Spatial networks have recently attracted great interest in various fields of research. While the traditional network-theoretic viewpoint is commonly restricted to their topological characteristics (often disregarding existing spatial constraints), this work takes a geometric perspective, which considers vertices and edges as objects in a metric space and quantifies the corresponding spatial distribution and alignment. For this purpose, we introduce the concept of edge anisotropy and define a class of measures characterizing the spatial directedness of connections. Specifically, we demonstrate that the local anisotropy of edges incident to a given vertex provides useful information about the local geometry of geophysical flows based on networks constructed from spatio-temporal data, which is complementary to topological characteristics of the same flow networks. Taken both structural and geometric viewpoints together can thus assist the identification of underlying flow structures from observations of scalar variables.
This work was financially supported by the German Research Foundation (DFG) via the DFG Graduate School 1536 (“Visibility and Visualization”), the European Commission via the Marie-Curie ITN LINC (P7-PEOPLE-2011-ITN, Grant No. 289447), the German Federal Ministry for Education and Research (BMBF) via the BMBF Young Investigator's Group CoSy-CC2 (“Complex Systems Approaches to Understanding Causes and Consequences of Past, Present and Future Climate Change, Grant No. 01LN1306A”) and the project GLUES, the Stordalen Foundation (via the Planetary Boundary Research Network PB.net), the Earth League's EarthDoc program, and the Volkswagen Foundation via the project “Recurrent extreme events in spatially extended excitable systems: Mechanism of their generation and termination” (Grant No. 85391). The presented research has greatly benefited from discussions with Emilio Hernández-Garcia and Cristóbal López. Parts of the network calculations have been performed using the Python package pyunicorn56 (see http://tocsy.pik-potsdam.de/pyunicorn.php). pyunicorn is freely available for download at https://github.com/pik-copan/pyunicorn