In this work, we develop an Aligned Entropic Reproducing Kernel (AERK) for graph classification. We commence by performing the Continuous-time Quantum Walk (CTQW) on each graph structure, and computing the Averaged Mixing Matrix (AMM) to describe how the CTQW visit all vertices from a starting vertex. More specifically, we show how this AMM matrix allows us to compute a quantum Shannon entropy for each vertex of a graph. For pairwise graphs, the proposed AERK kernel is defined by computing a reproducing kernel based similarity between the quantum Shannon entropies of their each pair of aligned vertices. The analysis of theoretical properties reveals that the proposed AERK kernel 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 problem of neglecting the structural differences between pairs of aligned vertices arising in existing vertex-based matching kernels. Moreover, unlike existing classical graph kernels that only focus on the global or local structural information of graphs, the proposed AERK kernel can simultaneously capture both global and local structural information through the quantum Shannon entropies, reflecting more precise kernel based similarity measures between pairs of graphs. The above theoretical properties explain the effectiveness of the proposed kernel. The experimental evaluation on standard graph datasets demonstrates that the proposed AERK kernel is able to outperform state-of-the-art graph kernels for graph classification tasks.

通过进行CTQW计算出平均混合矩阵（AMM），利用AMM矩阵计算出每个顶点的量子Shannon熵并定义了一种AERK核，可以同时捕获顶点匹配的结构对齐信息、顶点对应的结构差异以及全局和本地结构信息，使提出的AERK核在图形分类任务中具有优越性。

AERK：通过连续时间量子行走对齐熵再生核