Optimisation of the coalescent hyperbolic embedding of complex networks

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.

Learning Pattern Relation-Based Hyperbolic Embedding for Adverse Drug Reaction Extraction

Hyperbolic embedding has been recently developed to allow us to embed words in a Cartesian product of hyperbolic spaces, and its efficiency has been proved in several works of literature since the hierarchical structure is the natural form of texts. Such a hierarchical structure exhibits not only the syntactic structure but also semantic representation. This paper presents an approach to learn meaningful patterns by hyperbolic embedding and then extract adverse drug reactions from electronic medical records. In the experiments, the public source of data from MIMIC-III (Medical Information Mart for Intensive Care III) with over 58,000 observed hospital admissions of the brief hospital course section is used, and the result shows that the approach can construct a set of efficient word embeddings and also retrieve texts of the same relation type with the input. With the Poincaré embeddings model and its vector sum (PC-S), the authors obtain up to 82.3% in the precision at ten, 85.7% in the mean average precision, and 93.6% in the normalized discounted cumulative gain.

The hyperbolic geometry of financial networks

Based on data from the European banking stress tests of 2014, 2016 and the transparency exercise of 2018 we construct networks of European banks and demonstrate that the latent geometry of these financial networks can be well-represented by geometry of negative curvature, i.e., by hyperbolic geometry. Using two different hyperbolic embedding methods, hydra+ and Mercator, this allows us to connect the network structure to the popularity-vs-similarity model of Papdopoulos et al., which is based on the Poincaré disc model of hyperbolic geometry. We show that the latent dimensions of ‘popularity’ and ‘similarity’ in this model are strongly associated to systemic importance and to geographic subdivisions of the banking system, independent of the embedding method that is used. In a longitudinal analysis over the time span from 2014 to 2018 we find that the systemic importance of individual banks has remained rather stable, while the peripheral community structure exhibits more (but still moderate) variability. Based on our analysis we argue that embeddings into hyperbolic geometry can be used to monitor structural change in financial networks and are able to distinguish between changes in systemic relevance and other (peripheral) structural changes.

Hyperbolic matrix factorization reaffirms the negative curvature of the native biological space

Past research in systems biology has taken for granted the Euclidean geometry of biological space. This has not only drawn parallels to other fields but has also been convenient due to the ample statistical and numerical optimization tools available to address the core task and downstream machine learning problems. However, emerging theoretical studies now demonstrate that biological databases exhibit hierarchical topology, characterized by heterogeneous degree distribution and a high degree of clustering, thus contradicting the flat geometry assumption. Namely, since the number of nodes in hierarchical structures grows exponentially with node depth, the biological networks naturally reside in a hyperbolic space where the circle circumference and disk area are the exponential functions of the radius. To test these claims and assess potential benefits of the applications grounded in the above hypothesis, we have developed a mathematical framework and an accompanying computational procedure for matrix factorization and implied biological relationship inference in hyperbolic space. Not only does our study demonstrate a significant increase in the accuracy of hyperbolic embedding compared to Euclidean embedding, but it also shows that the latent dimension of an optimal hyperbolic embedding is by more than an order of magnitude smaller than the latent dimension of an optimal Euclidean embedding. We see this as additional evidence that hyperbolic geometry, rather than Euclidean, underlines the biological system.

Hyperbolic mapping of human proximity networks

Human proximity networks are temporal networks representing the close-range proximity among humans in a physical space. They have been extensively studied in the past 15 years as they are critical for understanding the spreading of diseases and information among humans. Here we address the problem of mapping human proximity networks into hyperbolic spaces. Each snapshot of these networks is often very sparse, consisting of a small number of interacting (i.e., non-zero degree) nodes. Yet, we show that the time-aggregated representation of such systems over sufficiently large periods can be meaningfully embedded into the hyperbolic space, using methods developed for traditional (non-mobile) complex networks. We justify this compatibility theoretically and validate it experimentally. We produce hyperbolic maps of six different real systems, and show that the maps can be used to identify communities, facilitate efficient greedy routing on the temporal network, and predict future links with significant precision. Further, we show that epidemic arrival times are positively correlated with the hyperbolic distance from the infection sources in the maps. Thus, hyperbolic embedding could also provide a new perspective for understanding and predicting the behavior of epidemic spreading in human proximity systems.

Hydra: a method for strain-minimizing hyperbolic embedding of network- and distance-based data

We introduce hydra (hyperbolic distance recovery and approximation), a new method for embedding network- or distance-based data into hyperbolic space. We show mathematically that hydra satisfies a certain optimality guarantee: it minimizes the ‘hyperbolic strain’ between original and embedded data points. Moreover, it is able to recover points exactly, when they are contained in a low-dimensional hyperbolic subspace of the feature space. Testing on real network data we show that the embedding quality of hydra is competitive with existing hyperbolic embedding methods, but achieved at substantially shorter computation time. An extended method, termed hydra+, typically outperforms existing methods in both computation time and embedding quality.