Public goods game across multilayer populations with different densities

Individuals may behave differently across multiple domains, which multiplex networks can model. The achievements in different domains shape specific individuals, who update strategy in different layers separately by the average payoff from multiple layers. Moreover, not all individuals have positions in all domains. For example, never online individuals are empty nodes in Internet relationships. This leads to different densities in different layers. In this work, we assume individuals play public goods games in multiple layers with different densities and update strategies in each layer separately. Monte Carlo simulations are conducted on two-layer square lattices. If one layer's density is zeros, the other layer's optimal density favoring cooperation is around the percolation threshold. If one layer's density is full, introducing the other full layer favors cooperation. However, if the introduced layer is partial, then cooperation is inhibited. Fully introducing the other layer also favors cooperation in the existing full layer. However, partially introducing it can even favor cooperation in the full layer more.