Generalized Shortest Path Kernel on Graphs

Abstract

Abstract. We consider the problem of classifying graphs using graph kernels. We define a new graph kernel, called the generalized shortest path kernel, based on the number and length of shortest paths between nodes. For our example classification problem, we consider the task of classifying random graphs from two well-known families, by the number of clusters they contain. We verify empirically that the generalized shortest path kernel outperforms the original shortest path kernel on a number of datasets. We give a theoretical analysis for explaining our experimental results. In particular, we estimate distributions of the expected feature vectors for the shortest path kernel and the generalized shortest path kernel, and we show some evidence explaining why our graph kernel outperforms the shortest path kernel for our graph classification problem.

A survey on graph kernels

“…Shortest-Path [24] IM + † + † Generalized Shortest-Path [25] IM + + † Graphlet [26] EX --Cycles and Trees [27] EX + ⋆ -Tree Pattern Kernel [28,29] IM + + ⋆ Ordered Directed Acyclic Graphs [30,31] EX + -GraphHopper [32] IM + † + Graph Invariant [33] IM + + Subgraph Matching [34] IM + + Weisfeiler-Lehman Subtree [35] EX + -Weisfeiler-Lehman Edge [35] EX + -Weisfeiler-Lehman Shortest-Path [35] EX + k-dim. Local Weisfeiler-Lehman Subtree [36] EX + -Neighborhood Hash Kernel [37] EX + -Propagation Kernel [38] EX + + Neighborhood Subgraph Pairwise Distance Kernel [39] EX + -Random Walk [22,23,40,3,41,42] IM + + Optimal Assignment Kernel [43] IM + + Weisfeiler-Lehman Optimal Assignment [44] IM + -Pyramid Match [45] IM + -Matchings of Geometric Embeddings [46] IM + + ⋆ Descriptor Matching Kernel [47] IM + + † Graphlet Spectrum [48] EX + -Multiscale Laplacian Graph Kernel [49] IM + + ⋆ † Global Graph Kernel [50] EX --Deep Graph Kernels [19] IM + -Smoothed Graph Kernels [51] IM + ⋆ -Hash Graph Kernel [52] EX + + Depth-based Representation Kernel [53] IM --Aligned Subtree Kernel [54] IM + -…”

“…Using this approach, the time complexity of the SP kernel is reduced to the time complexity of the Floyd-Warshall algorithm, which is in 𝑂(𝑛 3 ). In [25] the shortest-path is generalized by considering all shortest paths between two vertices.…”

Intrinsic-Extrinsic Convolution and Pooling for Learning on 3D Protein Structures

“…To our knowledge, no such convolution has considered using multiple distances in order to combine multiple intrinsic distances with one extrinsic distance. Other approaches used high-order neighborhoods [36] based on shortest-path [21] or Quantum Walks [54]. Our method differs from those since we define our neighborhood based on the Euclidean distance and compute the geodesic distances on a multi-graph as additional dimensions to our kernel.…”

Graphnet: Graph Clustering with Deep Neural Networks

“…Graph clustering, which aims to group a set of graphs into clusters, is a fundamental and crucial unsupervised method to explore graph structures. Traditional algorithms calculate various graph kernels to deal with graphs [1,2,3,4,5]. However, these kernel-based clustering methods suffer from the diagonal dominance issue [6], which impedes the information propagation across adjacent nodes, thus leading to poor graph clustering performance.…”