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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.
We study the percolation in coupled networks with both inner-dependency and inter-dependency links, where the inner- and inter-dependency links represent the dependencies between nodes in the same or different networks, respectively. We find that whe
In real networks, the dependency between nodes is ubiquitous; however, the dependency is not always complete and homogeneous. In this paper, we propose a percolation model with weak and heterogeneous dependency; i.e., dependency strengths could be di
Real data show that interdependent networks usually involve inter-similarity. Intersimilarity means that a pair of interdependent nodes have neighbors in both networks that are also interdependent (Parshani et al cite{PAR10B}). For example, the coupl
Previous work on undirected small-world networks established the paradigm that locally structured networks tend to have high density of short loops. On the other hand, many realistic networks are directed. Here we investigate the local organization o
The concept of temporal networks provides a framework to understand how the interaction between system components changes over time. In empirical communication data, we often detect non-Poissonian, so-called bursty behavior in the activity of nodes a