Machine learning meets complex networks via coalescent embedding in the hyperbolic space

Complex network topologies and hyperbolic geometry seem specularly connected (Papadopoulos et al. 2012), and one of the most fascinating and challenging problems of recent complex network theory is to map a given network to its hyperbolic space. The Popularity Similarity Optimization (PSO) model represents-at the moment-the climax of this theory (Papadopoulos et al. 2012). It suggests that the trade-off between node popularity and similarity is a mechanism to explain how complex network topologies emerge-as discrete samples-from the continuous world of hyperbolic geometry (Papadopoulos et al. 2015). The hyperbolic space seems appropriate to represent real complex networks. In fact, it preserves many of their fundamental topological properties, and can be exploited for real applications such as, among others, link prediction and community detection. Here, we observe for the first time that a topological-based machine learning class of algorithms-for nonlinear unsupervised dimensionality reduction-can directly approximate the network's node angular coordinates of the hyperbolic model into a two-dimensional space, according to a similar topological organization that we named angular coalescence. On the basis of this phenomenon, we propose a new class of algorithms that offers fast and accurate coalescent embedding of networks in the hyperbolic space even for graphs with thousands of nodes.