Multilayer time-variant network edges synchronization method

Abstract

In this work, we propose a method for measuring synchronization between the edges of a pair of networks with connections that vary over time. The product of this method is an Incidence-Fidelity network, indicating the intensity of edge synchronization during a time interval. Two distinct validations were performed: in the first, we employed time-varying multilayer graphs with random connections to validate the algorithm; and in the second, we used the coupled Rössler oscillator to validate the method. In the algorithm validation, we compared the algebraically predicted results with the results obtained by using random networks. In the method validation, we used the coupled Rössler oscillator to generate time series representing the existence of two edges, one in each layer of the time-varying multilayer graphs. We assessed the relationship between the pre-defined coupling values and the achieved synchronization indices. In the application of our method, we considered time series of 17 years of climate data for the Normalized Difference Vegetation Index (NDVI) and precipitation for all municipalities in the state of Bahia, Brazil. To do so, we first obtained the individual layers of the multilayer network using a motif synchronization method. From the individual layers, we generated the Interlayer Incidence-Fidelity network. The obtained Interlayer Incidence-Fidelity network for the NDVI and precipitation pair reflects a connection pattern that incorporates the configurations of the individual layers. This is because synchronizations are primarily distributed in the more interconnected regions of the individual layers. The Interlayer Incidence-Fidelity networks generated by our method can identify synchronization patterns in the evolution of relationships between networks over time. This method presents a promising tool for evaluating synchronization between networks, generating a weighted network that quantifies the different levels of temporal synchronization for each edge in the networks.