  Graph neural networks, which generalize deep neural network models to graph
structured data, have attracted increasing attention in recent years. They
usually learn node representations by transforming, propagating and aggregating
node features and have been proven to improve the performance of many graph
related tasks such as node classification and link prediction. To apply graph
neural networks for the graph classification task, approaches to generate the
\textit{graph representation} from node representations are demanded. A common
way is to globally combine the node representations. However, rich structural
information is overlooked. Thus a hierarchical pooling procedure is desired to
preserve the graph structure during the graph representation learning. There
are some recent works on hierarchically learning graph representation analogous
to the pooling step in conventional convolutional neural (CNN) networks.
However, the local structural information is still largely neglected during the
pooling process. In this paper, we introduce a pooling operator $\pooling$
based on graph Fourier transform, which can utilize the node features and local
structures during the pooling process. We then design pooling layers based on
the pooling operator, which are further combined with traditional GCN
convolutional layers to form a graph neural network framework $\m$ for graph
classification. Theoretical analysis is provided to understand $\pooling$ from
both local and global perspectives. Experimental results of the graph
classification task on $6$ commonly used benchmarks demonstrate the
effectiveness of the proposed framework.
