Threshold rules of spreading in binary-state networks lead to cascades. We study persistent cascade-recovery dynamics on quasi-robust networks, i.e., networks which are robust against small trigger but may collapse for larger one. It is observed that depending on the relative rate of triggering and recovery, the network falls into one of the two dynamical phases: collapsing or active phase. We devise an analytical framework which characterizes not only the critical behavior but also the temporal evolution of netwo…