Evolutionary public goods games on hypergraphs with heterogeneous multiplication factors

The spatial structure and social diversity playing a nontrivial role in the emergence and maintenance ofcooperation among selfish individuals have been verified. Their effects on the evolution of cooperation haveattracted great attention in recent years. Most of previous evolutionary game dynamics is based on pairwiseinteractions. However, the interactions often take place within groups of people in many real situations andcannot be described simply by dyads. The dynamics of evolutionary games in systems with higher-orderinteractions has not yet been explored as deserved. In this paper, we introduce heterogeneous multiplicationfactors into the spatial public goods game to investigate the cooperative behaviors on the hypergraphs. Inaddition to the original model in which all groups have the same multiplication factor, three types ofheterogeneous multiplication factor distributions including uniform, exponential and power-law distributions areconsidered. The numerical simulation results show that the increase of the order g of the uniform randomhypergraphs is conducive to the emergence and prosperity of the individuals' cooperative behavior no matterwhat types these distributions belong to. Furthermore, compared with the results of the original spatial publicgoods games on hypergraphs, the heterogeneous multiplication factors following three different distributions canremarkably promote the evolution of cooperation. In particular, for most of ranges of the average rescalingmultiplication factor , the highest cooperation level can be obtained under the power-law distribution, whilethe uniform distribution leads to the lowest cooperation level. We provide an explanation through investigatingthe number of cooperators in each group. In addition, to probe into the essence that influences the survival ofcooperative behaviors, we study the time series of the fraction of groups with different numbers of cooperators.Besides, we also investigate the influence of the number of hyperlinks on cooperation evolution. We find thatthe results are robust against the number of hyperlinks L, and the emergence of cooperative behaviors in publicgoods games on hypergraphs is hindered with the value of L increasing. To some extent, these results arehelpful in the better understanding of the evolutionary dynamics of the spatial public goods games on hypergraphs with social diversity