Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215 USA.
Computer Science Department & Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA.
Phys Rev E. 2017 Sep;96(3-1):032317. doi: 10.1103/PhysRevE.96.032317. Epub 2017 Sep 28.
Cascading failures in complex systems have been studied extensively using two different models: k-core percolation and interdependent networks. We combine the two models into a general model, solve it analytically, and validate our theoretical results through extensive simulations. We also study the complete phase diagram of the percolation transition as we tune the average local k-core threshold and the coupling between networks. We find that the phase diagram of the combined processes is very rich and includes novel features that do not appear in the models studying each of the processes separately. For example, the phase diagram consists of first- and second-order transition regions separated by two tricritical lines that merge and enclose a two-stage transition region. In the two-stage transition, the size of the giant component undergoes a first-order jump at a certain occupation probability followed by a continuous second-order transition at a lower occupation probability. Furthermore, at certain fixed interdependencies, the percolation transition changes from first-order → second-order → two-stage → first-order as the k-core threshold is increased. The analytic equations describing the phase boundaries of the two-stage transition region are set up, and the critical exponents for each type of transition are derived analytically.
核心分解渗流和相依网络。我们将这两种模型结合成一个通用模型，进行了分析求解，并通过广泛的模拟验证了我们的理论结果。我们还研究了渗流相变的完整相图，同时调整平均局部核心阈值和网络之间的耦合。我们发现，组合过程的相图非常丰富，包括在分别研究每个过程的模型中没有出现的新特征。例如，相图由第一和第二顺序相变区域组成，由两条合并并包围两阶段相变区域的三临界点线隔开。在两阶段相变中，在一定的占据概率下，巨配分函数经历一级跃迁，然后在较低的占据概率下发生连续的二级跃迁。此外，在某些固定的相依关系下，随着核心阈值的增加，渗流相变从一级→二级→两阶段→一级变化。建立了描述两阶段相变区域相边界的解析方程，并解析地推导了每种类型跃迁的临界指数。
