  The self-consistent probabilistic approach has proven itself powerful in
studying the percolation behavior of interdependent or multiplex networks
without tracking the percolation process through each cascading step. In order
to understand how directed dependency links impact criticality, we employ this
approach to study the percolation properties of networks with both undirected
connectivity links and directed dependency links. We find that when a random
network with a given degree distribution undergoes a second-order phase
transition, the critical point and the unstable regime surrounding the
second-order phase transition regime are determined by the proportion of nodes
that do not depend on any other nodes. Moreover, we also find that the triple
point and the boundary between first- and second-order transitions are
determined by the proportion of nodes that depend on no more than one node.
This implies that it is maybe general for multiplex network systems, some
important properties of phase transitions can be determined only by a few
parameters. We illustrate our findings using Erdos-Renyi (ER) networks.
