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Graph Neural Networks with Feature and Structure Aware Random Walk

Wei Zhuo, Guang Tan

TL;DR

This paper generalizes the graph Laplacian to digraph based on the proposed Feature-Aware PageRank algorithm, which simultaneously considers the graph directionality and long-distance feature similarity between nodes, and exploits the underlying long-distance correlations between nodes.

Abstract

Graph Neural Networks (GNNs) have received increasing attention for representation learning in various machine learning tasks. However, most existing GNNs applying neighborhood aggregation usually perform poorly on the graph with heterophily where adjacent nodes belong to different classes. In this paper, we show that in typical heterphilous graphs, the edges may be directed, and whether to treat the edges as is or simply make them undirected greatly affects the performance of the GNN models. Furthermore, due to the limitation of heterophily, it is highly beneficial for the nodes to aggregate messages from similar nodes beyond local neighborhood.These motivate us to develop a model that adaptively learns the directionality of the graph, and exploits the underlying long-distance correlations between nodes. We first generalize the graph Laplacian to digraph based on the proposed Feature-Aware PageRank algorithm, which simultaneously considers the graph directionality and long-distance feature similarity between nodes. Then digraph Laplacian defines a graph propagation matrix that leads to a model called {\em DiglacianGCN}. Based on this, we further leverage the node proximity measured by commute times between nodes, in order to preserve the nodes' long-distance correlation on the topology level. Extensive experiments on ten datasets with different levels of homophily demonstrate the effectiveness of our method over existing solutions in the task of node classification.

Graph Neural Networks with Feature and Structure Aware Random Walk

TL;DR

This paper generalizes the graph Laplacian to digraph based on the proposed Feature-Aware PageRank algorithm, which simultaneously considers the graph directionality and long-distance feature similarity between nodes, and exploits the underlying long-distance correlations between nodes.

Abstract

Graph Neural Networks (GNNs) have received increasing attention for representation learning in various machine learning tasks. However, most existing GNNs applying neighborhood aggregation usually perform poorly on the graph with heterophily where adjacent nodes belong to different classes. In this paper, we show that in typical heterphilous graphs, the edges may be directed, and whether to treat the edges as is or simply make them undirected greatly affects the performance of the GNN models. Furthermore, due to the limitation of heterophily, it is highly beneficial for the nodes to aggregate messages from similar nodes beyond local neighborhood.These motivate us to develop a model that adaptively learns the directionality of the graph, and exploits the underlying long-distance correlations between nodes. We first generalize the graph Laplacian to digraph based on the proposed Feature-Aware PageRank algorithm, which simultaneously considers the graph directionality and long-distance feature similarity between nodes. Then digraph Laplacian defines a graph propagation matrix that leads to a model called {\em DiglacianGCN}. Based on this, we further leverage the node proximity measured by commute times between nodes, in order to preserve the nodes' long-distance correlation on the topology level. Extensive experiments on ten datasets with different levels of homophily demonstrate the effectiveness of our method over existing solutions in the task of node classification.
Paper Structure (25 sections, 1 theorem, 20 equations, 4 figures, 6 tables)

This paper contains 25 sections, 1 theorem, 20 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Digraph Laplacian
  4. GNN on Non-homophilous Graphs
  5. Methodology
  6. DiLaplacian
  7. Feature-aware PageRank
  8. Combinatorial Laplacian
  9. DiL-GCN
  10. DiLaplacian with Commute Time
  11. Time Complexity
  12. Experiments
  13. Datasets
  14. Baselines
  15. Experimental Setting
  16. ...and 10 more sections

Key Result

Theorem 3.1

Given a combined graph $\tilde{G}_c$ and its Feature-Aware PageRank transition matrix $\tilde{\mathbf{P}}_{fpr}$, the DiLaplacian of $\tilde{G}_c$ is defined as $\tilde{\mathbf{T}}=\Pi(\tilde{\mathbf{D}}_c^{-1} - \tilde{\mathbf{P}}_{fpr})$. Then the fundamental matrix $\mathbf{Z}$ of $\tilde{G}_c$ c where the superscript $\dagger$ means Moore–Penrose pseudoinverse of the matrix.

Figures (4)

  • Figure 1: An illustration of the approximate similarity sorting. With the anchor node set $\mathcal{S}$, we compute the cosine similarity between all node features $\mathbf{X}$ and anchor node features $\mathbf{X}_s$ to obtain a similarity score vector. Then, we sort the similarity scores and return a sorted node index vector $\mathcal{B}$. We connect the nodes in order according to the score ranking in the $\mathcal{B}$ to generate a similarity sorting graph $G_s$. Note that $G_s$ is an undirected strongly connected graph, so $G_s$ is irreducible. At last, we combine $G_s$ and the input graph $G$ by adding all undirected edges in $G_s$ to $G$ (grey double-directional arrow), to generate an irreducible graph $G_c$.
  • Figure 2: Node classification performance under structural attack.
  • Figure 3: Values of $\alpha$ and $\beta$ when the validation set achieves the best accuracy.
  • Figure 4: Classification results under different $window\_size$.

Theorems & Definitions (1)

  • Theorem 3.1