Edge directionality properties in complex spherical networks

Spatially embedded networks have attracted increasing attention in the past decade. In this context, network characteristics have been introduced which explicitly take spatial information into account. Among others, edge directionality properties have recently gained particular interest. In this work, we investigate the applicability of mean edge direction, anisotropy, and local mean angle as geometric characteristics in complex spherical networks. By studying these measures, both analytically and numerically, we demonstrate the existence of a systematic bias in spatial networks where individual nodes represent different shares on a spherical surface, and we describe a strategy for correcting for this effect. Moreover, we illustrate the application of the mentioned edge directionality properties to different examples of real-world spatial networks in spherical geometry (with or without the geometric correction depending on each specific case), including functional climate networks, transportation, and trade networks. In climate networks, our approach highlights relevant patterns, such as large-scale circulation cells, the El Niño–Southern Oscillation, and the Atlantic Niño. In an air transportation network, we are able to characterize distinct air transportation zones, while we confirm the important role of the European Union for the global economy by identifying convergent edge directionality patterns in the world trade network.

Figure 1

(a, c, e) Uncorrected and (b, d, f) AOR corrected values of (a, b) local mean angle, (c, d) local anisotropy, and (e, f) zonal mean anisotropy for one realization of the benchmark network. Note the different color scales in panels (c) and (d).

Figure 2

(a) N.s.i. degree and (b) AOR corrected anisotropy of the aquaplanet SAT climate network. (c, d) Corresponding zonal means of (a, b) together with the meridonal wind at 925 hPa (rescaled to dimensionless units) to indicate the circulation cells. The edges of the Hadley ($H_{s/n}$), Ferrel ($F_{s/n}$) and Polar cells ($P_{s/n}$) are indicated, with subscripts specifying the hemisphere.

Figure 3

(a, b) Local anisotropy and (c, d) mean edge direction of the (a, c) SAT and (b, d) precipitation network from the ERA-Interim data for all nodes with $R_m>0.25$ and $k_m>10$. The color code indicates the cardinal direction. Nodes with degree zero are assigned zero anisotropy and represented by grey dots.

Figure 4

(a) Mean edge direction of the air transportation network for all nodes with $R_m>0.3$ and $k_m>25$. (b) Mean edge direction of the air transportation network over Eurasia. The color code indicates the cardinal direction.

Figure 5

Mean edge direction of the global (a) import and (b) export network of 2009 for all nodes with $R_m>0.3$ and $k_m>25$. The color code indicates the cardinal direction. The arrows point towards (import) or away from (export) the centers of the respective countries.

Figure 6

Mean edge direction of an excerpt of the global import network including Europe and northern Africa.

Figure 7

Schematic illustration of a spherical triangle $\mathrm{\Delta }ABC$ as utilized for the derivation of the course angle. Note that point $C$ is identified with the North Pole (${\varphi }_C={90}^{\circ }$).