  Most previous works study the evolution of cooperation in a structured
population by commonly employing an isolated single network. However, realistic
systems are composed of many interdependent networks coupled with each other,
rather than the isolated single one. In this paper, we consider a system
including two interacting networks with the same size, entangled with each
other by the introduction of probabilistic interconnections. We introduce the
public goods game into such system, and study how the probabilistic
interconnection influences the evolution of cooperation of the whole system and
the coupling effect between two layers of interdependent networks. Simulation
results show that there exists an intermediate region of interconnection
probability leading to the maximum cooperation level in the whole system.
Interestingly, we find that at the optimal interconnection probability the
fraction of internal links between cooperators in two layers is maximal. Also,
even if initially there are no cooperators in one layer of interdependent
networks, cooperation can still be promoted by probabilistic interconnection,
and the cooperation levels in both layers can more easily reach an agreement at
the intermediate interconnection probability. Our results may be helpful in
understanding the cooperative behavior in some realistic interdependent
networks and thus highlight the importance of probabilistic interconnection on
the evolution of cooperation.
