多层网络是近年来提出的新型复杂网络模型，在级联故障分析、信息传播、链路预测和网络同步等诸多领域均有广泛应用。多层网络的任意两层网络结构间往往存在关联性和耦合性，如何检测两层网络之间是否具有结构相关性并对其进行定量刻画是一个非常重要且亟待解决的问题。本文首先分3个层次总结并提出度量双层网络结构相关性的方法与统计量，其中第1层次是检测双层网络整体上的连接相关性，第2层次检测两层网络所有节点之间整体上的度度相关特性，第3层次是检测双层网络富节点之间的连接相关性。由于这3种相关性的计算和度量都依赖于网络统计量进行，而统计量的绝对数值往往没有意义，因此提出了以多种双层网络的零模型作为参照物，通过假设检验方法来量化双层网络之间的结构相关性，分析了这种结构相关性存在的内在机理。最后，使用一个实证双层网络——全球语言多层网络验证了本文研究范式的有效性。本文研究可检测出实证多层网络中任意两层之间复杂的耦合作用。
Recently the framework of multi-layer networks was proposed as a new model of complex networks, and was used in widespread applications in many fields, such as the cascading failure, the information spreading, the link prediction and the network synchronization. In multi-layer networks, the correlation and the coupling between two layers of network structures might exist, so it is a very significant issue how to detect the structural correlation and quantify the correlation between the two layers. In this study we summarize and propose methods to measure the structural correlation of double-layer networks in three levels. The first level is to detect the overall connection relationship of the whole double-layer network. The second level is to test the degree correlation characteristics between all nodes at different layers. At last, the third level is to look for the connection relationship between the rich nodes at different layers. Although the three kinds of correlations are all dependent on the network statistics, these statistics are all without units. Furthermore, the sizes and the structures of different networks see a great difference. Therefore, absolute numerical values of some statistics often are not important and we put forward a variety of null models for double-layer networks as a reference. Through the hypothesis testing methods, we can quantify the structure correlation in double-layer networks, and try to analyze the intrinsic mechanism of inducing this kind of structure correlation. Finally, we use an empirical double-layer network(the global language multi-layer network)to verify the effectiveness of our methodology. This methodology can be used to detect the complex coupling between the layers in an empirical double-layer network, and for the better understanding of multi-layer networks and for new applications based on the structure complexity of multi-layer networks.
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