Quantifying the spatial heterogeneity of geographical flows

Geographical flows describe the movements and connections of materials, energy, and information among locations and are commonly represented by origin-destination (OD) flows (flows for short). The spatial heterogeneity of such flows is characterized by their inhomogeneous distributions and offers a novel perspective for revealing the global spatial pattern of relevant geographical phenomena. In practice, comparing the spatial heterogeneity of different flow datasets is essential for gaining deeper insights into their global spatial patterns. This requires reliable quantification of the spatial heterogeneity of flows, a challenge that has been largely overlooked in previous studies. To address this gap, we first define the spatial heterogeneity of flows as the degree of deviation from complete spatial randomness (CSR). Based on this definition, we propose a benchmark spatial heterogeneity metric for flows called the normalized level of flow heterogeneity (NLFH*). Additionally, we propose nine nearest-neighbor (NN) distance-based statistics for flows by extending relevant methods for points. Simulation experiments and case studies involving tropical cyclone tracks and taxi OD data demonstrate that statistic NLFH*, along with two NN distance-based statistics of flows (FA-w and FH-xw), outperforms other statistics in quantifying the spatial heterogeneity of flows. Among them, FA-w and FH-xw are recommended for practical use due to their powerful performance and computational efficiency.