Optimisation of the coalescent hyperbolic embedding of complex networks

TL;DR

This paper introduces an optimized hyperbolic embedding method for complex networks, enhancing the accuracy of node coordinate inference by refining the angular coordinates in the ncMCE framework, which improves routing efficiency and reduces loss.

Contribution

It proposes a novel optimization of angular coordinates in the coalescent hyperbolic embedding, improving embedding quality and routing performance over existing methods.

Findings

Reduced logarithmic loss in hyperbolic embedding

Increased greedy routing score

Enhanced accuracy of node coordinate inference

Abstract

Several observations indicate the existence of a latent hyperbolic space behind real networks that makes their structure very intuitive in the sense that the probability for a connection is decreasing with the hyperbolic distance between the nodes. A remarkable network model generating random graphs along this line is the popularity-similarity optimisation (PSO) model, offering a scale-free degree distribution, high clustering and the small world property at the same time. These results provide a strong motivation for the development of hyperbolic embedding algorithms, that tackle the problem of finding the optimal hyperbolic coordinates of the nodes based on the network structure. A very promising recent approach for hyperbolic embedding is provided by the noncentered minimum curvilinear embedding (ncMCE) method, belonging to the family of coalescent embedding algorithms. This approach offers a high quality embedding at a low running time. In the present work we propose a further optimisation of the angular coordinates in this framework that seems to reduce the logarithmic loss and increase the greedy routing score of the embedding compared to the original version, thereby adding an extra improvement to the quality of the inferred hyperbolic coordinates.