Interrelation measurement based on the multi-layer limited penetrable horizontal visibility graph

Interrelation measurement is the basis of big data mining. This paper proposes an efficient method to measure the dynamic correlation and synchronicity relationship of multidimensional data using the microscopic topological structure of a multi-layer network. In order to measure the dynamic correlation between multidimensional data, multidimensional data are transformed into a time-varying multi-layer limited penetrable horizontal visibility graph network. On this basis, a time-varying correlation measurement index of multidimensional data based on the microscopic structure of the interlayer network connection is proposed. In addition, based on the degree distribution of each layer and information entropy theory, a time-varying information measurement index of multidimensional data is introduced. Further, to measure the synchronicity relationship between multidimensional data, time-delay parameters are defined, and a method to transform multidimensional data into a delay time-varying multi-layer limited penetrable horizontal visibility graph network is developed. A symmetrical and antisymmetrical combinations index is defined to measure the synchronicity relationship and to determine which system leads the others. Numerical simulation verifies the effectiveness of the proposed index and the proposed method's robustness to handle data disturbed by noise. Finally, an empirical analysis is conducted using the price data of the energy and carbon markets. The dynamic relationship between the crude oil future and gasoline future market is obtained. The dynamic information spillover effect between the carbon and energy markets is analyzed.