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.

multi-layer network  /  null model  /  random mixing of edges  /  structure correlation