Robustness of interdependent and interconnected clustered networks

In real world, most systems show significant clustering, and it is more practical to investigate the behaviors of clustered network. Previous studies are mostly focused on single clustered network and coupled clustered networks with dependency links. Here we study two clustered networks coupled with both interdependent and interconnected links by introducing generating function of the joint degree distribution. When the networks are fully dependent, we obtain the analytical solution of giant component P∞. We show rich phase transition phenomena and analyze their behaviors. We find that, as dependency coupling strength increases, the system changes from second order phase transition through hybrid transition to first order phase transition. For weak dependency coupling strength qA, corresponding to second order phase transition, we find that, clustering has almost no effect on the robustness of network, but for strong dependency coupling strength qA, corresponding to first order transition, the more clustered system is more vulnerable. At the same time, we notice that when the system is more clustered, the hybrid order region is almost unchangeable, the first order region becomes smaller, and the second order region is larger. Additionally, we can see that, the bigger the clustering coefficient c is, the bigger the second order region becomes. For the same c, the density of connectivity links between networks is higher, the second order region becomes smaller, and the density of connectivity links within each network is higher, the second order region becomes bigger.