Graphs, which encode pairwise relations between entities, are a kind of universal data structure for many real-world data, including social networks, transportation networks, and chemical molecules. Many important applications on these data can be treated as computational tasks on graphs. For example, friend recommendation in social networks can be regarded as a link prediction task and predicting properties of chemical compounds can be treated as a graph classification task. An essential step to facilitate these tasks is to learn vector representations either for nodes or the entire graphs. Given its great success of representation learning in images and text, deep learning offers great promise for graphs. However, compared to images and text, deep learning on graphs faces immense challenges. Graphs are irregular where nodes are unordered and each of them can have a distinct number of neighbors. Thus, traditional deep learning models cannot be directly applied to graphs, which calls for dedicated efforts for designing novel deep graph models. To help meet this pressing demand, we developed and investigated novel GNN algorithms to generalize deep learning techniques to graph-structured data. Two key operations in GNNs are the graph filtering operation, which aims to refine node representations; and the graph pooling operation, which aims to summarize node representations to obtain a graph representation. In this thesis, we provide deep understandings or develop novel algorithms for these two operations from new perspectives. For graph filtering operations, we propose a unified framework from the perspective of graph signal denoising, which demonstrates that most existing graph filtering operations are conducting feature smoothing. Then, we further investigate what information typical graph filtering operations can capture and how they can be understood beyond feature smoothing. For graph pooling operations, we study the procedure of pooling from the perspective of graph spectral theory and present a novel graph pooling operation. We also propose a technique to downsample nodes considering both mode importance and representativeness, which leads to a novel graph pooling operation.
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