Abstract
Many complex systems in the real world can be modeled as complex networks, which has captured in recent years enormous attention from researchers of diverse fields ranging from natural sciences to engineering. The extinction of species in ecosystems and the blackouts of power girds in engineering exhibit the vulnerability of complex networks, investigated by empirical data and analyzed by theoretical models. For studying the resilience of complex networks, three main factors should be focused on: the network structure, the network dynamics and the failure mechanism. In this review, we will introduce recent progress on the resilience of complex networks based on these three aspects. For the network structure, increasing evidence shows that biological and ecological networks are coupled with each other and that diverse critical infrastructures interact with each other, triggering a new research hotspot of “networks of networks” (NON), where a network is formed by interdependent or interconnected networks. The resilience of complex networks is deeply influenced by its interdependence with other networks, which can be analyzed and predicted by percolation theory. This review paper shows that the analytic framework for Energies 2015, 8 12188 NON yields novel percolation laws for n interdependent networks and also shows that the percolation theory of a single network studied extensively in physics and mathematics in the last 60 years is a specific limited case of the more general case of n interacting networks. Due to spatial constraints inherent in critical infrastructures, including the power gird, we also review the progress on the study of spatially-embedded interdependent networks, exhibiting extreme vulnerabilities compared to their non-embedded counterparts, especially in the case of localized attack. For the network dynamics, we illustrate the percolation framework and methods using an example of a real transportation system, where the analysis based on network dynamics is significantly different from the structural static analysis. For the failure mechanism, we here review recent progress on the spontaneous recovery after network collapse. These findings can help us to understand, realize and hopefully mitigate the increasing risk in the resilience of complex networks.
Highlights
The resilience of a network characterizes its ability to remain functional after node or link failures, the study of which is usually based on three main factors: the network structure [4,5,8,33,34], the network dynamics [35,36] and the failure mechanism [37,38,39]
The percolation threshold, pc, is the minimal fraction of remaining nodes that leads to the collapse of the network [4,43], which is usually predicted by percolation theory, a method from statistical physics [43]
In interdependent non-embedded networks, there is a critical coupling of interdependence above which a single failure can invoke cascading failures that may abruptly break down the system, while below which small-scale failures only lead to small damages on the system, and the percolation transition is continuous and not a collapse
Summary
Network science is an interdisciplinary research field, attracting enormous and increasing attention from many disciplines, ranging from computer science, physics, biology, social science to engineering [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. The resilience of a network characterizes its ability to remain functional after node or link failures, the study of which is usually based on three main factors: the network structure [4,5,8,33,34], the network dynamics [35,36] and the failure mechanism [37,38,39]. These papers [39,99] demonstrate the conditions for which a system will be in a functional state These three main factors (network structure, network dynamics and failure mechanism) play important roles in determining the resilience of realistic critical infrastructures, which may shed light on the study of the resilience of the power grid (an important system related to energies). We summarize the percolation transition and spontaneous recovery in dynamic systems respectively in Sections 4 and 5
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