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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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Many man-made networks support each other to provide efficient services and resources to the customers, despite that this support produces a strong interdependency between the individual networks. Thus an initial failure of a fraction $1-p$ of nodes in one network, exposes the system to cascade of failures and, as a consequence, to a full collapse of the overall system. Therefore it is important to develop efficient strategies to avoid the collapse by increasing the robustness of the individual networks against failures. Here, we provide an exact theoretical approach to study the evolution of the cascade of failures on interdependent networks when a fraction $alpha$ of the nodes with higher connectivity in each individual network are autonomous. With this pattern of interdependency we found, for pair of heterogeneous networks, two critical percolation thresholds that depend on $alpha$, separating three regimes with very different networks final sizes that converge into a triple point in the plane $p-alpha$. Our findings suggest that the heterogeneity of the networks represented by high degree nodes is the responsible of the rich phase diagrams found in this and other investigations.
In many real network systems, nodes usually cooperate with each other and form groups, in order to enhance their robustness to risks. This motivates us to study a new type of percolation, group percolation, in interdependent networks under attacks. In this model, nodes belonging to the same group survive or fail together. We develop a theoretical framework for this novel group percolation and find that the formation of groups can improve the resilience of interdependent networks significantly. However, the percolation transition is always of first order, regardless of the distribution of group sizes. As an application, we map the interdependent networks with inter-similarity structures, which attract many attentions very recently, onto the group percolation and confirm the non-existence of continuous phase transitions.
Computer viruses are evolving by developing spreading mechanisms based on the use of multiple vectors of propagation. The use of the social network as an extra vector of attack to penetrate the security measures in IP networks is improving the effectiveness of malware, and have therefore been used by the most aggressive viruses, like Conficker and Stuxnet. In this work we use interdependent networks to model the propagation of these kind of viruses. In particular, we study the propagation of a SIS model on interdependent networks where the state of each node is layer-independent and the dynamics in each network follows either a contact process or a reactive process, with different propagation rates. We apply this study to the case of existing multilayer networks, namely a Spanish scientific community of Statistical Physics, formed by a social network of scientific collaborations and a physical network of connected computers in each institution. We show that the interplay between layers increases dramatically the infectivity of viruses in the long term and their robustness against immunization.
Modern world builds on the resilience of interdependent infrastructures characterized as complex networks. Recently, a framework for analysis of interdependent networks has been developed to explain the mechanism of resilience in interdependent networks. Here we extend this interdependent network model by considering flows in the networks and study the systems resilience under different attack strategies. In our model, nodes may fail due to either overload or loss of interdependency. Under the interaction between these two failure mechanisms, it is shown that interdependent scale-free networks show extreme vulnerability. The resilience of interdependent SF networks is found in our simulation much smaller than single SF network or interdependent SF networks without flows.
In spite of the extensive previous efforts on traffic dynamics and epidemic spreading in complex networks, the problem of traffic-driven epidemic spreading on {em correlated} networks has not been addressed. Interestingly, we find that the epidemic threshold, a fundamental quantity underlying the spreading dynamics, exhibits a non-monotonic behavior in that it can be minimized for some critical value of the assortativity coefficient, a parameter characterizing the network correlation. To understand this phenomenon, we use the degree-based mean-field theory to calculate the traffic-driven epidemic threshold for correlated networks. The theory predicts that the threshold is inversely proportional to the packet-generation rate and the largest eigenvalue of the betweenness matrix. We obtain consistency between theory and numerics. Our results may provide insights into the important problem of controlling/harnessing real-world epidemic spreading dynamics driven by traffic flows.
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