Graph Neural Networks (GNNs) have received tremendous attention recently due to their power in handling graph data for different downstream tasks across different application domains. The key of GNN is its graph convolutional filters, and recently various kinds of filters are designed. However, there still lacks in-depth analysis on (1) Whether there exists a best filter that can perform best on all graph data; (2) Which graph properties will influence the optimal choice of graph filter; (3) How to design appropriate filter adaptive to the graph data. In this paper, we focus on addressing the above three questions. We first propose a novel assessment tool to evaluate the effectiveness of graph convolutional filters for a given graph. Using the assessment tool, we find out that there is no single filter as a `silver bullet' that perform the best on all possible graphs. In addition, different graph structure properties will influence the optimal graph convolutional filter's design choice. Based on these findings, we develop Adaptive Filter Graph Neural Network (AFGNN), a simple but powerful model that can adaptively learn task-specific filter. For a given graph, it leverages graph filter assessment as regularization and learns to combine from a set of base filters. Experiments on both synthetic and real-world benchmark datasets demonstrate that our proposed model can indeed learn an appropriate filter and perform well on graph tasks. Graph Neural Networks (GNNs) are a family of powerful tools for representation learning on graph data, which has been drawing more and more attention over the past several years. GNNs can obtain informative node representations for a graph of arbitrary size and attributes, and has shown great effectiveness in graph-related down-stream applications, such as node classification (Kipf & Welling, 2017) , graph classification (Wu et al., 2019b) , graph matching (Bai et al., 2019) , recommendation systems (Ying et al., 2018) , and knowledge graphs (Schlichtkrull et al., 2018) . As GNNs have superior performance in graph-related tasks, the question as to what makes GNNs so powerful is naturally raised. Note that GNNs adopt the concept of the convolution operation into graph domain. To obtain a representation of a specific node in a GNN, the node aggregates representations of its neighbors with a convolutional filter. For a task related to graph topology, the convolutional filter can help GNN nodes to get better task-specific representations (Xu et al., 2019) . Therefore, it is the filter that makes GNNs powerful, and thus the key to designing robust and accurate GNNs is to design proper graph convolutional filters. Recently, many GNN architectures are proposed (Zhou et al., 2018) with their own graph filter designs. However, none of them have properly answered the following fundamental questions of GNNs: (1) Is there a best filter that works for all graphs? (2) If not, what are the properties of graph structure that will influence the performance of graph convolutional filters? (3) Can we design an algorithm to adaptively find the appropriate filter for a given graph? In this paper, we focus on addressing the above three questions for semi-supervised node classification task. Inspired by studies in Linear Discriminant Analysis (LDA), we propose a Graph Filter Discriminant (GFD) Score metric to measure the power of a graph convolutional filter in discriminating node representations of different classes on a specific graph. We have analyzed all the existing GNNs' filters with this assessment method to answer the three aforementioned questions. We found that no single filter design can achieve optimal results on all possible graphs. In other words, for different graph data, we should adopt different graph convolutional filters to achieve optimal performance. We then experimentally and theoretically analyze how graph structure properties influence the optimal choice of graph convolutional filters. Based on all of our findings, we propose the Adaptive Filter Graph Neural Network (AF-GNN), which can adaptively learn a proper model for the given graph. We use the Graph Filter Discriminant Score (GFD) as a an extra loss term to guide the network to learn a good data-specific filter, which is a linear combination of a set of base filters. We show that the proposed Adaptive Filter can better capture graph topology and separate features on both real-world datasets and synthetic datasets. We highlight our main contributions as follows: • We propose an assessment tool: Graph Filter Discriminant Score, to analyze the effectiveness of graph convolutional filters. Using this tool, we find that no best filter can work for all graphs, the optimal choice of a graph convolutional filter depends on the graph data. • We propose Adaptive Filter Graph Neural Network that can adaptively learn a proper filter for a specific graph using the GFD Score as guidance. • We show that the proposed model can find better filters and achieve better performance compared to existing GNNs, on both real-word and newly created benchmark datasets. Understanding the graph convolutional filters in GNNs is very important, as it can help to determine whether a GNN will work on a given graph, and can provide important guidance for GNN design. In our paper, we focus on the semi-supervised node classification task. We first propose the Graph Filter Discriminant Score as an assessment tool for graph convolutional filter evaluation, and then apply this GFD Score to analyze a family of existing filters as a case study. Using this tool, we learn that no single fixed filter can produce optimal results on all graphs. We then develop a simple but powerful GNN model: Adapative Filter Graph Neural Network, which can learn to combine a family of filters and obtain a task-specific powerful filter. We also propose to add the negative GFD Score as an extra component to the objective function, it can act as a guidance for the model to learn a more effective filter. Experiments show that our approach outperforms many existing GNNs on both benchmark and synthetic graphs.
Propose an assessment framework to analyze and learn graph convolutional filter
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