The topic of finding effective strategies to restrain epidemic spreading in complex networks is of current interest. A widely used approach for epidemic containment is the fragmentation of the contact networks through immunization. However, due to the limitation of immune resources, we cannot always fragment the contact network completely. In this study, based on the size distribution of connected components for the network, we designed a risk indicator of epidemic outbreaks, the generalized Herfindahl–Hirschman index (GHI), which measures the upper bound of the expected infection’s prevalence (the fraction of infected nodes) in random outbreaks. An immunization approach based on minimizing GHI is developed to reduce the infection risk for individuals in the network. Experimental results show that our immunization strategy could effectively decrease the infection’s prevalence as compared to other existing strategies, especially against infectious diseases with higher infection rates or lower recovery rates. The findings provide an efficient and practicable strategy for immunization against epidemic diseases.

Networks can provide effective representations of the relationships between elements in complex systems through nodes and links. On this basis, relationships between multiple systems are often characterized as multilayer networks (or networks of networks). As a typical representative, a multiplex network is often used to describe a system in which there are many replaceable or dependent relationships among elements in different layers. This paper studies robustness measures for different types of multiplex networks by generalizing the natural connectivity calculated from the graph spectrum. Experiments on model and real multiplex networks show a close correlation between the robustness of multiplex networks consisting of connective or dependent layers and the natural connectivity of aggregated networks or intersections between layers. These indicators can effectively measure or estimate the robustness of multiplex networks according to the topology of each layer. Our findings shed new light on the design and protection of coupled complex systems.

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