Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with arbitrarily low distortion without using optimization. On WordNet, our combinatorial embedding obtains a mean-average-precision of 0.989 with only two dimensions, while Nickel et al.'s recent construction obtains 0.87 using 200 dimensions. We provide upper and lower bounds that allow us to characterize the precision-dimensionality tradeoff inherent in any hyperbolic embedding. To embed general metric spaces, we propose a hyperbolic generalization of multidimensional scaling (h-MDS). We show how to perform exact recovery of hyperbolic points from distances, provide a perturbation analysis, and give a recovery result that allows us to reduce dimensionality. The h-MDS approach offers consistently low distortion even with few dimensions across several datasets. Finally, we extract lessons from the algorithms and theory above to design a PyTorch-based implementation that can handle incomplete information and is scalable.
In this paper, we exhibit the tradeoffs between the (training) sample, computation and storage complexity for the problem of supervised classification using signal subspace estimation. Our main tool is the use of tensor subspaces, i.e. subspaces with a Kronecker structure, for embedding the data into lower dimensions. Among the subspaces with a Kronecker structure, we show that using subspaces with a hierarchical structure for representing data leads to improved tradeoffs. One of the main reasons for the improvement is that embedding data into these hierarchical Kronecker structured subspaces …
Detecting communities on graphs has received significant interest in recent literature. Current state-of-the-art community embedding approach called \textit{ComE} tackles this problem by coupling graph embedding with community detection. Considering the success of hyperbolic representations of graph-structured data in last years, an ongoing challenge is to set up a hyperbolic approach for the community detection problem. The present paper meets this challenge by introducing a Riemannian equivalent of \textit{ComE}. Our proposed approach combines hyperbolic embeddings with Riemannian K-means or…
Obtaining continuous representations of structural data such as directed acyclic graphs (DAGs) has gained attention in machine learning and artificial intelligence. However, embedding complex DAGs in which both ancestors and descendants of nodes are exponentially increasing is difficult. Tackling in this problem, we develop Disk Embeddings, which is a framework for embedding DAGs into quasi-metric spaces. Existing state-of-the-art methods, Order Embeddings and Hyperbolic Entailment Cones, are instances of Disk Embedding in Euclidean space and spheres respectively. Furthermore, we propose a nov…
Graduate School of Information Science and Technology, The University of Tokyo（东京大学信息科学与技术研究生院）
A hyperbolic space has been shown to be more capable of modeling complex networks than a Euclidean space. This paper proposes an explicit update rule along geodesics in a hyperbolic space. The convergence of our algorithm is theoretically guaranteed, and the convergence rate is better than the conventional Euclidean gradient descent algorithm. Moreover, our algorithm avoids the "bias" problem of existing methods using the Riemannian gradient. Experimental results demonstrate the good performance of our algorithm in the \Poincare embeddings of knowledge base data.
Learning graph representations via low-dimensional embeddings that preserve relevant network properties is an important class of problems in machine learning. We here present a novel method to embed directed acyclic graphs. Following prior work, we first advocate for using hyperbolic spaces which provably model tree-like structures better than Euclidean geometry. Second, we view hierarchical relations as partial orders defined using a family of nested geodesically convex cones. We prove that these entailment cones admit an optimal shape with a closed form expression both in the Euclidean and h…
Neural embeddings have been used with great success in Natural Language Processing (NLP). They provide compact representations that encapsulate word similarity and attain state-of-the-art performance in a range of linguistic tasks. The success of neural embeddings has prompted significant amounts of research into applications in domains other than language. One such domain is graph-structured data, where embeddings of vertices can be learned that encapsulate vertex similarity and improve performance on tasks including edge prediction and vertex labelling. For both NLP and graph based tasks, em…
Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers an exciting alternative, as it enables embeddings with much smaller distortion. However, extending GCNs to hyperbolic geometry presents several unique challenges because it is not clear how to define neural network operations, such as feature transformation and aggregation, in hyperbolic space. Furthermore, since input features are often Euclidean, it is unc…
The hyperbolic manifold is a smooth manifold of negative constant curvature. While the hyperbolic manifold is well-studied in the literature, it has gained interest in the machine learning and natural language processing communities lately due to its usefulness in modeling continuous hierarchies. Tasks with hierarchical structures are ubiquitous in those fields and there is a general interest to learning hyperbolic representations or embeddings of such tasks. Additionally, these embeddings of related tasks may also share a low-rank subspace. In this work, we propose to learn hyperbolic embeddi…
Recent work has demonstrated that embeddings of tree-like graphs in hyperbolic space surpass their Euclidean counterparts in performance by a large margin. Inspired by these results and scale-free structure in the word co-occurrence graph, we present an algorithm for learning word embeddings in hyperbolic space from free text. An objective function based on the hyperbolic distance is derived and included in the skip-gram negative-sampling architecture of word2vec. The hyperbolic word embeddings are then evaluated on word similarity and analogy benchmarks. The results demonstrate the potential …
Hong Kong University of Science and Technology（香港科技大学）；Ant Group（蚂蚁集团）
Learning the representation of data with hierarchical structures in the hyperbolic space attracts increasing attention in recent years. Due to the constant negative curvature, the hyperbolic space resembles tree metrics and captures the tree-like properties naturally, which enables the hyperbolic embeddings to improve over traditional Euclidean models. However, many real-world hierarchically structured data such as taxonomies and multitree networks have varying local structures and they are not trees, thus they do not ubiquitously match the constant curvature property of the hyperbolic space. …
University of Edinburgh（爱丁堡大学）；Samsung AI Centre（三星人工智能中心）
Hyperbolic embeddings have recently gained attention in machine learning due to their ability to represent hierarchical data more accurately and succinctly than their Euclidean analogues. However, multi-relational knowledge graphs often exhibit multiple simultaneous hierarchies, which current hyperbolic models do not capture. To address this, we propose a model that embeds multi-relational graph data in the Poincaré ball model of hyperbolic space. Our Multi-Relational Poincaré model (MuRP) learns relation-specific parameters to transform entity embeddings by Möbius matrix-vector multiplication…
Recent research on network embedding in hyperbolic space have proven successful in several applications. However, nodes in real world networks tend to interact through several distinct channels. Simple aggregation or ignorance of this multiplexity will lead to misleading results. On the other hand, there exists redundant information between different interaction patterns between nodes. Recent research reveals the analogy between the community structure and the hyperbolic coordinate. To learn each node's effective embedding representation while reducing the redundancy of multiplex network, we t…
Knowledge graph (KG) embeddings learn low-dimensional representations of entities and relations to predict missing facts. KGs often exhibit hierarchical and logical patterns which must be preserved in the embedding space. For hierarchical data, hyperbolic embedding methods have shown promise for high-fidelity and parsimonious representations. However, existing hyperbolic embedding methods do not account for the rich logical patterns in KGs. In this work, we introduce a class of hyperbolic KG embedding models that simultaneously capture hierarchical and logical patterns. Our approach combines h…
Efficient modeling of relational data arising in physical, social, and information sciences is challenging due to complicated dependencies within the data. In this work, we build off of semi-implicit graph variational auto-encoders to capture higher-order statistics in a low-dimensional graph latent representation. We incorporate hyperbolic geometry in the latent space through a Poincare embedding to efficiently represent graphs exhibiting hierarchical structure. To address the naive posterior latent distribution assumptions in classical variational inference, we use semi-implicit hierarchical…
Hong Kong University of Science and Technology（香港科技大学）；NVIDIA（英伟达）
The choice of geometric space for knowledge graph (KG) embeddings can have significant effects on the performance of KG completion tasks. The hyperbolic geometry has been shown to capture the hierarchical patterns due to its tree-like metrics, which addressed the limitations of the Euclidean embedding models. Recent explorations of the complex hyperbolic geometry further improved the hyperbolic embeddings for capturing a variety of hierarchical structures. However, the performance of the hyperbolic KG embedding models for non-transitive relations is still unpromising, while the complex hyperbo…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Basel（巴塞尔大学）
Hyperbolic space is a natural setting for mining and visualizing data with hierarchical structure. In order to compute a hyperbolic embedding from comparison or similarity information, one has to solve a hyperbolic distance geometry problem. In this paper, we propose a unified framework to compute hyperbolic embeddings from an arbitrary mix of noisy metric and non-metric data. Our algorithms are based on semidefinite programming and the notion of a hyperbolic distance matrix, in many ways parallel to its famous Euclidean counterpart. A central ingredient we put forward is a semidefinite charac…
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 recovers points exactly, when they are located on a hyperbolic submanifold 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 t…
University of Michigan（密歇根大学）；Yale University（耶鲁大学）；Department of Mathematics（数学系）；Department of Statistics and Data Science（统计与数据科学系）；Ann Arbor, MI, 48104（安阿伯，密歇根州，48104）；New Haven, CT, 06510（纽黑文，康涅狄格州，06510）
Given data, finding a faithful low-dimensional hyperbolic embedding of the data is a key method by which we can extract hierarchical information or learn representative geometric features of the data. In this paper, we explore a new method for learning hyperbolic representations by taking a metric-first approach. Rather than determining the low-dimensional hyperbolic embedding directly, we learn a tree structure on the data. This tree structure can then be used directly to extract hierarchical information, embedded into a hyperbolic manifold using Sarkar's construction \cite{sarkar}, or used a…
Hierarchical reinforcement learning deals with the problem of breaking down large tasks into meaningful sub-tasks. Autonomous discovery of these sub-tasks has remained a challenging problem. We propose a novel method of learning sub-tasks by combining paradigms of routing in computer networks and graph based skill discovery within the options framework to define meaningful sub-goals. We apply the recent advancements of learning embeddings using Riemannian optimisation in the hyperbolic space to embed the state set into the hyperbolic space and create a model of the environment. In doing so we …
Nickel and Kiela (2017) present a new method for embedding tree nodes in the Poincare ball, and suggest that these hyperbolic embeddings are far more effective than Euclidean embeddings at embedding nodes in large, hierarchically structured graphs like the WordNet nouns hypernymy tree. This is especially true in low dimensions (Nickel and Kiela, 2017, Table 1). In this work, we seek to reproduce their experiments on embedding and reconstructing the WordNet nouns hypernymy graph. Counter to what they report, we find that Euclidean embeddings are able to represent this tree at least as well as P…
University of Technology Sydney（悉尼科技大学）；Shandong University（山东大学）；The University of Queensland（昆士兰大学）
Hyperbolic space and hyperbolic embeddings are becoming a popular research field for recommender systems. However, it is not clear under what circumstances the hyperbolic space should be considered. To fill this gap, This paper provides theoretical analysis and empirical results on when and where to use hyperbolic space and hyperbolic embeddings in recommender systems. Specifically, we answer the questions that which type of models and datasets are more suited for hyperbolic space, as well as which latent size to choose. We evaluate our answers by comparing the performance of Euclidean space a…
Natural language text exhibits hierarchical structure in a variety of respects. Ideally, we could incorporate our prior knowledge of this hierarchical structure into unsupervised learning algorithms that work on text data. Recent work by Nickel & Kiela (2017) proposed using hyperbolic instead of Euclidean embedding spaces to represent hierarchical data and demonstrated encouraging results when embedding graphs. In this work, we extend their method with a re-parameterization technique that allows us to learn hyperbolic embeddings of arbitrarily parameterized objects. We apply this framework to …
We are concerned with the discovery of hierarchical relationships from large-scale unstructured similarity scores. For this purpose, we study different models of hyperbolic space and find that learning embeddings in the Lorentz model is substantially more efficient than in the Poincaré-ball model. We show that the proposed approach allows us to learn high-quality embeddings of large taxonomies which yield improvements over Poincaré embeddings, especially in low dimensions. Lastly, we apply our model to discover hierarchies in two real-world datasets: we show that an embedding in hyperbolic spa…
Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean geometry, mostly because of the absence of corresponding hyperbolic neural network layers. This makes it hard to use hyperbolic embeddings in downstream tasks. Here, we bridge this gap in a principled manner by combining the formalism of Möbius gyrovector spaces with the Riemannian geometry of the Poincaré model of hyperbolic spaces. As a result, we derive hyperbo…
Dartmouth College（达特茅斯学院）；Stony Brook University（石溪大学）；Jefferies Financial Group LLC（杰富瑞金融集团有限责任公司）
Recently, there has been an interest in embedding networks in hyperbolic space, since hyperbolic space has been shown to work well in capturing graph/network structure as it can naturally reflect some properties of complex networks. However, the work on network embedding in hyperbolic space has been focused on microscopic node embedding. In this work, we are the first to present a framework to embed the structural roles of nodes into hyperbolic space. Our framework extends struct2vec, a well-known structural role preserving embedding method, by moving it to a hyperboloid model. We evaluated ou…
University of Greenwich（格林威治大学）；Kyushu Institute of Technology（九州工业大学）；The Hong Kong Polytechnic University（香港理工大学）；The University of Tokyo（东京大学）
Hyperbolic ordinal embedding (HOE) represents entities as points in hyperbolic space so that they agree as well as possible with given constraints in the form of entity i is more similar to entity j than to entity k. It has been experimentally shown that HOE can obtain representations of hierarchical data such as a knowledge base and a citation network effectively, owing to hyperbolic space's exponential growth property. However, its theoretical analysis has been limited to ideal noiseless settings, and its generalization error in compensation for hyperbolic space's exponential representation …
Moscow Institute of Numerical Mathematics of the Russian Academy of Sciences（俄罗斯科学院莫斯科数值数学研究所）；Samsung AI Center Moscow（三星莫斯科人工智能中心）；Yandex（扬德克斯）
Computer vision tasks such as image classification, image retrieval and few-shot learning are currently dominated by Euclidean and spherical embeddings, so that the final decisions about class belongings or the degree of similarity are made using linear hyperplanes, Euclidean distances, or spherical geodesic distances (cosine similarity). In this work, we demonstrate that in many practical scenarios hyperbolic embeddings provide a better alternative.
Nanjing University（南京大学）；University of Pennsylvania（宾夕法尼亚大学）；University of Southern California（南加利福尼亚大学）；Alibaba Group（阿里巴巴集团）
Capturing associations for knowledge graphs (KGs) through entity alignment, entity type inference and other related tasks benefits NLP applications with comprehensive knowledge representations. Recent related methods built on Euclidean embeddings are challenged by the hierarchical structures and different scales of KGs. They also depend on high embedding dimensions to realize enough expressiveness. Differently, we explore with low-dimensional hyperbolic embeddings for knowledge association. We propose a hyperbolic relational graph neural network for KG embedding and capture knowledge associati…
Matrix Factorization (MF) is a common method for generating recommendations, where the proximity of entities like users or items in the embedded space indicates their similarity to one another. Though almost all applications implicitly use a Euclidean embedding space to represent two entity types, recent work has suggested that a hyperbolic Poincaré ball may be more well suited to representing multiple entity types, and in particular, hierarchies. We describe a novel method to embed a hierarchy of related music entities in hyperbolic space. We also describe how a parametric empirical Bayes app…
We introduce L-hydra (landmarked hyperbolic distance recovery and approximation), a method for embedding network- or distance-based data into hyperbolic space, which requires only the distance measurements to a few 'landmark nodes'. This landmark heuristic makes L-hydra applicable to large-scale graphs and improves upon previously introduced methods. As a mathematical justification, we show that a point configuration in d-dimensional hyperbolic space can be perfectly recovered (up to isometry) from distance measurements to just d+1 landmarks. We also show that L-hydra solves a two-stage strain…
University of Florida（佛罗里达大学）；National Taiwan University（国立台湾大学）
Hyperbolic neural networks have been popular in the recent past due to their ability to represent hierarchical data sets effectively and efficiently. The challenge in developing these networks lies in the nonlinearity of the embedding space namely, the Hyperbolic space. Hyperbolic space is a homogeneous Riemannian manifold of the Lorentz group. Most existing methods (with some exceptions) use local linearization to define a variety of operations paralleling those used in traditional deep neural networks in Euclidean spaces. In this paper, we present a novel fully hyperbolic neural network whic…
A Geometry-Aware Algorithm to Learn Hierarchical Embeddings in Hyperbolic Space
Alibaba Inc.（阿里巴巴集团）；University of California, Santa Barbara（加利福尼亚大学圣巴巴拉分校）；Carnegie Mellon University（卡内基梅隆大学）；University of California San Diego（加利福尼亚大学圣迭戈分校）；Purdue University（普渡大学）
Xidian University（西安电子科技大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Embedded topic models are able to learn interpretable topics even with large and heavy-tailed vocabularies. However, they generally hold the Euclidean embedding space assumption, leading to a basic limitation in capturing hierarchical relations. To this end, we present a novel framework that introduces hyperbolic embeddings to represent words and topics. With the tree-likeness property of hyperbolic space, the underlying semantic hierarchy among words and topics can be better exploited to mine more interpretable topics. Furthermore, due to the superiority of hyperbolic geometry in representing…
Indian Institute of Technology Bombay（印度理工学院孟买分校）；Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）
We consider the problem of multi-label classification where the labels lie in a hierarchy. However, unlike most existing works in hierarchical multi-label classification, we do not assume that the label-hierarchy is known. Encouraged by the recent success of hyperbolic embeddings in capturing hierarchical relations, we propose to jointly learn the classifier parameters as well as the label embeddings. Such a joint learning is expected to provide a twofold advantage: i) the classifier generalizes better as it leverages the prior knowledge of existence of a hierarchy over the labels, and ii) in …
Harbin Institute of Technology(Shenzhen)（哈尔滨工业大学（深圳））；The Chinese University of Hong Kong（香港中文大学）；Huawei Technologies co.ltd（华为技术有限公司）；Shenzhen University（深圳大学）
Recently, hyperbolic space has risen as a promising alternative for semi-supervised graph representation learning. Many efforts have been made to design hyperbolic versions of neural network operations. However, the inspiring geometric properties of this unique geometry have not been fully explored yet. The potency of graph models powered by the hyperbolic space is still largely underestimated. Besides, the rich information carried by abundant unlabelled samples is also not well utilized. Inspired by the recently active and emerging self-supervised learning, in this study, we attempt to enhanc…
University of Trento（特伦托大学）；Skolkovo Institute of Science and Technology（斯科尔科沃科技学院）；Skoltech（斯科尔科沃科技学院）；Yandex（杨德克斯）；AIRI（人工智能研究所）
Metric learning aims to learn a highly discriminative model encouraging the embeddings of similar classes to be close in the chosen metrics and pushed apart for dissimilar ones. The common recipe is to use an encoder to extract embeddings and a distance-based loss function to match the representations -- usually, the Euclidean distance is utilized. An emerging interest in learning hyperbolic data embeddings suggests that hyperbolic geometry can be beneficial for natural data. Following this line of work, we propose a new hyperbolic-based model for metric learning. At the core of our method is …
Nova School of Business and Economics, Universidade NOVA de Lisboa（新里斯本大学商业与经济学院）
In contrast to many other domains, recommender systems in health services may benefit particularly from the incorporation of health domain knowledge, as it helps to provide meaningful and personalised recommendations catering to the individual's health needs. With recent advances in representation learning enabling the hierarchical embedding of health knowledge into the hyperbolic Poincare space, this work proposes a content-based recommender system for patient-doctor matchmaking in primary care based on patients' health profiles, enriched by pre-trained Poincare embeddings of the ICD-9 codes …
Hyperbolic space can naturally embed hierarchies that often exist in real-world data and semantics. While high-dimensional hyperbolic embeddings lead to better representations, most hyperbolic models utilize low-dimensional embeddings, due to non-trivial optimization and visualization of high-dimensional hyperbolic data. We propose CO-SNE, which extends the Euclidean space visualization tool, t-SNE, to hyperbolic space. Like t-SNE, it converts distances between data points to joint probabilities and tries to minimize the Kullback-Leibler divergence between the joint probabilities of high-dimen…
Learning accurate drug representation is essential for tasks such as computational drug repositioning and prediction of drug side-effects. A drug hierarchy is a valuable source that encodes human knowledge of drug relations in a tree-like structure where drugs that act on the same organs, treat the same disease, or bind to the same biological target are grouped together. However, its utility in learning drug representations has not yet been explored, and currently described drug representations cannot place novel molecules in a drug hierarchy. Here, we develop a semi-supervised drug embedding …
Department of Computer Science, Stanford University（斯坦福大学计算机科学系）；Institute for Computational and Mathematical Engineering, Stanford University（斯坦福大学计算与数学工程研究所）；Google Research, NY（谷歌研究院（纽约））
Similarity-based Hierarchical Clustering (HC) is a classical unsupervised machine learning algorithm that has traditionally been solved with heuristic algorithms like Average-Linkage. Recently, Dasgupta reframed HC as a discrete optimization problem by introducing a global cost function measuring the quality of a given tree. In this work, we provide the first continuous relaxation of Dasgupta's discrete optimization problem with provable quality guarantees. The key idea of our method, HypHC, is showing a direct correspondence from discrete trees to continuous representations (via the hyperboli…
Advanced Telecommunications Research Institute International（先进电信研究所国际）；RIKEN Center for Advanced Intelligence Project（RIKEN高级人工智能项目中心）；University of Trento（特伦特大学）；Nanyang Technological University（南洋理工大学）
Seoul National University（首尔大学）；University of Birmingham（伯明翰大学）
Most of the existing literature regarding hyperbolic embedding concentrate upon supervised learning, whereas the use of unsupervised hyperbolic embedding is less well explored. In this paper, we analyze how unsupervised tasks can benefit from learned representations in hyperbolic space. To explore how well the hierarchical structure of unlabeled data can be represented in hyperbolic spaces, we design a novel hyperbolic message passing auto-encoder whose overall auto-encoding is performed in hyperbolic space. The proposed model conducts auto-encoding the networks via fully utilizing hyperbolic …
Jilin University（吉林大学）；University of Technology Sydney（悉尼科技大学）；Nanjing Forestry University（南京林业大学）
Signed network embedding methods aim to learn vector representations of nodes in signed networks. However, existing algorithms only managed to embed networks into low-dimensional Euclidean spaces whereas many intrinsic features of signed networks are reported more suitable for non-Euclidean spaces. For instance, previous works did not consider the hierarchical structures of networks, which is widely witnessed in real-world networks. In this work, we answer an open question that whether the hyperbolic space is a better choice to accommodate signed networks and learn embeddings that can preserve…
Fully Hyperbolic Rotation for Knowledge Graph Embedding
College of Computer Science, Inner Mongolia University（内蒙古大学计算机学院）；National and Local Joint Engineering Research Center of Intelligent Information Processing Technology for Mongolian（蒙古语智能信息处理技术国家地方联合工程研究中心）；Inner Mongolia Key Laboratory of Multilingual Artificial Intelligence Technology（内蒙古多语言人工智能技术重点实验室）
University of Stuttgart（斯图加特大学）；University of Bonn（波恩大学）；Monash University（莫纳什大学）；University of Southampton（南安普顿大学）
Recent knowledge graph (KG) embeddings have been advanced by hyperbolic geometry due to its superior capability for representing hierarchies. The topological structures of real-world KGs, however, are rather heterogeneous, i.e., a KG is composed of multiple distinct hierarchies and non-hierarchical graph structures. Therefore, a homogeneous (either Euclidean or hyperbolic) geometry is not sufficient for fairly representing such heterogeneous structures. To capture the topological heterogeneity of KGs, we present an ultrahyperbolic KG embedding (UltraE) in an ultrahyperbolic (or pseudo-Riemanni…
Technische Universität Dresden（德累斯顿工业大学）；Tsinghua University（清华大学）
Analysis of 'big data' characterized by high-dimensionality such as word vectors and complex networks requires often their representation in a geometrical space by embedding. Recent developments in machine learning and network geometry have pointed out the hyperbolic space as a useful framework for the representation of this data derived by real complex physical systems. In the hyperbolic space, the radial coordinate of the nodes characterizes their hierarchy, whereas the angular distance between them represents their similarity. Several studies have highlighted the relationship between the an…
