Analysis of percolation behaviors of clustered networks with partial support–dependence relations

We carry out a study of percolation behaviors of clustered networks with partial support–dependence relations by adopting two different attacking strategies, attacking only one network and both networks, which help to further understand real coupled networks. For two different attacking strategies we find that the system changes from a second-order phase transition to a first-order phase transition as coupling strength q increases. We also notice that the first-order region becomes smaller and the second-order region becomes larger as average degree or clustering coefficient increases. And, as the average supported degree approaches infinity, coupled clustered networks become independent and only the second-order transition is observed, which is similar to q=0. Furthermore, we find that clustering coefficient has a significant impact on robustness of the system for strong coupling strength, but for weak coupling strength it has little influence, especially for attacking both networks. The study implies that we can obtain a more robust network by reducing clustering coefficient and increasing average degree for strong coupling strength. However, for weak coupling strength, a more robust network is obtained only by increasing average degree for the same support average degree. Additionally, we find that for attacking both networks the system becomes more vulnerable and difficult to defend compared to attacking only one network.