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Boolean Product Graph Neural Networks

Ziyan Wang, Bin Liu, Ling Xiang

TL;DR

This paper proposes a novel Boolean product-based graph residual connection in GNNs to link the latent graph and the original graph and computes the Boolean product between the latent graph and the original graph at each layer to correct the learning process.

Abstract

Graph Neural Networks (GNNs) have recently achieved significant success, with a key operation involving the aggregation of information from neighboring nodes. Substantial researchers have focused on defining neighbors for aggregation, predominantly based on observed adjacency matrices. However, in many scenarios, the explicitly given graphs contain noise, which can be amplified during the messages-passing process. Therefore, many researchers have turned their attention to latent graph inference, specifically learning a parametric graph. To mitigate fluctuations in latent graph structure learning, this paper proposes a novel Boolean product-based graph residual connection in GNNs to link the latent graph and the original graph. It computes the Boolean product between the latent graph and the original graph at each layer to correct the learning process. The Boolean product between two adjacency matrices is equivalent to triangle detection. Accordingly, the proposed Boolean product graph neural networks can be interpreted as discovering triangular cliques from the original and the latent graph. We validate the proposed method in benchmark datasets and demonstrate its ability to enhance the performance and robustness of GNNs.

Boolean Product Graph Neural Networks

TL;DR

This paper proposes a novel Boolean product-based graph residual connection in GNNs to link the latent graph and the original graph and computes the Boolean product between the latent graph and the original graph at each layer to correct the learning process.

Abstract

Graph Neural Networks (GNNs) have recently achieved significant success, with a key operation involving the aggregation of information from neighboring nodes. Substantial researchers have focused on defining neighbors for aggregation, predominantly based on observed adjacency matrices. However, in many scenarios, the explicitly given graphs contain noise, which can be amplified during the messages-passing process. Therefore, many researchers have turned their attention to latent graph inference, specifically learning a parametric graph. To mitigate fluctuations in latent graph structure learning, this paper proposes a novel Boolean product-based graph residual connection in GNNs to link the latent graph and the original graph. It computes the Boolean product between the latent graph and the original graph at each layer to correct the learning process. The Boolean product between two adjacency matrices is equivalent to triangle detection. Accordingly, the proposed Boolean product graph neural networks can be interpreted as discovering triangular cliques from the original and the latent graph. We validate the proposed method in benchmark datasets and demonstrate its ability to enhance the performance and robustness of GNNs.
Paper Structure (21 sections, 1 theorem, 12 equations, 5 figures, 4 tables, 1 algorithm)

This paper contains 21 sections, 1 theorem, 12 equations, 5 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Model
  4. Background
  5. The Proposed Model
  6. Boolean Matrix Multiplication
  7. Network Architecture
  8. Probabilistic Boolean product
  9. Loss Function
  10. Experiments
  11. Datasets and Setup
  12. Baselines
  13. Performance
  14. Results
  15. Robustness
  16. ...and 6 more sections

Key Result

Proposition 1

Given two graph $\mathcal{G}_1=(\mathcal{V}, \mathcal{E}_1, \mathbf{A})$, $\mathcal{G}_2=(\mathcal{V}, \mathcal{E}_2, \mathbf{B})$ that share the same node set $\mathcal{V}$, where $\mathcal{E}_1$, $\mathcal{E}_2$ are edge sets, $\mathbf{A}$ and $\mathbf{B} \in \{0,1\}^{n \times n}$ are adjacent mat

Figures (5)

  • Figure 1: The interpretation of Boolean product between two adjacent matrices $\mathbf{A}$ and $\mathbf{B}$. The graphs defined on $\mathbf{A}$ and $\mathbf{B}$ share a same set of nodes, and all are aligned. Suppose that there is no edge between node $i$ and $j$ from the perspectives of both matrices $\mathbf{A}$ and $\mathbf{B}$, that is $\mathbf{A}_{ij}=0$ and $\mathbf{B}_{ij}=0$. Panel (a) demonstrates the set of shared neighbors $\mathcal{N}_i(\mathbf{A}) \cap \mathcal{N}_j(\mathbf{B}):=\{1,\cdots,k\}$ of node $i$ from view $\mathbf{A}$ and node $j$ from view $\mathbf{B}$. Panel (b) visualizes the Boolean product resulting matrix $\mathbf{A}\diamond \mathbf{B}$ (as shown in \ref{['eq:booleanFusion']}). We observe $[\mathbf{A}\diamond \mathbf{B}]_{ij}=1$ once we observe that the shared neighbor set $\{1,\cdots,k\}$ of nodes $i$ and $j$ is not null. It yields a new edge (red dash line) between node $i$ and $j$ as the overall relationship from the points of view of both $\mathbf{A}$ and $\mathbf{B}$.
  • Figure 2: The basic architecture of $l$-layer of BPGNN.
  • Figure 3: Test accuracy ($\pm$ standard deviation ) in percent for the edge addition and deletion on Cora and CiteSeer.
  • Figure 4: The ratio of two nodes in the test set sharing the same label in different probability interval.
  • Figure 5: Time requirements for matrix multiplication under both sparsity and non-sparsity computed using the torch.sparse.mm and torch.matmul functions in PyTorch, respectively.

Theorems & Definitions (1)

  • Proposition 1