AEGK: Aligned Entropic Graph Kernels Through Continuous-Time Quantum Walks

In this work, we develop a family of Aligned Entropic Graph Kernels (AEGK) for graph classification. We commence by performing the Continuous-time Quantum Walk (CTQW) on each graph structure, and compute the Averaged Mixing Matrix (AMM) to describe how the CTQW visits all vertices from a starting vertex. More specifically, we show how this AMM matrix allows us to compute a quantum Shannon entropy of each vertex for either un-attributed or attributed graphs. For pairwise graphs, the proposed AEGK kernels are defined by computing the kernel-based similarity between the quantum Shannon entropies of their pairwise aligned vertices. The analysis of theoretical properties reveals that the proposed AEGK kernels cannot only address the shortcoming of neglecting the structural correspondence information between graphs arising in most existing R-convolution graph kernels, but also overcome the problems of neglecting the structural differences and vertex-attributed information arising in existing vertex-based matching kernels. Moreover, unlike most existing classical graph kernels that only focus on the global or local structural information of graphs, the proposed AEGK kernels can simultaneously capture both global and local structural characteristics through the quantum Shannon entropies, reflecting more precise kernel-based similarity measures between pairwise graphs. The above theoretical properties explain the effectiveness of the proposed AEGK kernels. Experimental evaluations demonstrate that the proposed kernels can outperform state-of-the-art graph kernels and deep learning models for graph classification.