Graph Neural Networks (GNNs) have recently received tremendous attention due to their power in handling graph data for different downstream tasks across different application domains. Many GNN models have been proposed, which mainly differ in their graph filter design. However, most of these models believe there is a best filter for all the graph data. Instead, we attempt to provide in-depth analysis on (1) Whether there exists an optimal filter that performs the best on all graph data; (2) Which graph properties should be considered for finding the optimal graph filter; and (3) How to design appropriate filters that adapt to a given graph. In this paper, we focus on addressing the above three questions for the semi-supervised node classification task. We propose a novel assessment tool, called Graph Filter Discriminant Score (GFD), for evaluating the effectiveness of graph convolutional filters for a given graph in terms of node classification. Using the assessment tool, we find out that there is no single filter as a "silver bullet" that performs the best on all possible graphs, and graphs with different properties are in favor of different graph convolutional filters. Based on these findings, we develop Adaptive Filter Graph Neural Network (AFGNN), a simple but powerful model that can adaptively learn data-specific filters. For a given graph, AFGNN leverages graph filter assessment as an extra loss term and learns to combine a set of base filters. Experiments on both synthetic and real-world benchmark datasets have demonstrated that our proposed model has the flexibility in learning an appropriate filter and consistently provides state-of-the-art performance across all the datasets.
In this paper, the authors raise and address three questions about graph neural network: (1) Whether there is a best filter that works for all graphs. (2) Which properties of the graph will influence the performance of the graph filter. (3) How to design a method to adaptively find the optimal filter for a given graph.
The paper proposes an assessment method called the Graph Filter Discriminant Score based on the Fisher Score. It measures how well the graph convolutional filter discriminate node representations of different classes in the graph by comparing the Fisher score before and after the filter.
Based on the GFD scores of different normalization strategy and different order of the graph filter in the experiments on synthetic data, the authors answer the first two questions: (1) There is no optimal normalization for all graphs. (2) row normalization performs better with lower power-law coefficient, but works worse with imbalanced label classes and large density gap.
For the third question, the authors propose a learnable linear combination of a limited family of graph convolutional filters as the layer of model AFGNN, which can learn the optimal arguments of the combination based on the FGD score.
The paper focuses on a significant topic and proposes an assessment tool for the graph filters. Based on that, it also introduces a model to choose filters from a family of filters for any specific graph.
The description of preliminaries is clear.
The observations of the impact of the graph properties on the filter choice are interesting and explanations are provided.
The results of the test accuracy on both bench mark and synthetic datasets demonstrate the good performance of the proposed model.
It is good that the paper provides proof for the claim that the graph convolutional can help the non-linear separable data to be linearly separable, so it is reasonable to use Fisher score. However, does this claim support the second term in equation (3), where the Fisher score is used to evaluate before the filters applied?
The presentation of the last paragraph of “graph filter discriminant score” in page 4 can be improved. The Figure references seem incorrect and confusing.
The analysis of the influence of label ratio seems not accurate enough.
For the GFD score comparison in Figure 4, why choose order 1,3,7 for density and different order 2,3,6 for density gap?
What is the meaning of the symbol psi(l)?
It would be better if the explanation of the raining loss section is more detailed and clear.
What is “AFGNN_P” in the experiment analysis?
It could be interesting to see the comparison of time between the proposed method and the GAT.
For the graph filter discriminate analysis, is it fair to compare the learned layer with the other base filter using the GFD score? Since the learned layer is picked with highest GFD score. Maybe one or two sentences on this will be helpful.
The writing of the paper must be improved. Too many typos and grammar problems will impair the presentation and the reader can be distracted.
Minor comments:
The layout of the sub caption of Figure 1 can be improved.
The usage of capital letter in the phrase “density gap” is inconsistent.
“As shown in figure” instead of “As is shown in figure”.
Many sentences miss article.
There are many typos in the writing.
For example, “Note that for given (feature)”, “…make the representation of nodes in different (class) more separable.”, “Noted that there are some other (variant) of GNN filters that (does) not fall into…” in page 4.
“Here we give (a) empirical explanation to this phenomenon”, “this normalization strategy (take) into account…”, “Thus even in the case that the other two (doesn’t) perform well…” in page 5.
“…a very important factor that (influence) the choice of normalization strategy”, “when power-law coefficient (decrease)”, “when the (sizes) of each class (become) more imbalanced”,  “This is because column normalization better (leverage) …” , “in a similar manner (with) label ratio”, “when the density or density gap (increase)”, “high-order filters can help gather… and thus (makes) the representations…”, “when the density gap (increase)” in page 6.
These can be continued but it is obvious that this paper needs proofreading.