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

Analysis of 'big data' characterized by high-dimensionality such as word vectors and complex networks requires often their representation in a geometrical space by embedding. Recent developments in machine learning and network geometry have pointed out the hyperbolic space as a useful framework for the representation of this data derived by real complex physical systems. In the hyperbolic space, the radial coordinate of the nodes characterizes their hierarchy, whereas the angular distance between them represents their similarity. Several studies have highlighted the relationship between the angular coordinates of the nodes embedded in the hyperbolic space and the community metadata available. However, such analyses have been often limited to a visual or qualitative assessment. Here, we introduce the angular separation index (ASI), to quantitatively evaluate the separation of node network communities or data clusters over the angular coordinates of a geometrical space. ASI is particularly useful in the hyperbolic space - where it is extensively tested along this study - but can be used in general for any assessment of angular separation regardless of the adopted geometry. ASI is proposed together with an exact test statistic based on a uniformly random null model to assess the statistical significance of the separation. We show that ASI allows to discover two significant phenomena in network geometry. The first is that the increase of temperature in 2D hyperbolic network generative models, not only reduces the network clustering but also induces a 'dimensionality jump' of the network to dimensions higher than two. The second is that ASI can be successfully applied to detect the intrinsic dimensionality of network structures that grow in a hidden geometrical space.