CIDG-I method can combine with its adjoint method CIDG-II which is also a energy-preserving method to form a new method, namely CIDG-C method. The CIDG-C method is symmetrical and can conserve the Hamiltonian energy directly and exactly. With comparison to the well-used Boris method, numerical experiments indicate that the CIDG-C method holds advantage over the Boris method in terms of energy-conserving.

Understanding the evolution of cooperation is pivotal in biology and social science. Public resources sharing is a common scenario in the real world. In our study, we explore the evolutionary dynamics of cooperation on a regular graph with degree $k$, introducing the presence of a third strategy, namely the benevolence, who does not evolve over time, but provides a fixed benefit to all its neighbors. We find that the presence of the benevolence can foster the development of cooperative behavior and it follows a simple rule: $b/c > k - p_S(k-1)$. Our results provide new insights into the evolution of cooperation in structured populations.