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

Alessandro Muscoloni,Josephine Maria Thomas,Carlo Vittorio Cannistraci,Ginestra Bianconi,Sara Ciucci

doi:10.1038/s41467-017-01825-5

Alessandro Muscoloni, Josephine Maria Thomas + Show 3 more

Open Access

https://doi.org/10.1038/s41467-017-01825-5

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Abstract

Physicists recently observed that realistic complex networks emerge as discrete samples from a continuous hyperbolic geometry enclosed in a circle: the radius represents the node centrality and the angular displacement between two nodes resembles their topological proximity. The hyperbolic circle aims to become a universal space of representation and analysis of many real networks. Yet, inferring the angular coordinates to map a real network back to its latent geometry remains a challenging inverse problem. Here, we show that intelligent machines for unsupervised recognition and visualization of similarities in big data can also infer the network angular coordinates of the hyperbolic model according to a geometrical organization that we term “angular coalescence.” Based on this phenomenon, we propose a class of algorithms that offers fast and accurate “coalescent embedding” in the hyperbolic circle even for large networks. This computational solution to an inverse problem in physics of complex systems favors the application of network latent geometry techniques in disciplines dealing with big network data analysis including biology, medicine, and social science.

Highlights

Physicists recently observed that realistic complex networks emerge as discrete samples from a continuous hyperbolic geometry enclosed in a circle: the radius represents the node centrality and the angular displacement between two nodes resembles their topological proximity
Here we clarify that minimum-curvilinearity-based algorithms compress the information in one unique dimension because they learn nonlinear similarities by means of the minimum spanning tree (MST), providing a hierarchical-based mapping that is fundamentally different from the manifold-based of ISO
We propose coalescent embedding: a class of unsupervised topological-based nonlinear machine learning algorithms for dimension reduction that offer a fast and accurate embedding in the hyperbolic space even for large graphs, this is the main product of the work

Summary

Introduction

Physicists recently observed that realistic complex networks emerge as discrete samples from a continuous hyperbolic geometry enclosed in a circle: the radius represents the node centrality and the angular displacement between two nodes resembles their topological proximity. We show that intelligent machines for unsupervised recognition and visualization of similarities in big data can infer the network angular coordinates of the hyperbolic model according to a geometrical organization that we term “angular coalescence.” Based on this phenomenon, we propose a class of algorithms that offers fast and accurate “coalescent embedding” in the hyperbolic circle even for large networks. Manifold machine learning for unsupervised nonlinear dimensionality reduction is an important subclass of topological machine learning algorithms They learn nonlinear similarities/ proximities (that can be interpreted as dissimilarities/distances) between points (samples) distributed over a hidden manifold in a multidimensional feature space, in order to preserve, embed (map) and visualize them in a two-dimensional reduced space[26]. These methods are already used in network biology for instance to predict node similarities in protein interaction networks[4, 8], it was likely for us to envisage their usage for network embedding in the hyperbolic space

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