In this work, we propose an interdependent, multilayer network model and percolation process that matches infrastructures better than previous models by allowing some nodes to survive when their interdependent neighbors fail. We consider a node-to-link failure propagation mechanism and establish weak interdependence across layers via a tolerance parameter $alpha$ which quantifies the likelihood that a node survives when one of its interdependent neighbors fails. We measure the robustness of any individual layer by the final size of its giant component. Analytical and numerical results show that weak interdependence produces a striking phenomenon: layers at different positions within the multilayer system experience distinct percolation transitions. Especially, layers with high super degree values percolate in an abrupt manner, while those with low super degree values exhibit both continuous and abrupt transitions. This novel phenomenon we call emph{mixed percolation transitions} has significant implications for network robustness. Previous results that do not consider cascade tolerance and layer super degree may be under- or over-estimating the vulnerability of real systems. Moreover, since $alpha$ represents a generic measure of various risk management strategies used to buffer infrastructure assets from cascades, our model reveals how nodal protection activities influence failure dynamics in interdependent, multilayer systems.
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