Public goods games on random hyperbolic graphs with mixing

Understanding the evolution of cooperation in structured populations remains one of the fundamental challenges of the 21st century, with far-reaching implications for the wellbeing of modern human societies. Studies over the past two decades showed that the structure of the network of contacts plays a crucial role in determining whether cooperation prevails or not. An important step to more realistic networks was made with the shift from regular grids and lattices to complex social networks at the turn of the century. Real networks exhibit a high mean local clustering coefficient, short average path lengths, and community structure. Recent studies have revealed that random geometric graphs in hyperbolic spaces exhibit properties that are frequently found in real networks. We here study the public goods game on random geometric graphs in hyperbolic spaces, and we consider assortative and disassortative mixing with different frequencies. We show that in hyperbolic spaces heterogeneous networks promote the evolution of public cooperation in comparison to the more homogeneous networks. We also confirm that assortative and disassortative mixing on random hyperbolic networks both impair the evolutionary success of cooperators, regardless of the network architecture. The differences between the two mixing protocols are most expressed at low mixing frequencies, whilst at high mixing frequencies the two almost converge.