Robustness of interdependent higher-order networks.

In real complex systems, interactions occur not only between a pair of nodes, but also in groups of three or more nodes, which can be abstracted as higher-order structures in the networks. The simplicial complex is one of a model to represent systems with both low-order and higher-order structures. In this paper, we study the robustness of interdependent simplicial complexes under random attacks, where the complementary effects of the higher-order structure are introduced. When a higher-order node in a 2-simplex fails, its dependent node in the other layer survives with a certain probability due to the complementary effects from the 2-simplex. By using the percolation method, we derive the percolation threshold and the size of the giant component when the cascading failure reaches its steady state. The simulation results agree well with analytical predictions. We find that the type of phase transition changes from the first-order to the second-order when the complementary effect of the higher-order structure on the dependent node increases or the number of 2-simplices in the interdependent simplicial complex increases. While the interlayer coupling strength increases, the type of phase transition changes from the second-order to the first-order. In particular, even if the higher-order interactions do not provide complementary effects for dependent nodes, the robustness of the interdependent heterogeneous simplicial complex is higher than that of the ordinary interdependent network with the same average degree due to the existence of 2-simplices in the system. This study furthers our understanding in the robustness of interdependent higher-order networks.