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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 the 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.
Complex networks have recently attracted a rising interest for studying dynamical patterns in geophysical flows such as in the atmosphere and ocean. For this purpose, two distinct approaches have been proposed based on either (i) correlations between values of a certain variable measured at different parts of the flow domain (correlation-based flow networks) or (ii) transition probabilities of passively advected tracers between different parts of the fluid domain (Lagrangian flow networks). So far, the investigations on both types of flow networks have mostly addressed classical topological network characteristics, disregarding the fact that such networks are naturally embedded in some physical space and, hence, have intrinsic restrictions to their structural organization. In this paper, we introduce a novel concept to obtain a complementary geometric characterization of the local network patterns based on the anisotropy of edge orientations. For two prototypical model systems of different complexity, we demonstrate that the geometric characterization of correlation-based flow networks derived from scalar observables can actually provide additional and useful information contributing to the identification of the underlying flow patterns that are often not directly accessible. In this spirit, the proposed approach provides a prospective diagnostic tool for geophysical as well as technological flows
