Percolation on interacting networks with feedback-dependency links.

Abstract

When real networks are considered, coupled networks with connectivity and feedback-dependency links are not rare but more general. Here, we develop a mathematical framework and study numerically and analytically the percolation of interacting networks with feedback-dependency links. For the case that all degree distributions of intra- and inter- connectivity links are Poissonian, we find that for a low density of inter-connectivity links, the system undergoes from second order to first order through hybrid phase transition as coupling strength increases. It implies that the average degree k of inter-connectivity links has a little influence on robustness of the system with a weak coupling strength, which corresponds to the second order transition, but for a strong coupling strength corresponds to the first order transition. That is to say, the system becomes robust as k increases. However, as the average degree k of each network increases, the system becomes robust for any coupling strength. In addition, we find that one can take less cost to design robust system as coupling strength decreases by analyzing minimum average degree kmin of maintaining system stability. Moreover, for high density of inter-connectivity links, we find that the hybrid phase transition region disappears, the first order region becomes larger and second order region becomes smaller. For the case of two coupled scale-free networks, the system also undergoes from second order to first order through hybrid transition as the coupling strength increases. We find that for a weak coupling strength, which corresponds to the second order transitions, feedback dependency links have no effect on robustness of system relative to no-feedback condition, but for strong coupling strength which corresponds to first order or hybrid phase transition, the system is more vulnerable under feedback condition comparing with no-feedback condition. Thus, for designing resilient system, designers should try to avoid the feedback dependency links, because the existence of feedback-dependency links makes the system extremely vulnerable and difficult to defend.