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Multi-scale Wasserstein Shortest-path Graph Kernels for Graph Classification

Wei Ye, Hao Tian, Qijun Chen

TL;DR

A novel graph kernel called the multiscale Wasserstein shortest-path graph kernel (MWSP), at the heart of which is the multiscale shortest-path node feature map, of which each element denotes the number of occurrences of the shortest path around a node.

Abstract

Graph kernels are conventional methods for computing graph similarities. However, the existing R-convolution graph kernels cannot resolve both of the two challenges: 1) Comparing graphs at multiple different scales, and 2) Considering the distributions of substructures when computing the kernel matrix. These two challenges limit their performances. To mitigate both of the two challenges, we propose a novel graph kernel called the Multi-scale Wasserstein Shortest-Path graph kernel (MWSP), at the heart of which is the multi-scale shortest-path node feature map, of which each element denotes the number of occurrences of the shortest path around a node. The shortest path is represented by the concatenation of all the labels of nodes in it. Since the shortest-path node feature map can only compare graphs at local scales, we incorporate into it the multiple different scales of the graph structure, which are captured by the truncated BFS trees of different depths rooted at each node in a graph. We use the Wasserstein distance to compute the similarity between the multi-scale shortest-path node feature maps of two graphs, considering the distributions of shortest paths. We empirically validate MWSP on various benchmark graph datasets and demonstrate that it achieves state-of-the-art performance on most datasets.

Multi-scale Wasserstein Shortest-path Graph Kernels for Graph Classification

TL;DR

A novel graph kernel called the multiscale Wasserstein shortest-path graph kernel (MWSP), at the heart of which is the multiscale shortest-path node feature map, of which each element denotes the number of occurrences of the shortest path around a node.

Abstract

Graph kernels are conventional methods for computing graph similarities. However, the existing R-convolution graph kernels cannot resolve both of the two challenges: 1) Comparing graphs at multiple different scales, and 2) Considering the distributions of substructures when computing the kernel matrix. These two challenges limit their performances. To mitigate both of the two challenges, we propose a novel graph kernel called the Multi-scale Wasserstein Shortest-Path graph kernel (MWSP), at the heart of which is the multi-scale shortest-path node feature map, of which each element denotes the number of occurrences of the shortest path around a node. The shortest path is represented by the concatenation of all the labels of nodes in it. Since the shortest-path node feature map can only compare graphs at local scales, we incorporate into it the multiple different scales of the graph structure, which are captured by the truncated BFS trees of different depths rooted at each node in a graph. We use the Wasserstein distance to compute the similarity between the multi-scale shortest-path node feature maps of two graphs, considering the distributions of shortest paths. We empirically validate MWSP on various benchmark graph datasets and demonstrate that it achieves state-of-the-art performance on most datasets.
Paper Structure (18 sections, 9 equations, 5 figures, 4 tables)

This paper contains 18 sections, 9 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Notations
  5. Wasserstein Distance
  6. Multi-scale Wasserstein Shortest-path Graph Kernels
  7. Shortest-Path Graph Feature Map
  8. Multi-scale Shortest-path Graph Feature Map
  9. Computing Graph Kernels
  10. Algorithm
  11. Experimental Evaluation
  12. Experimental Setup
  13. Results
  14. Classification Accuracy
  15. Ablation Study
  16. ...and 3 more sections

Figures (5)

  • Figure 1: A graph of a post on Reddit. Each node represents a user. Two users are connected by an edge if at least one of them responds to the other's comment. The image is drawn with Gephi.
  • Figure 2: Illustration of the shortest paths in graphs. $\Sigma=\{1,2,3,4\}$.
  • Figure 3: Truncated BFS trees of depth two rooted at each node in the undirected labeled graph $\mathcal{G}_1$.
  • Figure 4: Truncated BFS trees of depth two rooted at each node in the undirected labeled graph $\mathcal{G}_2$.
  • Figure 5: Parameter sensitivity of MWSP. $d$ represents the depth of the truncated BFS tree rooted at each node, which is used for the extraction of shortest paths whose maximal length is $d$. $k$ represents the depth of the truncated BFS tree rooted at each node, which is used for capturing multiple different scales of the graph structure around each node (for augmenting graph nodes). Each entry in the heatmap is the mean classification accuracy over the 10 folds.

Theorems & Definitions (3)

  • Definition 1: Shortest-path Representation
  • Definition 2: Shortest-path Graph Feature Map
  • Definition 3: Shortest-path Node Feature Map