Evolutionary public goods games on hypergraphs with heterogeneous multiplication factors

Abstract

In recent years, the effects of spatial structure and social diversity on the evolution of cooperation have been attracted great attention. Both of them have been verified to play a nontrivial role in the emergence and maintenance of cooperation among selfish individuals. And most previous evolutionary game dynamics are based on pairwise interactions. However, interactions can often take place within groups of people in many real situations and cannot be described simply by dyads. While the dynamics of evolutionary games in systems with higher-order interactions have not yet been explored as deserved. In this paper, we introduce heterogeneous multiplication factors into the spatial public goods game to investigate the cooperative behaviors on the hypergraphs. In addition to the original model in which all groups have the same multiplication factor, three types of heterogeneous multiplication factor distributions including uniform, exponential and power-law distributions are considered. The numerical simulation results show that, the increase of the order <inline-formula><tex-math id="M3">\begin{document}$g$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212436_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212436_M3.png"/></alternatives></inline-formula> of the uniform random hypergraphs is conducive to the emergence and prosperity of the individuals' cooperative behavior no matter what types of the distributions are. Furthermore, compared with the results of the original spatial public goods games on hypergraphs, the heterogeneous multiplication factors following three different distributions can remarkably promote the evolution of cooperation. In particular, for most ranges of the average rescaling multiplication factor <inline-formula><tex-math id="M4">\begin{document}$r_0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212436_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212436_M4.png"/></alternatives></inline-formula>, the highest cooperation level can be obtained under the power-law distribution, while the uniform distribution induces to the lowest enhancement. We provide an explanation through investigating the number of cooperators in each group. In addition, to probe into the essence that influences the survival of cooperative behaviors, we have studied the time series of the fraction of groups with different numbers of cooperators. Besides, we also study the impact of the number of hyperlinks on the evolution of cooperation. We find that the results are robust against the number of hyperlinks <inline-formula><tex-math id="M5">\begin{document}$L$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212436_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212436_M5.png"/></alternatives></inline-formula>, and the emergence of cooperative behaviors in public goods games on hypergraphs is hindered with increasing of <inline-formula><tex-math id="M6">\begin{document}$L$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212436_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212436_M6.png"/></alternatives></inline-formula>. To some extent, these results are helpful for better understanding the evolutionary dynamics of the spatial public goods games on hypergraphs with social diversity.