Wasserstein Non-Negative Matrix Factorization for Multi-Layered Graphs and its Application to Mobility Data

Multi-layered graphs are popular in mobility studies because transportation data include multiple modalities, such as railways, buses, and taxis. Another example of a multi-layered graph is the time series of mobility when periodicity is considered. The graphs are analyzed using standard signal processing methods such as singular value decomposition and tensor analysis, which can estimate missing values. However, their feature extraction abilities are insufficient for optimizing mobility networks. This study proposes a method that combines the Wasserstein non-negative matrix factorization (W-NMF) with line graphs to obtain low-dimensional representations of multi-layered graphs. A line graph is defined as the dual graph of a graph, where the vertices correspond to the edges of the original graph, and the edges correspond to the vertices. Thus, the shortest path length between two vertices in the line graph corresponds to the distance between the edges in the original graph. Through experiments using synthetic and benchmark datasets, we show that the performance and robustness of our method are superior to conventional methods. Additionally, we apply our method to real-world taxi origin—destination data as a mobility dataset and discuss the findings.