title: Percolation of networks with directed dependency links.
abstract: 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 Erdős-Rényi networks.
authors.affiliation: School of Data and Computer Science, Sun Yat-sen University, Guangzhou 510006, China.|School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China.
authors.forename: Dunbiao
authors.initials: D
authors.lastname: Niu
authors.affiliation: Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215 USA.
authors.forename: Xin
authors.initials: X
authors.lastname: Yuan
authors.affiliation: School of Data and Computer Science, Sun Yat-sen University, Guangzhou 510006, China.
authors.forename: Minhui
authors.initials: M
authors.lastname: Du
authors.affiliation: Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215 USA.
authors.forename: H Eugene
authors.initials: HE
authors.lastname: Stanley
authors.affiliation: School of Data and Computer Science, Sun Yat-sen University, Guangzhou 510006, China.|Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 611731, China.
authors.forename: Yanqing
authors.initials: Y
authors.lastname: Hu