Embedding Heterogeneous Information Network in Hyperbolic Spaces

Heterogeneous information network (HIN) embedding, aiming to project HIN into a low-dimensional space, has attracted considerable research attention. Most of the existing HIN embedding methods focus on preserving the inherent network structure and semantic correlations in Euclidean spaces. However, one fundamental problem is whether the Euclidean spaces are the intrinsic spaces of HIN? Recent researches find the complex network with hyperbolic geometry can naturally reflect some properties, e.g., hierarchical and power-law structure. In this article, we make an effort toward embedding HIN in hyperbolic spaces. We analyze the structures of three HINs and discover some properties, e.g., the power-law distribution, also exist in HINs. Therefore, we propose a novel HIN embedding model HHNE. Specifically, to capture the structure and semantic relations between nodes, HHNE employs the meta-path guided random walk to sample the sequences for each node. Then HHNE exploits the hyperbolic distance as the proximity measurement. We also derive an effective optimization strategy to update the hyperbolic embeddings iteratively. Since HHNE optimizes different relations in a single space, we further propose the extended model HHNE++. HHNE++ models different relations in different spaces, which enables it to learn complex interactions in HINs. The optimization strategy of HHNE++ is also derived to update the parameters of HHNE++ in a principle manner. The experimental results demonstrate the effectiveness of our proposed models.

The inherent community structure of hyperbolic networks

A remarkable approach for grasping the relevant statistical features of real networks with the help of random graphs is offered by hyperbolic models, centred around the idea of placing nodes in a low-dimensional hyperbolic space, and connecting node pairs with a probability depending on the hyperbolic distance. It is widely appreciated that these models can generate random graphs that are small-world, highly clustered and scale-free at the same time; thus, reproducing the most fundamental common features of real networks. In the present work, we focus on a less well-known property of the popularity-similarity optimisation model and the $${\mathbb {S}}^1/{\mathbb {H}}^2$$ S 1 / H 2 model from this model family, namely that the networks generated by these approaches also contain communities for a wide range of the parameters, which was certainly not an intention at the design of the models. We extracted the communities from the studied networks using well-established community finding methods such as Louvain, Infomap and label propagation. The observed high modularity values indicate that the community structure can become very pronounced under certain conditions. In addition, the modules found by the different algorithms show good consistency, implying that these are indeed relevant and apparent structural units. Since the appearance of communities is rather common in networks representing real systems as well, this feature of hyperbolic models makes them even more suitable for describing real networks than thought before.

Hyperbolic distance versus quasihyperbolic distance in plane domains

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct examples where the spaces are not quasiisometrically equivalent.

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.

Hyperbolic disc embedding of functional human brain connectomes using resting state fMRI

The brain presents a real complex network of modular, small-world, and hierarchical nature, which are features of non-Euclidean geometry. Using resting-state functional magnetic resonance imaging (rs-fMRI), we constructed a scale-free binary graph for each subject, using internodal time-series correlation of regions-of-interest (ROIs) as a proximity measure. The resulted network could be embedded onto manifolds of various curvature and dimensions. While maintaining the fidelity of embedding (low distortion, high mean average precision), functional brain networks were found to be best represented in the hyperbolic disc. Using a popularity-similarity optimization model (PSOM) on the hyperbolic plane, we reduced the dimension of the network into 2-D hyperbolic space and were able to efficiently visualize the internodal connections of the brain, preserving proximity as distances and angles on the PSOM discs. Each individual PSOM disc revealed decentralized nature of information flow and anatomic relevance. Using the hyperbolic distance on the PSOM disc, we could detect the anomaly of network in autistic spectrum disorder (ASD) subjects. This procedure of embedding grants us a reliable new framework for studying functional brain networks and the possibility of detecting anomalies of the network in the hyperbolic disc on an individual scale.

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.

Conformal Invariants in Simply Connected Domains

This paper studies the numerical computation of several conformal invariants of simply connected domains in the complex plane including, the hyperbolic distance, the reduced modulus, the harmonic measure, and the modulus of a quadrilateral. The used method is based on the boundary integral equation with the generalized Neumann kernel. Several numerical examples are presented. The performance and accuracy of the presented method is validated by considering several model problems with known analytic solutions.