Spatially embedded networks have become a standard tool in Earth and social sciences. Nodes in such networks are often tied to the nodes of a regular spherical grid, as in climate data, where each cell represents its own patch of the surface. However, the area of cells near the poles is smaller than near the equator, a long-known issue for topological measures. Directional characteristics, previously introduced for Euclidean networks, had not yet been tested on spherical networks with non-uniform area representation. This work stands apart: the nearest studies in the archive concern weak AI agents, urban transportation networks, and weight geometry, but do not address spherical edge directionality.
The authors analytically derive an expression for the bearing of an edge on a sphere in terms of the latitudes and longitudes of its nodes. To test for bias, they construct a synthetic network on a regular angular grid with a resolution of 1.875° × 1.875° (18,432 nodes), where connections appear with a probability depending only on the angular distance. They compare local anisotropy and mean edge direction before and after applying weights proportional to the cell area of the neighboring node. They then apply the measures to functional climate networks from ERA-Interim reanalysis data and aquaplanet simulations, as well as to the OpenFlights aviation network and the Eora world trade network for 2009.
On the synthetic network without correction, anisotropy increases from the equator toward the poles, peaking around ±60° latitude, and the mean edge direction in the Northern Hemisphere points north, while in the Southern Hemisphere it points south. After introducing weights proportional to the cosine of the neighboring node's latitude, the zonal mean anisotropy drops by a factor of one hundred, and the angle distribution becomes uniform. In the climate network based on surface temperature, the correction reveals zones of low anisotropy in the centers of the Hadley and Ferrel cells, while in the precipitation network it uncovers the structure of El Niño and the Atlantic Niño with opposite edge directions. In the aviation network, the mean directions of outbound flights in Europe point south, while in East Asia they converge toward central China. In the trade network, imports and exports exhibit different geometric patterns, with EU countries forming a zone of convergent import directions.
This work extends the concept of edge anisotropy to the sphere and shows for the first time that non-uniform cell areas create a systematic bias even for completely random connections. Unlike neighboring studies that examine topology or weight geometry in Euclidean space, here an explicit correction is proposed for spherical geometry. This paves the way for correctly analyzing the directionality of flows in climate, transportation, and economic networks where nodes have different representative areas.
The method can be extended to networks embedded on arbitrary non-Euclidean surfaces, where the notion of bearing requires generalization. For climate networks, seasonal analysis and tracking the temporal dynamics of directionality are promising. Boundary effects will also need to be accounted for when studying local regions rather than the entire sphere.
The method is applicable in climatology for identifying atmospheric teleconnections, in transportation logistics for analyzing flows, and in economic geography for studying trade links.
Test the correction on irregular spherical grids with arbitrary node placement. Develop a generalization of directionality measures for surfaces other than the sphere.
For the theory of complex networks, the question of correctly defining geometric measures on curved manifolds remains open. This work solves the specific case of the sphere, but the general problem of measure invariance under surface partitioning is not closed.
In the aviation network, airports on the west coast of the Americas have a mean outbound flight direction toward the east, while Asian airports point west, reflecting the asymmetry of transpacific traffic.