Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Abstract and 1. Introduction Related Work


Preliminaries and Notations


Differentiable Structural Information
4.1. A New Formulation
4.2. Properties
4.3. Differentiability & Deep Graph Clustering


LSEnet
5.1. Embedding Leaf Nodes
5.2. Learning Parent Nodes
5.3. Hyperbolic Partitioning Tree


Experiments
6.1. Graph Clustering
6.2. Discussion on Structural Entropy


Conclusion, Broader Impact, and References Appendix Related Work Related Work Related Work Related Work Preliminaries and Notations Preliminaries and Notations Preliminaries and Notations Preliminaries and Notations Differentiable Structural Information
4.1. A New Formulation
4.2. Properties
4.3. Differentiability & Deep Graph Clustering Differentiable Structural Information Differentiable Structural Information Differentiable Structural Information 4.1. A New Formulation 4.1. A New Formulation 4.2. Properties 4.2. Properties 4.3. Differentiability & Deep Graph Clustering 4.3. Differentiability & Deep Graph Clustering LSEnet
5.1. Embedding Leaf Nodes
5.2. Learning Parent Nodes
5.3. Hyperbolic Partitioning Tree LSEnet LSEnet LSEnet 5.1. Embedding Leaf Nodes 5.1. Embedding Leaf Nodes 5.2. Learning Parent Nodes 5.2. Learning Parent Nodes 5.3. Hyperbolic Partitioning Tree 5.3. Hyperbolic Partitioning Tree Experiments
6.1. Graph Clustering
6.2. Discussion on Structural Entropy Experiments Experiments 6.1. Graph Clustering 6.1. Graph Clustering 6.2. Discussion on Structural Entropy 6.2. Discussion on Structural Entropy Conclusion, Broader Impact, and References Appendix Conclusion, Broader Impact, and References Appendix Conclusion, Broader Impact, and References Appendix Conclusion, Broader Impact, and References Appendix A. Proofs A. Proofs B. Hyperbolic Space B. Hyperbolic Space C. Technical Details C. Technical Details D. Additional Results D. Additional Results B. Hyperbolic Space B.1. Riemannian Manifold B.1. Riemannian Manifold B.2. Models of Hyperbolic Space B.2. Models of Hyperbolic Space B.3. Tree and Hyperbolic Space B.3. Tree and Hyperbolic Space We denote the embedding of node i of graph G in H is ϕi . The following definitions and theorem are from (Sarkar, 2011). Definition B.1 (Delaunay Graph). Given a set of vertices in H their Delaunay graph is one where a pair of vertices are neighbors if their Voronoi cells intersect. Definition B.1 Definition B.2 (Delaunay Embedding of Graphs). Given a graph G, its Delaunay embedding in H is an embedding of the vertices such that their Delaunay graph is G. Definition B.2 Following the above Theorem, we know that a tree can be embedded into hyperbolic space with an arbitrarily low distortion. B.4. Midpoint in the Manifold B.4. Midpoint in the Manifold Let (M, g) be a Riemannian manifold, and x1, x2, ..., xn are points on the manifold. The Frechet variance at point ´ µ ∈ M of these points are given by If there is a point p locally minimizes the variance, it is called Frechet mean. Generally, if we assign each ´ xi a weight wi , the Frechet mean can be formulated as Note that, d is the canonical distance in the manifold. Remark. In fact, the geometric centroid in Theorem 5.2 is also equivalent to the gyro-midpoint in Poincare ball model of ´ hyperbolic space. Remark Authors:
(1) Li Sun, North China Electric Power University, Beijing 102206, China (ccesunli@ncepu.edu);
(2) Zhenhao Huang, North China Electric Power University, Beijing 102206, China;
(3) Hao Peng, Beihang University, Beijing 100191, China;
(4) Yujie Wang, North China Electric Power University, Beijing 102206, China;
(5) Chunyang Liu, Didi Chuxing, Beijing, China;
(6) Philip S. Yu, University of Illinois at Chicago, IL, USA. Authors: Authors: (1) Li Sun, North China Electric Power University, Beijing 102206, China (ccesunli@ncepu.edu); (2) Zhenhao Huang, North China Electric Power University, Beijing 102206, China; (3) Hao Peng, Beihang University, Beijing 100191, China; (4) Yujie Wang, North China Electric Power University, Beijing 102206, China; (5) Chunyang Liu, Didi Chuxing, Beijing, China; (6) Philip S. Yu, University of Illinois at Chicago, IL, USA. This paper is available on arxiv under CC BY-NC-SA 4.0 Deed (Attribution-Noncommercial-Sharelike 4.0 International) license. This paper is available on arxiv under CC BY-NC-SA 4.0 Deed (Attribution-Noncommercial-Sharelike 4.0 International) license. available on arxiv available on arxiv

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Curved Space Geometry: Embedding Trees and Finding Midpoints in Hyperbolic Models

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