Core structure: The coupling failure procedure in multiplex networks

Multiplex networks are a special class of multilayered networks in which a fixed set of nodes is connected by different types of links. The core organization, the residual graph from recursively removing dead-end nodes and their nearest neighbor, plays a significant role in a wide range of typical problems. However previous study about core structure ignored the fact that many real systems are coupled together. In this paper, we firstly generalize the pruning algorithm for multiplex networks, such that leaves in multiplex networks are redefined. The extended pruning algorithm corresponds to a coupling failure procedure in multiplex networks. We develop an analytical approach to study the structural stability of multiplex networks under this scenario. Specifically, by implementing the rate-equation approach to this generalized pruning process, we can analyze a set of cores and describe the birth point and their structures. Moreover, to clarify the general results, we solve the rate equation in four different distribution multiplex networks with two layers: Erdös Rényi, exponentially distributed, purely power-law distributed, and static model networks. Finally, we explore the possible origin for the unique transition behavior in multiplex networks.