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 4. Differentiable Structural Information In this section, we first establish a new formulation of structural information (Li & Pan, 2016), and then give a continuous relaxation to the differentiable realm, yielding a new objective and an optimization approach for graph clustering without the predefined cluster number. We start with the classic formulation as follows. Definition 4.1 (H-dimensional Structural Entropy (Li & Pan, 2016)). Given a weighted graph G = (V, E) with weight function w and a partitioning tree T of G with height H, the structural information of G with respect to each non-root node α of T is defined as Definition 4.1 In the partitioning tree T with root node λ, each tree node α is associated with a subset of V, denoted as module Tα, and the immediate predecessor of it is written as α −. The module of the leaf node is a singleton of the graph node. The scalar gα is the total weights of graph edges with exactly one endpoint in module Tα. Then, the H-dimensional structural information of G by T is given as, Traversing all possible partitioning trees of G with height H, H-dimensional structural entropy of G is defined as H-dimensional structural entropy where T ∗ is the optimal tree of G which encodes the self-organization and minimizes the uncertainty of the graph. The structural information in Eq. (2) is formulated node-wisely, and cannot be optimized via gradient-based methods. 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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LSEnet: A Smarter Way to Organize Data Using Curved Space

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