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Collective behavior, from murmurations to synchronized beating of heart cells, governs some of the most beautiful and important aspects of nature. Likewise, cooperation—the act of sacrificing personal benefits for the common good—is one of the pillars of social evolution, and it is the basis for the emergence of collective organized actions from single-cell organisms to modern human societies. Here we merge these two phenomena into a single model, considering an ensemble of networked oscillators, where each oscillator can be either a cooperator or a defector, and with only cooperators contributing to synchrony. At the same time, the value of the order parameter in the neighborhood of each oscillator is considered as an effective local temperature which determines the strategy updating procedure in the evolutionary game. The emergence of cooperation is thus intertwined with that of synchronization, producing a novel and fascinating dynamics which includes a double explosive transition. [ABSTRACT FROM AUTHOR]

During the past decades, pattern formulation with reaction–diffusion systems has attracted great research interest. Complex networks, from single-layer networks to more complicated multiplex networks, have made great contribution to the development of this area, especially with emergence of Turing patterns. While among vast majority of existing works on multiplex networks, they only take into account the simple case with ordinary diffusion, which is termed as self-diffusion. However, cross-diffusion, as a significant phenomenon, reveals the direction of species' movement, and is widely found in chemical, biological and physical systems. Therefore, we study the pattern formulation on multiplex networks with the presence of both self-diffusion and cross-diffusion. Of particular interest, heterogeneous patterns with abundant characteristics are generated, which cannot arise in other systems. Through linear analysis, we theoretically derive the Turing instabilities region. Besides, our numerical experiments also generate diverse patterns, which verify the theoretical prediction in our work and show the impact of cross-diffusion on pattern formulation on multiplex networks. [ABSTRACT FROM AUTHOR]