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We investigate robustness of correlated networks against propagating attacks modeled by a susceptible-infected-removed model. By Monte-Carlo simulations, we numerically determine the first critical infection rate, above which a global outbreak of disease occurs, and the second critical infection rate, above which disease disintegrates the network. Our result shows that correlated networks are robust compared to the uncorrelated ones, regardless of whether they are assortative or disassortative, when a fraction of infected nodes in an initial state is not too large. For large initial fraction, disassortative network becomes fragile while assortative network holds robustness. This behavior is related to the layered network structure inevitably generated by a rewiring procedure we adopt to realize correlated networks.
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With increasing threats by large attacks or disasters, the time has come to reconstruct network infrastructures such as communication or transportation systems rather than to recover them as before in case of accidents, because many real networks are extremely vulnerable. Thus, we consider self-healing mechanisms by rewirings (reuse or addition of links) to be sustainable and resilient networks even against malicious attacks. In distributed local process for healing, the key strategies are the extension of candidates of linked nodes and enhancing loops by applying a message-passing algorithm inspired from statistical physics. Simulation results show that our proposed combination of ring formation and enhancing loops is particularly effective in comparison with the conventional methods, when more than half damaged links alive or are compensated from reserved ones.
In a recent work [Proc. Natl. Acad. Sci. USA 108, 3838 (2011)], the authors proposed a simple measure for network robustness under malicious attacks on nodes. With a greedy algorithm, they found the optimal structure with respect to this quantity is an onion structure in which high-degree nodes form a core surrounded by rings of nodes with decreasing degree. However, in real networks the failure can also occur in links such as dysfunctional power cables and blocked airlines. Accordingly, complementary to the node-robustness measurement ($R_{n}$), we propose a link-robustness index ($R_{l}$). We show that solely enhancing $R_{n}$ cannot guarantee the improvement of $R_{l}$. Moreover, the structure of $R_{l}$-optimized network is found to be entirely different from that of onion network. In order to design robust networks resistant to more realistic attack condition, we propose a hybrid greedy algorithm which takes both the $R_{n}$ and $R_{l}$ into account. We validate the robustness of our generated networks against malicious attacks mixed with both nodes and links failure. Finally, some economical constraints for swapping the links in real networks are considered and significant improvement in both aspects of robustness are still achieved.
In todays global economy, supply chain (SC) entities have become increasingly interconnected with demand and supply relationships due to the need for strategic outsourcing. Such interdependence among firms not only increases efficiency but also creates more vulnerabilities in the system. Natural and human-made disasters such as floods and transport accidents may halt operations and lead to economic losses. Due to the interdependence among firms, the adverse effects of any disruption can be amplified and spread throughout the systems. This paper aims at studying the robustness of SC networks against cascading failures. Considering the upper and lower bound load constraints, i.e., inventory and cost, we examine the fraction of failed entities under load decrease and load fluctuation scenarios. The simulation results obtained from synthetic networks and a European supply chain network [1] both confirm that the recovery strategies of surplus inventory and backup suppliers often adopted in actual SCs can enhance the system robustness, compared with the system without the recovery process. In addition, the system is relatively robust against load fluctuations but is more fragile to demand shocks. For the underload-driven model without the recovery process, we found an occurrence of a discontinuous phase transition. Differently from other systems studied under overload cascading failures, this system is more robust for power-law distributions than uniform distributions of the lower bound parameter for the studied scenarios.
The characterization of various properties of real-world systems requires the knowledge of the underlying network of connections among the systems components. Unfortunately, in many situations the complete topology of this network is empirically inaccessible, and one has to resort to probabilistic techniques to infer it from limited information. While network reconstruction methods have reached some degree of maturity in the case of single-layer networks (where nodes can be connected only by one type of links), the problem is practically unexplored in the case of multiplex networks, where several interdependent layers, each with a different type of links, coexist. Even the most advanced network reconstruction techniques, if applied to each layer separately, fail in replicating the observed inter-layer dependencies making up the whole coupled multiplex. Here we develop a methodology to reconstruct a class of correlated multiplexes which includes the World Trade Multiplex as a specific example we study in detail. Our method starts from any reconstruction model that successfully reproduces some desired marginal properties, including node strengths and/or node degrees, of each layer separately. It then introduces the minimal dependency structure required to replicate an additional set of higher-order properties that quantify the portion of each nodes degree and each nodes strength that is shared and/or reciprocated across pairs of layers. These properties are found to provide empirically robust measures of inter-layer coupling. Our method allows joint multi-layer connection probabilities to be reliably reconstructed from marginal ones, effectively bridging the gap between single-layer properties and truly multiplex information.
Many real-world networks depend on other networks, often in non-trivial ways, to maintain their functionality. These interdependent networks of networks are often extremely fragile. When a fraction $1-p$ of nodes in one network randomly fails, the damage propagates to nodes in networks that are interdependent and a dynamic failure cascade occurs that affects the entire system. We present dynamic equations for two interdependent networks that allow us to reproduce the failure cascade for an arbitrary pattern of interdependency. We study the rich club effect found in many real interdependent network systems in which the high-degree nodes are extremely interdependent, correlating a fraction $alpha$ of the higher degree nodes on each network. We find a rich phase diagram in the plane $p-alpha$, with a triple point reminiscent of the triple point of liquids that separates a non-functional phase from two functional phases.
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