Giant components in directed multiplex networks

Generating functions for message passing on weighted networks: Directed bond percolation and susceptible, infected, recovered epidemics

We study the SIR (susceptible, infected, removed/recovered) model on directed graphs with heterogeneous transmission probabilities within the message-passing approximation. We characterize the percolation transition, predict cluster size distributions, and suggest vaccination strategies. All predictions are compared to numerical simulations on real networks. The percolation threshold that we predict is a rigorous lower bound to the threshold on real networks. For large, locally treelike networks, our predictions agree very well with the numerical data.

Multiple phase transitions in networks of directed networks

The robustness in real-world complex systems with dependency connectivities differs from that in isolated networks. Although most complex network research has focused on interdependent undirected systems, many real-world networks-such as gene regulatory networks and traffic networks-are directed. We thus develop an analytical framework for examining the robustness of networks made up of directed networks of differing topologies. We use it to predict the phase transitions that occur during node failures and to generate the phase diagrams of a number of different systems, including treelike and random regular (RR) networks of directed Erdős-Rényi (ER) networks and scale-free networks. We find that the the phase transition and phase diagram of networks of directed networks differ from those of networks of undirected networks. For example, the RR networks of directed ER networks show a hybrid phase transition that does not occur in networks of undirected ER networks. In addition, system robustness is affected by network topology in networks of directed networks. As coupling strength q increases, treelike networks of directed ER networks change from a second-order phase transition to a first-order phase transition, and RR networks of directed ER networks change from a second-order phase transition to a hybrid phase transition, then to a first-order phase transition, and finally to a region of collapse. We also find that heterogenous network systems are more robust than homogeneous network systems. We note that there are multiple phase transitions and triple points in the phase diagram of RR networks of directed networks and this helps us understand how to increase network robustness when designing interdependent infrastructure systems.

Message passing theory for percolation models on multiplex networks with link overlap

Multiplex networks describe a large variety of complex systems, including infrastructures, transportation networks, and biological systems. Most of these networks feature a significant link overlap. It is therefore of particular importance to characterize the mutually connected giant component in these networks. Here we provide a message passing theory for characterizing the percolation transition in multiplex networks with link overlap and an arbitrary number of layers M. Specifically we propose and compare two message passing algorithms that generalize the algorithm widely used to study the percolation transition in multiplex networks without link overlap. The first algorithm describes a directed percolation transition and admits an epidemic spreading interpretation. The second algorithm describes the emergence of the mutually connected giant component, that is the percolation transition, but does not preserve the epidemic spreading interpretation. We obtain the phase diagrams for the percolation and directed percolation transition in simple representative cases. We demonstrate that for the same multiplex network structure, in which the directed percolation transition has nontrivial tricritical points, the percolation transition has a discontinuous phase transition, with the exception of the trivial case in which all the layers completely overlap.

Cooperative epidemics on multiplex networks

The spread of one disease, in some cases, can stimulate the spreading of another infectious disease. Here, we treat analytically a symmetric coinfection model for spreading of two diseases on a two-layer multiplex network. We allow layer overlapping, but we assume that each layer is random and locally loopless. Infection with one of the diseases increases the probability of getting infected with the other. Using the generating function method, we calculate exactly the fraction of individuals infected with both diseases (so-called coinfected clusters) in the stationary state, as well as the epidemic spreading thresholds and the phase diagram of the model. With increasing cooperation, we observe a tricritical point and the type of transition changes from continuous to hybrid. Finally, we compare the coinfected clusters in the case of cooperating diseases with the so-called "viable" clusters in networks with dependencies.

Breakdown of interdependent directed networks

Increasing evidence shows that real-world systems interact with one another via dependency connectivities. Failing connectivities are the mechanism behind the breakdown of interacting complex systems, e.g., blackouts caused by the interdependence of power grids and communication networks. Previous research analyzing the robustness of interdependent networks has been limited to undirected networks. However, most real-world networks are directed, their in-degrees and out-degrees may be correlated, and they are often coupled to one another as interdependent directed networks. To understand the breakdown and robustness of interdependent directed networks, we develop a theoretical framework based on generating functions and percolation theory. We find that for interdependent Erdős-Rényi networks the directionality within each network increases their vulnerability and exhibits hybrid phase transitions. We also find that the percolation behavior of interdependent directed scale-free networks with and without degree correlations is so complex that two criteria are needed to quantify and compare their robustness: the percolation threshold and the integrated size of the giant component during an entire attack process. Interestingly, we find that the in-degree and out-degree correlations in each network layer increase the robustness of interdependent degree heterogeneous networks that most real networks are, but decrease the robustness of interdependent networks with homogeneous degree distribution and with strong coupling strengths. Moreover, by applying our theoretical analysis to real interdependent international trade networks, we find that the robustness of these real-world systems increases with the in-degree and out-degree correlations, confirming our theoretical analysis.