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Centrality Graph Shift Operators for Graph Neural Networks

Yassine Abbahaddou, Fragkiskos D. Malliaros, Johannes F. Lutzeyer, Michalis Vazirgiannis

TL;DR

This work proposes and studies Centrality GSOs (CGSOs), which normalize adjacency matrices by global centrality metrics such as the PageRank, $k$-core or count of fixed length walks, and outlines how this CGSO can act as the message passing operator in any Graph Neural Network.

Abstract

Graph Shift Operators (GSOs), such as the adjacency and graph Laplacian matrices, play a fundamental role in graph theory and graph representation learning. Traditional GSOs are typically constructed by normalizing the adjacency matrix by the degree matrix, a local centrality metric. In this work, we instead propose and study Centrality GSOs (CGSOs), which normalize adjacency matrices by global centrality metrics such as the PageRank, $k$-core or count of fixed length walks. We study spectral properties of the CGSOs, allowing us to get an understanding of their action on graph signals. We confirm this understanding by defining and running the spectral clustering algorithm based on different CGSOs on several synthetic and real-world datasets. We furthermore outline how our CGSO can act as the message passing operator in any Graph Neural Network and in particular demonstrate strong performance of a variant of the Graph Convolutional Network and Graph Attention Network using our CGSOs on several real-world benchmark datasets.

Centrality Graph Shift Operators for Graph Neural Networks

TL;DR

This work proposes and studies Centrality GSOs (CGSOs), which normalize adjacency matrices by global centrality metrics such as the PageRank, -core or count of fixed length walks, and outlines how this CGSO can act as the message passing operator in any Graph Neural Network.

Abstract

Graph Shift Operators (GSOs), such as the adjacency and graph Laplacian matrices, play a fundamental role in graph theory and graph representation learning. Traditional GSOs are typically constructed by normalizing the adjacency matrix by the degree matrix, a local centrality metric. In this work, we instead propose and study Centrality GSOs (CGSOs), which normalize adjacency matrices by global centrality metrics such as the PageRank, -core or count of fixed length walks. We study spectral properties of the CGSOs, allowing us to get an understanding of their action on graph signals. We confirm this understanding by defining and running the spectral clustering algorithm based on different CGSOs on several synthetic and real-world datasets. We furthermore outline how our CGSO can act as the message passing operator in any Graph Neural Network and in particular demonstrate strong performance of a variant of the Graph Convolutional Network and Graph Attention Network using our CGSOs on several real-world benchmark datasets.

Paper Structure

This paper contains 36 sections, 4 theorems, 38 equations, 5 figures, 16 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Graph Shift Operators
  4. Graph Neural Networks
  5. CGSO: Centrality Graph Shift Operators
  6. Mathematical Formulation
  7. CGNN: Centrality Graph Neural Network
  8. A Spectral Clustering Perspective of CGSOs
  9. Spectral Clustering on Stochastic Block Barabási–Albert Models
  10. Centrality Recovery in Spectral Clustering
  11. Experimental Evaluation
  12. Experimental Results
  13. Conclusion and Limitations
  14. Datasets and Implementation Details
  15. Statistics of the Node Classification Datasets
  16. ...and 21 more sections

Key Result

Proposition 3.1

The following properties of operator $\mathbf{M}_{G}$ hold.

Figures (5)

  • Figure 1: Result for the spectral clustering task on the Cora graph dataset_node_classification with core numbers considered as clusters. We report the values of the Adjusted Mutual Information (AMI) in percentage for different combinations of the exponents $(e_2, e_3)$ in $\mathbf{V}^{e_2} \mathbf{A} \mathbf{V}^{e_3}.$
  • Figure 2: Dirichlet Energy variation with layers in (a) Cora and (b) Chamelon.
  • Figure 3: The adjacency matrix of the synthetic graph generated from an SBBAM with 3 blocks.
  • Figure 4: The left figure represents the $k$-core distributions of three different BA models with the hyperparameters $r=5,10$ and $15$ serving as blocks of of our SBBAM. The right figure represents the $k$-core distribution of the SBBAM.
  • Figure 5: Result for the spectral clustering task on Cora graph with core numbers considered as clusters. We report the values of the Adjusted Rand Information (ARI) in % different combination of the exponents $(e_2,e_3)$ in $\mathbf{V}^{e_2} \mathbf{A} \mathbf{V}^{e_3}.$

Theorems & Definitions (10)

  • Definition 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Definition 3.3
  • Proposition 3.4
  • Lemma 4.1
  • proof : Proof of Proposition \ref{['prop:spectral_properties']}
  • proof : Proof of Proposition \ref{['prop:eigenvalues']}
  • proof : Proof of Proposition \ref{['prop:cheeger']}
  • proof : Proof of Proposition \ref{['lem:BA_average_degree']}