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 5.3. Hyperbolic Partitioning Tree In the principle of structural information minimization, the optimal partitioning tree is constructed to describe the selforganization of the graph (Li & Pan, 2016; Wu et al., 2023) The overall procedure of training LSEnet is summarized in Algorithm 1. Specifically, in Lines 1-9, we learn level-wise assignments and node embeddings in a bottom-up manner. In Lines 10-17, the optimal partitioning tree is given via a top-down process in which we avoid empty leaf nodes and fix single chains in the relaxed tree (case one and case two in Theorem 4.8). Further details of the procedure are shown in Algorithm 2 given in Appendix C.2. In brief, LSEnet combines the advantages of both structural entropy and hyperbolic space, and uncovers the clustering structures without predefined cluster number K. In brief, LSEnet combines the advantages of both structural entropy and hyperbolic space, and uncovers the clustering structures without predefined cluster number K. 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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Self-Organizing Networks: Training Hyperbolic Partitioning Trees with LSEnet

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