In a variety of scientific fields, one might need to reduce the dimensionality of a problem in order to make it treatable or to be able to detect patterns that are hardly seen in high-dimensional data. Network embedding techniques, for instance, have been developed to encode large network structures into spaces with lower, tunable dimensions. The main challenge with these techniques is to find an optimal reduced dimension that is small enough to be efficient, but large enough to keep all of the necessary properties of the whole network. In addition, ideally, the embedding has to be applicable for a variety of tasks. Previous studies have proposed methods that select the embedding dimension via maximization of the performance for the selected tasks and that require setting an initial spatial embedding to start with. However, having this prior knowledge is not straightforward, especially if the data is characterized by high complexity.
In a recent work, M. Ángeles Serrano and colleagues introduced a method for the detection of the intrinsic dimensionality of high-dimensional networks without the need for an initial spatial embedding. The method is based on a multidimensional geometric model in hyperbolic space that describes the connectivity of the network and that exploits the densities of edge cycles to extract information about dimensionality. The authors showed that original networks operate in a reduced region of a high-dimensional space and can be represented in hyperbolic geometry with ultra-low dimensionality, which was even lower when compared to the results obtained by other approaches. Interestingly, by testing the approach on some real networks, domain-specific dependencies were observed: tissue-specific bio-molecular networks in the cell were found to be extremely low-dimensional, connectomes in the brain were close to three dimensions, and social networks required more than three dimensions for a faithful description. In addition to reducing the computational cost for analyzing high-dimensional network data, this work paves the way for a better detection of critical transitions and patterns, and for the development of improved techniques for dimensionality reduction of complex relational datasets.
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