Our current world is linked by a complex mesh of networks where information, people and goods flow. These networks are interdependent each other, and present structural and dynamical features different from those observed in isolated networks. While examples of such "dissimilar" properties are becoming more abundant, for example diffusion, robustness and competition, it is not yet clear where these differences are rooted in. Here we show that the composition of independent networks into an interconnected network of networks undergoes a structurally sharp transition as the interconnections are formed. Depending of the relative importance of inter and intra-layer connections, we find that the entire interdependent system can be tuned between two regimes: in one regime, the various layers are structurally decoupled and they act as independent entities; in the other regime, network layers are indistinguishable and the whole system behave as a single-level network. We analytically show that the transition between the two regimes is discontinuous even for finite size networks. Thus, any real-world interconnected system is potentially at risk of abrupt changes in its structure that may reflect in new dynamical properties.
Percolation in Interdependent and Interconnected Networks: Abrupt Change from Second to First Order Transition
两个耦合网络系统的鲁棒性研究仅限于依赖耦合（S。布尔德列夫等例如，Nature，2010）并且仅用于连接耦合（E. A. Leicht和R. M. D'Souza，arxiv：09070894）.在这里，我们研究，使用渗流的方法，一个更现实的耦合网络系统，相互依赖和相互联系的链接存在。我们发现了丰富而不寻常的相变现象，包括混合的一级和二级的混合相变，不连续性，如巨分量的一阶跃迁，接着是连续减少到零，如二阶跃迁。此外，我们发现不寻常的不连续的变化，从二阶到一阶过渡作为两个网络之间的依赖耦合的函数。
