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Analysis of 'big data' characterized by high-dimensionality such as wordvectors and complex networks requires often their representation in ageometrical space by embedding. Recent developments in machine learning andnetwork geometry have pointed out the hyperbolic space as a useful frameworkfor the representation of this data derived by real complex physical systems.In the hyperbolic space, the radial coordinate of the nodes characterizes theirhierarchy, whereas the angular distance between them represents theirsimilarity. Several studies have highlighted the relationship between theangular coordinates of the nodes embedded in the hyperbolic space and thecommunity metadata available. However, such analyses have been often limited toa visual or qualitative assessment. Here, we introduce the angular separationindex (ASI), to quantitatively evaluate the separation of node networkcommunities or data clusters over the angular coordinates of a geometricalspace. ASI is particularly useful in the hyperbolic space - where it isextensively tested along this study - but can be used in general for anyassessment of angular separation regardless of the adopted geometry. ASI isproposed together with an exact test statistic based on a uniformly random nullmodel to assess the statistical significance of the separation. We show thatASI allows to discover two significant phenomena in network geometry. The firstis that the increase of temperature in 2D hyperbolic network generative models,not only reduces the network clustering but also induces a 'dimensionalityjump' of the network to dimensions higher than two. The second is that ASI canbe successfully applied to detect the intrinsic dimensionality of networkstructures that grow in a hidden geometrical space.

Angular separability of data clusters or network communities in geometrical space and its relevance to hyperbolic embedding