The pyramid quantized Weisfeiler–Lehman graph representation

Abstract

Graphs are flexible and powerful representations for non-vectorial structured data. Graph kernels have been shown to enable efficient and accurate statistical learning on this important domain, but many graph kernel algorithms have high order polynomial time complexity. Efficient graph kernels rely on a discrete node labeling as a central assumption. However, many real world domains are naturally described by continuous or vector valued node labels. In this paper, we propose an efficient graph representation and comparison scheme for large graphs with continuous vector labels, the pyramid quantized Weisfeiler–Lehman graph representation. Our algorithm considers statistics of subtree patterns with discrete labels based on the Weisfeiler–Lehman algorithm and uses a pyramid quantization strategy to determine a logarithmic number of discrete labelings that results in a representation that guarantees a multiplicative error bound on an approximation to the optimal partial matching. As a result, we approximate a graph representation with continuous vector labels as a sequence of graphs with increasingly granular discrete labels. We evaluate our proposed algorithm on two different tasks with real datasets, on a fMRI analysis task and on the generic problem of 3D shape classification. Source code of the implementation can be downloaded from https://web.imis.athena-innovation.gr/~kgkirtzou/Projects/WLpyramid.html