Multiple phase transition in the non-symmetrical interdependent networks

Analysis of the robustness of interdependent networks has attracted much attention in recent years. In practice the interdependent networks usually are non-symmetrical in terms of size. However, the behavior of interdependent networks in this case has not been well addressed. We adopt a simplified self-consistent probabilities method to provide a straightforward framework to study the percolation behavior of such non-symmetrical interdependent networks. We define γ as the ratio of the sizes of two layers within an interdependent network to characterize the unsymmetrical property. We find a rich phase diagram in the plane composed of critical threshold, pc, and γ. As γ increases from zero to infinity, the phase transition behavior of the giant component in network under random attack shows a change from second-order (0<γ≤γc1), to a first-order (γc1<γ≤γc2), then through hybrid transition (γc2<γ≤γc3) again to a second-order transition (γc3<γ<∞). Furthermore, we find γc1 tends to move close to 1 as the average degree k increases. In contrast, γc2 and γc3 do not vary with k. Moreover, we also explore the mechanism of the different types of phase transition and illustrate the process by intuition pictures, which is meaningful for better understanding the hybrid phase transition forming. Our research brings insight to understand the vulnerability of non-symmetrical interdependent networks and may useful for network optimization and design.