Is Graph Convolution Always Beneficial For Every Feature?

## Abstract

Abstract Graph Neural Networks (GNNs) have demonstrated strong capabilities in processing structured data. While traditional GNNs typically treat each feature dimension equally during graph convolution, we raise an important question: Is the graph convolution operation equally beneficial for each feature dimension? If not, the convolution operation on certain feature dimensions can possibly lead to harmful effects, even worse than the convolution-free models. In prior studies, to assess the impacts of graph convolution on features, people proposed metrics based on feature homophily to measure feature consistency with the graph topology. However, these metrics have shown unsatisfactory alignment with GNN performance and have not been effectively employed to guide feature selection in GNNs. To address these limitations, we introduce a novel metric, Topological Feature Informativeness (TFI), to distinguish between GNN-favored and GNN-disfavored features, where its effectiveness is validated through both theoretical analysis and empirical observations. Based on TFI, we propose a simple yet effective Graph Feature Selection (GFS) method, which processes GNN-favored and GNN-disfavored features separately, using GNNs and non-GNN models. Compared to original GNNs, GFS significantly improves the extraction of useful topological information from each feature with comparable computational costs. Extensive experiments show that after applying GFS to 8 8 8 8 baseline and state-of-the-art (SOTA) GNN architectures across 10 10 10 10 datasets, 83.75 83.75 83.75 83.75 % of the GFS-augmented cases show significant performance boosts. Furthermore, our proposed TFI metric outperforms other feature selection methods. These results validate the effectiveness of both GFS and TFI. Additionally, we demonstrate that GFS’s improvements are robust to hyperparameter tuning, highlighting its potential as a universal method for enhancing various GNN architectures.

## 1 Introduction

Graph Neural Networks (GNNs) are widely used for processing graph-structured data, such as recommendation systems (Wu et al., ; ), social networks (Li et al., ; Awasthi et al., ; Luan et al., ), telecommunication (Lu et al., ) and bio-informatics (Zhang et al., ; Kang et al., ; Hua et al., ). Although graph convolution has been shown effective to enrich node features with topological information through message propagation, the performance gain is found to be restricted by the assumption of homophily, i.e., similar nodes are more likely to be connected in a graph (McPherson et al., ). On the other hand, when a graph exhibits low homophily, i.e., heterophily, the graph convolution operation can lead to performance degradation and sometimes even underperform convolution-free models, such as Multi-Layer Perceptrons (MLPs) (Zhu et al., ; Luan et al., ; ). Therefore, to measure the impact of graph convolution operation, label-based homophily metrics (Pei et al., ; Zhu et al., ) are proposed to measure the label consistency along graph topology. However, they neglect the effects on node features, which is crucial for graph learning. Feature homophily metrics (Yang et al., ; Jin et al., ) are then proposed to measure the feature consistency along the graph.

Although these existing metrics can capture the feature similarity between connected nodes, they overlook that different feature dimensions may exhibit different levels of compatibility with graph structures, and thus may gain different amounts of benefits or negative impacts from graph convolution. For example, as illustrated in Figure 1 (right), the GNN-favored feature exhibits uniform values among intra-class nodes while differing across inter-class nodes. This characteristic enables graph convolution operation to effectively improve the distinguishability among nodes from different classes, as demonstrated in (Luan et al., ). Conversely, graph convolution on GNN-disfavored features may hinder the learning process of GNNs. This example raises a crucial question: How can we determine whether graph convolution is beneficial or not for a specific feature?

Figure: Figure 1: Improvements in GNN performance at node level and feature level. Different colors denote node labels, while the direction and magnitude of arrows denote node features.
Refer to caption: x1.png

To address this issue, in this paper, we propose Topological Feature Informativeness (TFI), which measures the mutual information between each dimension of aggregated node features and labels. TFI can identify features that are either favored or disfavored by GNNs, provably providing an upper bound on the performance gap between graph convolution and convolution-free models, which is then supported by both theoretical analysis and empirical observations. Additionally, TFI overcomes the “good heterophily” issue, which is a serious misalignment of existing feature homophily metrics and GNN performance. Besides, TFI can work well in sparse label scenarios.

Motivated by the principle of “feed the right features to the right model” Luan et al. (), we propose a simple yet effective method called Graph Feature Selection (GFS). GFS first uses TFI to identify GNN-favored and GNN-disfavored features. Then, to enhance the extraction of useful information, the GNN-favored features are processed by GNNs, while MLPs handle the GNN-disfavored features. Last, a final linear layer fuses the embeddings from both models to obtain the final node representation. GFS can be seamlessly integrated into almost any GNN architecture, improving overall model performance with comparable computational costs. Our experiments on real-world datasets demonstrate that GFS significantly boosts the performance of $8$ GNNs across $10$ datasets in node classification tasks and the improvement is robust to hyperparameter tuning. Moreover, we demonstrate that TFI outperforms other statistical and optimization-based metrics for feature selection in GNNs, validating its superiority. Besides, we surprisingly find that GFS is much more effective on node embeddings encoded by Pretrained Large Models (PLMs) than other methods. This implies that the advantages of PLMs in understanding graph-structured data might be rooted in their ability to disentangle the topology-aware and topology-agnostic information into separate feature dimensions. In summary, our main contributions are as follows.

- •
We introduce a novel metric, Topological Feature Informativeness (TFI), to distinguish between GNN-favored and GNN-disfavored features. We validate its effectiveness through both empirical observations and theoretical analysis.
- •
We propose Graph Feature Selection (GFS) based on TFI, a simple yet powerful method that significantly boosts GNN performance. To the best of our knowledge, this is the first study to address the feature selection problem based on GNN-favored and GNN-disfavored disentanglement.
- •
Our extensive experiments demonstrate that applying GFS to $8$ baseline and state-of-the-art (SOTA) GNN architectures across $10$ datasets yields a significant performance boost in $83.75\%$ ($67$ out of $80$) of the cases.

## 2 Preliminary

We define a graph as $\mathcal{G}=\{\mathcal{V},\mathcal{E}\}$, where $\mathcal{V}$ is the node set and $\mathcal{E}$ is the edge set. For an undirected graph, its structure can be represented by an adjacency matrix $\bm{A}\in\mathbb{R}^{N\times N}$, where $N$ is the number of nodes, $A_{u,v}=A_{v,u}=1$ indicates the presence of an edge between nodes $u$ and $v$, i.e., $e_{uv},e_{vu}\in\mathcal{E}$, otherwise $A_{u,v}=A_{v,u}=0$. The node classification task on graph aims to predict node labels $\bm{Y}\in\mathbb{R}^{N}$ by utilizing node features $\bm{X}\in\mathbb{R}^{N\times M}$ and topological information, where $X_{u,m}$ denotes the $m$-th feature of node $u$. The node degree matrix is denoted as $\bm{D}\in\mathbb{R}^{N\times N}$, where $D_{u}=\sum_{v\in\mathcal{V}}A_{u,v}$ is the degree of node $u$. The adjacency matrix can be normalized as $\bm{\hat{A}}=\bm{D}^{-\frac{1}{2}}\bm{A}\bm{D}^{-\frac{1}{2}}$. The neighbors of node $u$ are denoted by $\mathcal{N}_{u}=\{v|e_{uv}\in\mathcal{E}\}$.

#### Graph Neural Networks.

Graph Neural Networks (GNNs), such as GCN (Kipf & Welling, ), GAT (Velicković et al., ), and GraphSage (Hamilton et al., ), can effectively capture useful topological information from neighbors by message propagation mechanism. Specifically, for a node $u$, its embedding $z_{u}^{l}$ in the $l$-th layer of a GNN can be expressed as:

$$ $z_{u}^{l}=\text{UPDATE}\left(z_{u}^{l-1},\text{AGGREGATE}(z_{u}^{l-1},\{z_{v}^ {l-1}|v\in\mathcal{N}_{u}\})\right)$ (1) $$

where $\text{AGGREGATE}(\cdot)$ represents the aggregation function, and $\text{UPDATE}(\cdot)$ is the update function based on the ego node and the aggregated neighbor representations from the previous layer. For the first layer, the initial embedding is set as input node features.

The representative graph convolution can be formulated as $\bm{H^{l}}=\sigma(\bm{\hat{A}}\bm{H^{l-1}}\bm{W^{l-1}})$, where $\bm{H^{l}}$ and $\bm{W^{l}}$ denote the node embeddings and learnable parameter matrix at the $l$-th layer, and $\sigma(\cdot)$ is an activation function. To boost GNN performance, some architectures concatenate ego and neighborhood node embeddings as the input of the next layer (Abu-El-Haija et al., ; Zhu et al., ), i.e., $[\bm{\hat{A}}\bm{H^{l-1}},\bm{H^{l-1}}]$ or adding the aggregated node embeddings with the ego node embeddings (Bo et al., ; Luan et al., ), i.e., $(\bm{\hat{A}}+\bm{I})\bm{H^{l-1}}$, which has been shown to improve GNN performance (Platonov et al., ), especially in the scenario of heterophily (Luan et al., ; Lu et al., ).

#### Label Homophily.

Homophily, a concept originating from social networks, is defined as the tendency of similar nodes are more likely to connect with each other (Khanam et al., ). In graph learning, homophily is used to measure whether graph convolution on certain graph is beneficial for GNN or not. Higher homophily implies that the topological structure can provide useful information, which typically leads to better GNN performance. The commonly used label-based homophily metrics are defined as follows:

$$ $\displaystyle\leavevmode\resizebox{422.77661pt}{}{$h_{\text{edge}}(\mathcal{G} ,\bm{Y})=\frac{\big{|}\{e_{uv}\mid e_{uv}\in\mathcal{E},Y_{u}=Y_{v}\}\big{|}}{ |\mathcal{E}|},\ h_{\text{node}}(\mathcal{G},\bm{Y})=\frac{1}{|\mathcal{V}|} \sum_{v\in\mathcal{V}}\frac{\big{|}\{u\mid u\in\mathcal{N}_{v},Y_{u}=Y_{v}\} \big{|}}{\big{|}\mathcal{N}_{v}\bigl{|}}$}$ (2) $$

Generally, these metrics only measure label consistency across graph topology, but they overlook the feature aspect, which also plays a critical role in GNN performance (Zheng et al., ). Therefore, some studies focus on the feature consistency across graph topology.

#### Feature Homophily.

To extend the conventional definition of label homophily to node features, feature homophily metrics are introduced. The general form of feature homophily is,

$$ $h(\mathcal{G},\bm{X_{:,m}})=\frac{1}{\eta(\bm{X_{:,m}})}\sum_{e_{uv}\in \mathcal{E}}\text{sim}(X_{u,m},X_{v,m})$ (3) $$

where $\eta(\cdot)$ is a normalization function and $\text{sim}(\cdot)$ is a similarity metric. The key difference between different feature homophily metrics lies in the choice of the similarity function $\text{sim}(\cdot)$, which could be cosine similarity (Jin et al., ), dot-product (Yang et al., ), or Euclidean distance (Chen et al., ). Please refer to Appendix A for more detailed introduction to homophily metrics.

#### Mutual Information.

Mutual Information measures the amount of information obtained about one random variable given another variable (Reza, ). More specifically, given variable $X$ and $Y$, the mutual information can be expressed as(^1^11For a discrete variable ${Y}$ and a continuous variable ${X}$, mutual information is estimated based on entropy using k-nearest neighbor distances, following Kraskov et al. (); Ross ().):

$$ $I({X};{Y})=\sum_{y\in\mathcal{Y}}\sum_{x\in\mathcal{X}}p(x,y)\ \log{\frac{p(x, y)}{p(x)p(y)}}$ (4) $$

where $p(x,y)$ is joint probability, and $p(x)$ and $p(y)$ are marginal probability distributions.

## 3 Topological Feature Informativeness

As illustrated in Figure 1, not all features in graphs benefit from graph convolution. This raises the question: how can we determine which features are beneficial for graph convolution? Previous studies (Zheng et al., ; Luan et al., ; Wang et al., ) have demonstrated that it is the stable topological patterns of intra-class nodes that enable effective message aggregation from neighbors, rather than relying solely on consistency-based homophily metrics. To address the limitations of feature homophily metrics, we propose TFI (Topological Feature Informativeness) to measure the informativeness of neighbors in graphs given the feature $\mathbf{X_{:,m}}$, which can be expressed as:

$$ $\text{TFI}_{m}=I(Y;\tilde{X}_{:,m})$ (5) $$

where $\bm{\tilde{X}_{:,m}}=(\bm{\hat{A}})^{k}\bm{X_{:,m}}$ is the aggregated node features from k-hop neighbors. Here $\bm{\hat{A}}$ is the adjacency matrix without self-loop, and we set $k=1$ in $\bm{\tilde{X}_{:,m}}$ by default.

The proposed $\text{TFI}_{m}$ leverages mutual information to quantify the degree of dependence between variables, thereby revealing both linear and non-linear relationships. Then, we conduct a theoretical analysis to determine how $\text{TFI}_{m}$ can effectively measure whether specific features in graphs benefit from graph convolution.

- Theorem1.
Given a graph $\mathcal{G}=\{\mathcal{V},\mathcal{E}\}$ with $C$ classes and $\text{TFI}_{m}$ measured on $m$-th feature, the prediction accuracy $P_{A}$ of a classifier on node labels $\bm{Y}$ with aggregated features $\bm{\tilde{X}_{:,m}}$ is upper bounded by:
$P_{A}\leq\frac{\text{TFI}_{m}+\log 2}{\log C}$
(6)

where the proof is given in Appendix B.

From Theorem 1, we can see that $\text{TFI}_{m}$ can measure the effect of graph convolution on feature dimension $m$ by relating it to model performance. More specifically, a higher $\text{TFI}_{m}$ means the prediction accuracy of a classifier with aggregated node feature $m$ has a higher upper bound, which indicates that the graph convolution operation is more likely to be beneficial to feature $m$. To guide feature selection with $\text{TFI}_{m}$, we next explain how it relates to the performance gap between graph-convoluted and convolution-free features.

- Theorem2.
The gap between $I(Y;\tilde{X}_{:,m},X_{:,m})$ and $I(Y;X_{:,m})$ is upper bounded by $\text{TFI}_{m}$:
$I(Y;\tilde{X}_{:,m},X_{:,m})-I(Y;X_{:,m})=I(Y;\tilde{X}_{:,m}|X_{:,m})\leq I(Y
;\tilde{X}_{:,m})=\text{TFI}_{m}$
(7)

In Theorem 2, the mutual information $I(Y;X_{:,m})$ measures how well the ego node features capture the relevant information of labels, without making assumptions about the underlying data distribution. This makes $I(Y;X_{:,m})$ a non-parametric measure of MLP performance. On the other hand, $I(Y;\tilde{X}_{:,m},X_{:,m})$ serves as a non-parametric measure of GNN performance, which utilizes both the ego node features $\bm{X_{:,m}}$ and the aggregated features $\bm{\tilde{X}_{:,m}}$ for prediction (^2^22The combination of $\tilde{X}_{:,m}$ and $X_{:,m}$ could be either concatenating node embeddings, i.e., $[\bm{\hat{A}}\bm{H^{l-1}},\bm{H^{l-1}}]$ or adding the aggregated node embeddings with the ego node embeddings, i.e., $(\bm{\hat{A}}+\bm{I})\bm{H^{l-1}}$ as introduced in Section 2). The subtraction of $I(Y;\tilde{X}_{:,m},X_{:,m})$ and $I(Y;X_{:,m})$ indicates the performance gap between graph-convoluted and convolution-free features, which is upper bounded by $\text{TFI}_{m}$.

Figure: Figure 2: The performance gap between GCN and MLP with the increase of TFI.
Refer to caption: x2.png

To verify the effectiveness of the above claims about TFI, we conduct experiments on real-world benchmark datasets. Specifically, we first compute $\text{TFI}_{m}$ for all features in the graph, sort them, and divide them evenly into 10 bins. Next, we train GCN and MLP only with the features in a bin respectively, and report the performance difference of GCN and MLP. As shown in Figure 2, GCN underperforms MLP in bins with low TFI and outperforms MLP in bins with high TFI. This observation is consistent with Theorem 1 and 2, and indicates that TFI can effectively identifies GNN-favored and GNN-disfavored features.

Note that the estimation of $\text{TFI}_{m}$ does not require all of the labels $\bm{Y}$. In a semi-supervised setting, $\text{TFI}_{m}$ can be calculated using only the training labels and features, i.e., $I({Y}_{\text{train}};{\tilde{X}_{\text{train},m}})$, without the need for pseudo labels. In the next section, we will introduce how to use $\text{TFI}_{m}$ for feature selection in GNNs.

## 4 Graph Feature Selection

In this section, we propose GFS (Graph Feature Selection), a TFI-based feature selection method. It is composed of three main components: (1) GNN-favored feature selection, (2) feature embedding, and (3) feature fusion.

Figure: Figure 3: Framework of GFS with TFI.
Refer to caption: x3.png

As illustrated in Figure 3, for each feature dimension $m$, we first measure its corresponding $\text{TFI}_{m}$ using Eq. (5). Since a higher $\text{TFI}_{m}$ indicates stronger GNN performance compared to MLP, we apply a threshold to select the set of GNN-favored features as $\{\bm{X_{\mathcal{G}}}|\bm{X_{:,m}}\in\bm{X_{\mathcal{G}}},\text{TFI}_{m}\geq
\delta(r)\}$, where $\delta(r)$ is the threshold corresponding to the top $r$ percentile of all TFI values in features, and $r\in(0,1)$. The GNN-disfavored features are the remaining $1-r$ of features, defined as $\bm{X_{\neg\mathcal{G}}}=\bm{X}\backslash\{\bm{X_{\mathcal{G}}}\}$.

The GNN-favored features $\bm{X_{\mathcal{G}}}$ are expected to be more suitable for GNNs, whereas the GNN-disfavored features $\bm{X_{\neg\mathcal{G}}}$ are better suited for MLPs. Thus, we feed $\bm{X_{\mathcal{G}}}$ into a GCN (or any other GNN architecture) and $\bm{X_{\neg\mathcal{G}}}$ into an MLP to better leverage information from both neighbors and ego nodes. Specifically, in $l$-th layer of GCN or MLP, we have:

$$ $\begin{split}\text{GCN: }\bm{H^{l}_{\mathcal{G}}}=\sigma(\bm{\hat{A}}\bm{H^{l- 1}_{\mathcal{G}}}\bm{W^{l-1}_{1}}),\ \text{MLP: }\bm{H^{l}_{\neg\mathcal{G}}}= \sigma(\bm{H^{l-1}_{\neg\mathcal{G}}}\bm{W^{l-1}_{2}})\end{split}$ (8) $$

where $\bm{H^{0}_{\mathcal{G}}}=\bm{X^{0}_{\mathcal{G}}}$ and $\bm{H^{0}_{\neg\mathcal{G}}}=\bm{X^{0}_{\neg\mathcal{G}}}$. During message passing, the representations of $\bm{H^{l}_{\mathcal{G}}}$ and $\bm{H^{l}_{\neg\mathcal{G}}}$ are derived independently, ensuring that $\bm{X^{l}_{\mathcal{G}}}$ and $\bm{X^{l}_{\neg\mathcal{G}}}$ can be encoded by the most appropriate model without interference.

Lastly, to combine the node information from $\bm{X_{\mathcal{G}}}$ and $\bm{X_{\neg\mathcal{G}}}$ together, we concatenate the node representation from the last layer $L$ of GNN and MLP, and feed them into a linear layer:

$$ $\bm{H}=[\bm{H^{L}_{\mathcal{G}}},\bm{H^{L}_{\neg\mathcal{G}}}]\bm{W}+\bm{b}$ (9) $$

where $\bm{H}$ is the final node representation, $\bm{W}$ is the weight matrix, and $\bm{b}$ is the bias, respectively. This final node representation is then used for downstream graph tasks.

#### Complexity Analysis.

To compare the time complexity of GNN+GFS against standard GNNs, we use GCN as the backbone GNN and assume the embedding size $F$ is consistent across all layers. First, estimating and sorting TFI introduces a complexity of $O(KC+N(1+\log N))$, where $K$ is a parameter related to mutual information estimation, $C$ is the number of classes, and $N$ is the number of nodes. Both $K$ and $C$ are much smaller than $N$ or the number of edges $\left|\mathcal{E}\right|$, making this overhead negligible compared to GNN training complexity as below.

After feature selection, we obtain GNN-favored features $\bm{X_{\mathcal{G}}}\in\mathbb{R}^{N\times\lceil rM\rfloor}$ and GNN-disfavored features $\bm{X_{\neg\mathcal{G}}}\in\mathbb{R}^{N\times\lceil(1-r)M\rfloor}$, where $\lceil\cdot\rfloor$ is the floor function. For the first layer of GCN+GFS, the time complexity is $O(\left|\mathcal{E}\right|\lceil rM\rfloor+N\lceil rM\rfloor F)$ for the GNN channel and $O(N\lceil(1-r)M\rfloor F)$ for the MLP channel. The sum of these terms, $O(\left|\mathcal{E}\right|\lceil rM\rfloor+NM^{2})$, is smaller than the complexity of a standard GCN, $O(\left|\mathcal{E}\right|M+NM^{2})$, for $r<1$. This is because GFS reduces the dimension of features that input into GCN.

In subsequent layers, GCN+GFS has a complexity of $O(\left|\mathcal{E}\right|F+2NF^{2})$, slightly higher than the complexity of GCN $O(\left|\mathcal{E}\right|F+NF^{2})$ by an additional $NF^{2}$ term due to the MLP layers. However, this additional cost $O(NF^{2})$ is smaller than the $O(\left|\mathcal{E}\right|F)$ introduced by graph convolutions. Thus, the overall complexity of GCN+GFS is comparable to that of the original GCN.

## 5 Experiments

To verify the effectiveness of our proposed Graph Feature Selection (GFS) and Topological Feature Informativeness (TFI), we answer the following research questions with experimental evaluations.

RQ1: To what extent does GFS enhance the performance of GNNs?

RQ2: Can GFS universally improve GNN performance without the need for hyperparameter tuning?

RQ3: How does TFI compare to other statistic-based and optimization-based metrics in the context of GFS?

RQ4: Do the GNN-favored and GNN-disfavored features identified by TFI truly align better with GNNs or MLPs, respectively?

### 5.1 Experimental Setups

#### Data Preparation.

The datasets used in our experiments include Children, Computers, Fitness, History, and Photo from Yan et al. (), and Amazon-Ratings, Minesweeper, Questions, Roman-Empire, and Tolokers from Platonov et al. (). These datasets exhibit varying levels of label homophily across different domains. The dataset statistics and descriptions are shown in Appendix C.1. For all datasets, we randomly split the data into training, validation, and test sets in a ratio of $50\%:25\%:25\%$ for 10 runs. Note that the node features in datasets from Yan et al. () are encoded by Pretrained Language Models (PLMs).

#### Baselines.

We implement GFS and compare its performance with $5$ baseline GNNs, GCN (Kipf & Welling, ), GAT (Velicković et al., ), GraphSAGE (SAGE) (Hamilton et al., ), SGC (Wu et al., ) and APPNP (Gasteiger et al., ), and $3$ SOTA GNNs, Graph Transformer (GT) (Shi et al., ), ACM-GCN (Luan et al., ) and FAGCN (Bo et al., ). The GFS-augmented GNN is denoted as GNN+GFS throughout this paper. To enhance performance, we incorporate skip connections (He et al., ) and layer normalization (Ba et al., ) for all the methods, as recommended in Platonov et al. (); Luo et al. (). The detailed descriptions of these GNNs are introduced in Appendix C.2. Additionally, we compare our proposed TFI with other statistic-based metrics, including $h_{GE}$ (Jin et al., ), $h_{attr}$ (Yang et al., ), $h_{sim-cos}$ (Chen et al., ), $h_{sim-euc}$ (Chen et al., ), and $h_{CTF}$ (Lee et al., ). We also evaluate optimization-based metrics such as $\theta_{Soft}$ and $\theta_{Hard}$ within GFS. Detailed definitions of these statistic-based metrics and optimization-based metrics can be found in Appendix A and Appendix C.4, respectively.

#### Training.

During training, we utilize the Adam optimizer (Kingma & Ba, ) to update model parameters. Each model is trained for $1000$ epochs, with the epoch demonstrating the best performance on the validation set selected for testing. Model performance is evaluated on node classification tasks, measuring accuracy for multi-class datasets and AUC-ROC for binary-class datasets. The searching space for the ratio $r$ is $\{0.1,0.2,\dots,0.9\}$ in GFS(^3^33To prevent the behavior of GNN+GFS from closely resembling that of traditional GNNs, we exclude $r=1.0$ from our experiments. However, in practice, $r=1.0$ may occur in GNN+GFS when all features are graph-favored, which would make GNN+GFS comparable to GNNs in the worst-case scenario.) and $k=1$ for the number of the hop of neighbors in TFI(^4^44In Appendix D.4, we examine the influence of the number of k-hop neighbors on TFI. Our findings indicate that $k=1$ is sufficient to achieve strong model performance.). See Appendix C.3 for more training details.

### 5.2 Performance (RQ1)

The results and comparisons of the model performance are shown in Table 1, where the performance gap between GNN+GFS and GNN is denoted as $\Delta$. We observe that, (1) GFS significantly boosts GNN performance on $83.75\%(67/80)$ cases of the GFS-augmented GNNs, with an average increase of $2.2\%$ in accuracy or AUC-ROC on node classification tasks with various homophily levels. This demonstrate the effectiveness of our proposed method. (2) We notice that the improvement of GFS varies by dataset, e.g., in Children and Computers, GCN+GFS achieves a 6% average performance improvement, while in Minesweeper and Questions, the improvement is less significant. This may be because most features in Minesweeper and Questions are already GNN-favored. Nevertheless, GFS remains effective on most datasets. (3) We find that GFS demonstrates greater improvements on datasets encoded by Pretrained Language Models (PLMs) (50/50 cases) than those encoded by some traditional methods (37/50 cases), e.g., one-hot, fasttext or statistics-based methods. This suggests that, compared to traditional feature encoding methods, which is widely used in graph learning, PLMs potentially enable the disentanglement of topology-aware and topology-agnostic information into separate feature dimensions, further enhancing the effectiveness of GNN+GFS. This is a surprising discovery and to our knowledge, we are the first to reveal this phenomenon. It can help explain why PLMs can assist the learning of graph models (Ye et al., ). It would be interesting to analyze each node feature encoded by PLMs in the future, but we will not conduct a deeper discussion in this paper.

**Table 1: The performance of GFS on GNN baselines. The improvement is highlighted as bold if there is an increase, i.e., $\Delta>0$, after applying GFS.**
| Model | Children | Comp. | Fitness | History | Photo | Amazon. | Mines. | Questions | Roman. | Tolokers |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| GCN | 53.36±1.08 | 83.38±1.78 | 87.56±0.79 | 84.13±0.29 | 84.00±0.72 | 51.40±0.49 | 89.93±0.52 | 76.24±1.48 | 75.51±0.65 | 84.19±0.75 |
| GCN+GFS | 59.29±0.31 | 90.23±0.18 | 91.77±0.16 | 84.99±0.38 | 87.11±0.42 | 53.32±0.81 | 89.99±0.60 | 76.50±1.38 | 83.30±0.57 | 85.01±0.91 |
| $\Delta$ | +5.93 | +6.85 | +4.21 | +0.86 | +3.11 | +1.92 | +0.06 | +0.26 | +7.79 | +0.82 |
| GAT | 53.54±0.49 | 85.91±0.69 | 88.18±0.81 | 83.83±0.27 | 85.22±0.45 | 51.08±0.60 | 90.26±0.60 | 77.20±0.96 | 79.74±0.51 | 83.41±0.47 |
| GAT+GFS | 57.37±0.37 | 89.93±0.24 | 92.26±0.11 | 84.49±0.34 | 87.45±0.30 | 52.07±0.44 | 90.22±0.64 | 77.03±1.11 | 83.00±0.83 | 83.62±0.68 |
| $\Delta$ | +3.83 | +4.02 | +4.08 | +0.66 | +2.23 | +0.99 | -0.04 | -0.17 | +3.26 | +0.21 |
| SAGE | 54.68±0.84 | 86.08±0.50 | 88.65±1.22 | 84.06±0.42 | 85.08±0.64 | 53.80±0.56 | 90.74±0.59 | 74.91±1.06 | 80.48±0.77 | 82.77±0.38 |
| SAGE+GFS | 59.14±0.33 | 89.63±0.23 | 93.03±0.11 | 84.63±0.30 | 86.59±0.49 | 53.52±0.53 | 90.71±0.62 | 75.37±1.33 | 81.44±0.63 | 82.70±0.47 |
| $\Delta$ | +4.46 | +3.55 | +4.38 | +0.57 | +1.51 | -0.28 | -0.03 | +0.46 | +0.96 | -0.07 |
| GT | 50.96±1.33 | 85.63±1.16 | 87.37±1.38 | 83.61±0.43 | 82.67±1.13 | 51.30±0.73 | 90.11±0.57 | 77.57±1.09 | 81.08±0.42 | 82.46±0.98 |
| GT+GFS | 55.59±0.22 | 89.44±0.20 | 91.94±0.08 | 84.02±0.28 | 87.04±0.37 | 52.18±0.62 | 89.93±0.64 | 77.33±1.07 | 83.27±0.46 | 82.69±0.92 |
| $\Delta$ | +4.63 | +3.81 | +4.57 | +0.41 | +4.37 | +0.88 | -0.18 | -0.24 | +2.19 | +0.23 |
| SGC | 51.25±0.52 | 81.92±0.67 | 84.22±1.07 | 82.92±0.39 | 82.33±0.59 | 46.15±1.20 | 73.95±1.30 | 63.84±1.25 | 57.15±0.44 | 71.75±1.01 |
| SGC+GFS | 52.85±0.38 | 83.90±0.38 | 86.74±0.21 | 83.48±0.30 | 83.61±0.36 | 47.80±0.52 | 75.43±1.55 | 70.67±1.29 | 71.91±0.58 | 76.61±0.80 |
| $\Delta$ | +1.60 | +1.98 | +2.52 | +0.56 | +1.28 | +1.65 | +1.48 | +6.83 | +14.76 | +4.86 |
| APPNP | 50.63±0.89 | 83.67±0.90 | 86.76±1.18 | 83.37±0.26 | 82.20±1.41 | 48.73±0.61 | 81.61±0.79 | 75.29±1.06 | 71.48±0.65 | 79.82±1.17 |
| APPNP+GFS | 54.29±0.37 | 87.61±0.24 | 90.42±0.10 | 84.12±0.30 | 85.82±0.27 | 49.06±0.59 | 81.13±1.22 | 73.42±1.02 | 72.35±0.69 | 81.39±1.04 |
| $\Delta$ | +3.66 | +3.94 | +3.66 | +0.75 | +3.62 | +0.33 | -0.48 | -1.87 | +0.87 | +1.57 |
| ACMGCN | 54.60±0.50 | 85.94±0.72 | 89.10±0.98 | 84.21±0.22 | 84.19±0.42 | 51.70±0.57 | 90.47±0.64 | 76.66±1.27 | 81.44±0.60 | 83.14±1.01 |
| ACMGCN+GFS | 57.61±0.46 | 88.19±0.22 | 92.29±0.20 | 84.60±0.31 | 85.37±0.37 | 51.64±0.36 | 90.45±0.59 | 75.99±1.27 | 81.40±0.62 | 83.36±0.93 |
| $\Delta$ | +3.01 | +2.25 | +3.19 | +0.39 | +1.18 | -0.06 | -0.02 | -0.67 | -0.04 | +0.22 |
| FAGCN | 50.43±0.86 | 79.92±0.99 | 83.10±0.45 | 82.04±0.62 | 80.67±0.77 | 46.08±0.52 | 78.22±2.86 | 58.60±2.08 | 62.02±2.98 | 73.07±0.70 |
| FAGCN+GFS | 52.62±0.54 | 83.56±0.67 | 85.76±0.53 | 83.43±0.36 | 82.79±0.52 | 47.57±0.38 | 84.20±1.75 | 71.09±1.21 | 71.15±1.05 | 77.44±1.33 |
| $\Delta$ | +2.19 | +3.64 | +2.66 | +1.39 | +2.12 | +1.49 | +5.98 | +12.49 | +9.13 | +4.37 |

### 5.3 Sensitivity To Hyperparameters (RQ2)

Some recent GNNs (Li et al., ; Liu et al., ) introduce more and more hyperparameters, which could be fragile to hyperparameter tuning (Luan et al., ). To investigate the sensitivity of the superiority of GFS over GNNs to ratio $r$ and other model hyperparameters, we compare the performance between GCN and GCN+GFS under varying settings. First, we observe how model performance responds to changes in ratio $r$, while fixing all other hyperparameters. Figure 4 shows the response of GCN+GFS to $r$ on $4$ datasets, where GFS collapses to normal GCN when $r=1.0$ or MLP when $r=0.0$, as all features are sent to GNN or MLP, respectively. On Children, Computer, and Roman-Empire, GCN+GFS with $r=0.1$ to $0.9$ consistently outperforms GCN ($r=1.0$) and MLP ($r=0.0$), indicating the performance gain obtained from GFS is robust to hyperparameter $r$. We also observe that although on Tolokers GCN+GFS cannot significantly outperform GCN, sending all the features into GCN ($r=1.0$) is comparable to sending $40\%$ ($r=0.4$) to $90\%$ ($r=0.9$) features with the highest TFI in GCN+GFS. These results highlight the necessity of GFS, as neither $r=1.0$ (all features to GCN) nor $r=0.0$ (all features to MLP) yields optimal results on most datasets, which reduces the need for convoluting all the features.

Figure: Figure 4: The performance of GCN+GFS is shown as the ratio $r$ of GNN-favored features in TFI increases. The point representing the best performance is highlighted as $\bigstar$.
Refer to caption: x4.png

Second, we analyze how other hyperparameters affect GFS performance. Figure 5 shows how GCN, MLP, and GCN+GFS respond to the changes of number of layers, dimension of hidden embeddings, learning rate, weight decay, and dropout rate on Computers (homophilic graph) and Amazon-Ratings (heterophilic graph). Results indicate that GCN+GFS consistently outperforms GCN and MLP across various hyperparameter values. This property allows GFS to be easily integrated into other GNNs without much hyperparameter tuning. Please refer to Appendix D.2 for more results on the sensitivity of GFS to hyperparameters on the other datasets.

Figure: Figure 5: Response of GCN+GFS, GCN, and MLP to $5$ hyperparameters on Computers and Amazon-Ratings.
Refer to caption: x5.png

### 5.4 Effectiveness of TFI (RQ3)

We show the superiority of TFI by comparing GCN+GFS with other metric-based feature selection methods, e.g., statistic-based metrics $h_{GE}$ Jin et al. (), $h_{attr}$ Yang et al. (), $h_{sim-cos}$, $h_{sim-euc}$ Chen et al. (), and $h_{CTF}$ Lee et al. () and optimization-based methods e.g., $\theta_{Soft}$ and $\theta_{Hard}$, where feature selection process is learned in an end-to-end manner. All methods follow the same hyperparameter tuning as in GFS. Table 2 shows the performance of different metrics in feature selection for GCN, where TFI achieves the best performance in average rank across $10$ datasets. This is reasonable because other statistic-based metrics suffer from explaining “good heterophily” issue and TFI overcomes the issue by the measurement of mutual information Platonov et al. (). Besides, the optimization-based metrics underperform TFI, indicating the feature selection cannot be effectively handled automatically during the optimization process of GNNs. In addition, we find that even for the metrics that are inferior to TFI, e.g., $h_{attr}$, $h_{sim-euc}$, and $h_{CTF}$, applying them within our GFS framework can still improve the performance of baseline GCN, indicating the necessity of feature selection in GNNs.

**Table 2: The performance of feature selection on feature homophily metrics and optimization-based methods.**
| Metrics | Children | Comp. | Fitness | History | Photo | Amazon. | Mines. | Questions | Roman. | Tolokers | Avg. Rank |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| TFI | 59.29±0.31 | 90.23±0.18 | 91.77±0.16 | 84.99±0.38 | 87.11±0.42 | 53.32±0.81 | 89.99±0.60 | 76.50±1.38 | 83.30±0.57 | 85.01±0.91 | 1.40 |
| $h_{GE}$ | 55.00±0.25 | 83.72±1.32 | 87.06±0.58 | 83.88±0.26 | 83.36±0.94 | 51.71±0.53 | 89.75±0.68 | 75.73±1.53 | 79.11±0.48 | 83.31±0.77 | 6.70 |
| $h_{attr}$ | 55.15±0.32 | 84.62±0.61 | 87.88±0.53 | 84.22±0.25 | 83.22±0.52 | 51.46±0.70 | 90.18±0.70 | 76.40±1.39 | 78.90±0.58 | 84.04±0.82 | 4.70 |
| $h_{sim-cos}$ | 55.15±0.51 | 84.11±0.84 | 87.09±0.98 | 84.07±0.31 | 83.38±1.01 | 51.68±0.39 | 89.87±0.53 | 75.88±1.07 | 78.98±0.66 | 83.69±0.82 | 5.95 |
| $h_{sim-euc}$ | 56.32±0.26 | 89.08±0.20 | 87.52±1.04 | 85.00±0.38 | 86.71±0.30 | 51.35±0.49 | 90.19±0.69 | 76.51±1.17 | 78.96±0.73 | 84.03±0.83 | 3.35 |
| $h_{CTF}$ | 55.04±0.51 | 84.80±0.87 | 88.69±2.25 | 84.32±0.30 | 83.46±0.85 | 51.76±0.57 | 89.97±0.68 | 76.40±1.37 | 78.70±0.38 | 83.38±0.89 | 4.55 |
| Soft | 57.04±1.37 | 85.35±3.53 | 89.54±1.32 | 84.47±0.92 | 84.83±3.28 | 51.70±0.62 | 89.93±0.60 | 75.94±1.28 | 79.20±0.46 | 84.03±0.84 | 3.50 |
| Hard | 45.28±0.79 | 66.92±4.70 | 69.48±4.14 | 80.98±0.70 | 72.09±3.72 | 42.77±0.57 | 86.62±2.25 | 75.56±1.29 | 76.96±0.61 | 79.13±0.94 | 8.90 |
| None | 53.36±1.08 | 83.38±1.78 | 87.56±0.79 | 84.13±0.29 | 84.00±0.72 | 51.40±0.49 | 89.93±0.52 | 76.24±1.48 | 75.51±0.65 | 84.19±0.75 | 5.95 |

Since TFI is not an unsupervised measurement, we investigate how the percentage of supervision in TFI (percentage of node labels used to calculate TFI in Eq. (5)) influences the model performance of GCN+GFS. As shown in Figure 6, as the supervision percentage increases, the model performance first increases and then stabilizes after 30%. This indicates that $30\%$ of the label supervision achieves similar results as full labels in most datasets and it highlights the effectiveness of TFI in sparse label scenarios. Compared to some other methods (Zheng et al., ; Li et al., ) that require pseudo labels during training, TFI requires less preprocessing in semi-supervised node classification settings. Even with only $10\%$ supervision, TFI can still enhance GCN+GFS performance on most datasets compared to the original GCN. Please refer to Appendix D.3 for more results on the influence of the supervision percentage in TFI on GFS performance.

Figure: Figure 6: Influence of the percentage of the supervision in TFI on the model performance of GCN+GFS.
Refer to caption: x6.png

Based on TFI, we extend the feature selection from the raw node features to node embeddings, which may also be GNN-favored or GNN-disfavored. Specifically, we train a GCN or MLP on the training set and then get the node embeddings from the last layer. After that, we train GCN+GFS on these node embeddings following the same process as on the original node features. As shown in Figure 8, applying GCN+GFS on either the original node features, $\bm{X}$, or pretrained node embeddings, $\text{MLP}(\bm{X})$ and $\text{GCN}(\bm{X},\bm{A})$, improves GCN performance on most datasets. Furthermore, for some datasets, such as Computers, Questions, and Amazon-ratings, the performance of GCN+GFS could be further improved using pretrained node embeddings compared with original node features. Additionally, it would be interesting to explore the impact of graph feature selection on dynamically updated node embeddings in future research.

### 5.5 GNN-favored and GNN-disfavored Features (RQ4)

To validate whether features selected by a high TFI are truly GNN-favored, we swapped features by feeding GNN-favored features into MLP and GNN-disfavored features into GNN. As shown in Figure 8, the performance of GCN+GFS drops in all datasets after swapping GNN-favored and GNN-disfavored features. Especially in some datasets, such as Minesweeper, there is a $40\%$ drop in performance. These results show that TFI reliably identifies GNN-favored and GNN-disfavored features.

Figure: Figure 7: GCN+GFS on node features $\bm{X}$ and pretrained node embeddings $\text{GCN}(\bm{X},\bm{A})$ and $\text{MLP}(\bm{X})$.
Refer to caption: x7.png

## 6 Conclusion and Future Works

In this paper, we study feature selection to enhance GNN performance by identifying GNN-favored and GNN-disfavored features. To answer whether each feature in graphs is GNN-favored or not, previous feature homophily metrics suffer from “good heterophily”. To address this issue, we introduce Topological Feature Informativeness (TFI), which measures the mutual information between aggregated node features and labels. This metric effectively quantifies the performance gap between GNNs and Multi-Layer Perceptrons (MLPs), as supported by our empirical observations and theoretical analyses. We then propose a simple yet effective method, Graph Feature Selection (GFS), to incorporate TFI to improve the GNN performance. Our extensive experiments demonstrate that applying GFS to $8$ GNN architectures across 10 datasets yields a significant performance boost in $83.75\%$ ($67$ out of $80$) of the cases, with an average increase of $2.2\%$ in accuracy or AUC-ROC on node classification tasks. We also show that the performance gains from GFS are robust to hyperparameter tuning, indicating its potential as a universal method for enhancing various types of GNNs.

Admittedly, the features selected by TFI introduce a new hyperparameter, ratio $r$, in GFS. Although GFS improves GNNs performance across a large range of $r$, the best performance point remains unknown due to the high complexity of the optimization process in GNNs and MLP. Therefore, it is interesting to explore the auto-selection of ratio $r$ in the future. Furthermore, since TFI is a supervised, statistic-based metric and other types of metrics may also enhance GNN performance under GFS, it would be valuable to investigate more types of metrics, both supervised and unsupervised, for feature selection in graphs with GNNs in the future.

## Appendix A Related Work

The proposed Graph Feature Selection (GFS) in this paper mainly relates with feature selection and GNNs efficacy. Thus, in this section, we first introduce feature selection methods on non-GNNs models and GNNs. Then, we introduce the studies on the effectiveness of GNNs from a graph level, node level, or feature level.

#### Feature Selection.

Feature selection is crucial for improving model performance, preventing overfitting, and reducing computational complexity (Li et al., ). To preserve data similarity, Laplacian Score (He et al., ) and SPEC (Zhao & Liu, ) are proposed to select features that best preserve the data manifold structure. To maximize the correlation between feature and class labels, MIM (Lewis, ) is proposed based on information theory. To minimize the fitting errors in classification tasks, Hara & Maehara (); Zhu et al. () propose to select features with larger weights in models with $l_{1}$-norm regularization. To reduce feature redundancy, T-Score (Davis et al., ) and Chi-Square Score (Liu & Setiono, ) are proposed to access whether features could distinguish different classes. All of the aforementioned methods only consider data in Euclidean space, which cannot be applied for non-Euclidean data, such as graph-structured data. To address this, Ye & Liu (); Kim & Xing () use a Graph Lasso regularizer to select features consistent across connected nodes.

While effective in traditional machine learning, these methods do not identify GNN-favored or GNN-disfavored features, as discussed in this paper. Our approach does not simply discard GNN-disfavored features; instead, it strategically utilizes them with a non-GNN model, driven by the principle of aligning the right features with the right model.

#### Graph Homophily.

Even if traditional GNNs (Kipf & Welling, ; Velicković et al., ) are believed to perform well on graph-related tasks, the performance of GNNs could be inferior than non-GNNs models in some graphs. Therefore, metrics of graph homophily, such as edge homophily (Zhu et al., ), node homophily (Pei et al., ), and class homophily (Lim et al., ), are proposed to determine when GNNs perform well. The definitions of these homophily metrics are given as:

$$ $\displaystyle\leavevmode\resizebox{422.77661pt}{}{$h_{\text{edge}}(\mathcal{G} ,\bm{Y})=\frac{\big{|}\{e_{uv}\mid e_{uv}\in\mathcal{E},Y_{u}=Y_{v}\}\big{|}}{ |\mathcal{E}|},\ h_{\text{node}}(\mathcal{G},\bm{Y})=\frac{1}{|\mathcal{V}|} \sum_{v\in\mathcal{V}}\frac{\big{|}\{u\mid u\in\mathcal{N}_{v},Y_{u}=Y_{v}\} \big{|}}{\big{|}\mathcal{N}_{v}\bigl{|}}$}$ (10) $$

$$ $h_{class}(\mathcal{G},\bm{Y})=\frac{1}{C-1}\sum_{c=1}^{C}\bigg{[}\frac{\sum_{u \in\mathcal{V},Y_{u}=c}\big{|}\{v\mid v\in\mathcal{N}_{u},Y_{u}=Y_{v}\}\big{|} }{\sum_{u\in\{u|Y_{u}=c\}}d_{u}}-\frac{N_{c}}{N}\bigg{]}_{+}$ (11) $$

$$ $h_{adj}(\mathcal{G},\bm{Y})=\frac{h_{edge}(\mathcal{G},\bm{Y})-\sum_{c=1}^{C} \frac{D^{2}_{c}}{(2|\mathcal{E}|)^{2}}}{1-\sum_{c=1}^{C}\frac{D^{2}_{c}}{(2| \mathcal{E}|)^{2}}}$ (12) $$

Generally, these metrics measure the label consistency along the graph topology using an indicator function to determine if the connected nodes share the same labels. However, these metrics suffering from the ”good heterophily” (Ma et al., ), leading to a misalignment with GNNs performance. Therefore, new metrics, such as label informativeness (Platonov et al., ), aggregation homophily (Luan et al., ), and classifier-based homophily (Luan et al., ) are proposed to mitigate this deficiency. Nevertheless, all these metrics neglect the feature aspect in graphs, which also constitute an important role in GNNs performance (Zheng et al., ). Therefore, some feature homophily metrics, such as generalized edge homophily (Jin et al., ) is proposed to measure the feature consistency across the graph topology:

$$ $h_{GE}(\mathcal{G},\bm{X})=\frac{1}{|\mathcal{E}|}\sum_{e_{uv}\in\mathcal{E}} \frac{\bm{X}_{u}\bm{X}_{v}}{\left\lVert\bm{X}_{u}\right\rVert\left\lVert\bm{X} _{v}\right\rVert}$ (13) $$

Similarly, local similarity (Chen et al., ) is proposed to measure feature homophily at the node level by cosine similarity or Euclidean similarity:

$$ $\begin{split}h_{LS-cos}(\mathcal{G},\bm{X})&=\frac{1}{|\mathcal{V}|}\sum_{u\in \mathcal{V}}\frac{1}{d_{u}}\sum_{v\in\mathcal{N}_{u}}\frac{\bm{X}_{u}\bm{X}_{v }}{\left\lVert\bm{X}_{u}\right\rVert\left\lVert\bm{X}_{v}\right\rVert}\\ h_{LS-euc}(\mathcal{G},\bm{X})&=\frac{1}{|\mathcal{V}|}\sum_{u\in\mathcal{V}} \frac{1}{d_{u}}\sum_{v\in\mathcal{N}_{u}}(-\left\lVert\bm{X}_{u}-\bm{X}_{v} \right\rVert_{2})\\ \end{split}$ (14) $$

Attribute homophily (Yang et al., ) also consider the feature homophily but with a different normalization on each feature:

$$ $\begin{split}h_{attr,m}(\mathcal{G},\bm{X}_{:,m})&=\frac{1}{\sum_{u\in\mathcal {V}}X_{u,m}}\sum_{u\in\mathcal{V}}\left(X_{u,m}\frac{\sum_{v\in\mathcal{N}_{u} }X_{v,m}}{d_{u}}\right)\\ h_{attr}(\mathcal{G},\bm{X})&=\sum_{m=1}^{M}h_{attr,m}(\mathcal{G},\bm{X}_{:,m })\\ \end{split}$ (15) $$

Class-controlled feature homophily (Lee et al., ) examines the relationship between graph topology and feature dependence by analyzing the difference in expected distances between a node and its neighbors compared to random nodes, defined as follows:

$$ $\begin{split}h_{CF}(\mathcal{G},\bm{X},\bm{Y})&=\frac{1}{\left|\mathcal{V} \right|}\sum_{u\in\mathcal{V}}\frac{1}{d_{u}}\sum_{v\in\mathcal{N}_{u}}\Bigl{( }\bm{d}(v,\mathcal{V}\backslash\{u\})-\bm{d}(v,\{u\})\Bigl{)}\\ \bm{d}(u,\mathcal{V}^{\prime})&=\frac{1}{\left|\mathcal{V}^{\prime}\right|} \sum_{v\in\mathcal{V}^{\prime}}\left\lVert(\bm{X}_{u}|\bm{Y})-(\bm{X}_{v}|\bm{ Y})\right\rVert\\ \bm{X}_{u}|\bm{Y}&=\bm{X}_{u}-\left(\frac{\sum_{Y_{u}=Y_{v}}\bm{X}_{v}}{\left| \{v|Y_{u}=Y_{v},v\in\mathcal{V}\}\right|}\right)\end{split}$ (16) $$

where $\bm{X}_{u}|\bm{Y}$ represents class-controlled features and $\bm{d}(\cdot)$ denotes a distance function.

All of these feature homophily metrics measure the feature consistency, which still suffer from the “good heterophily” issue as similar as node homophily or edge homophily. Our proposed Topological Feature Informativeness (TFI) addresses this limitation by the mutual information, which better aligns with GNNs performance on each feature in graphs.

#### Heterophily-oriented GNNs

For graphs with low homophily (heterophily), GNNs are likely to fail and even worse than non-GNNs models (Luan et al., ). Therefore, many approaches (Wang et al., ; Yang et al., ; Zheng et al., ; Lu et al., ; Luan et al., ) are proposed to improve the performance of GNNs on heterophilous graphs, which can be categorized into graph level, node level, or feature level-based GNNs.

For the graph level, FAGCN (Bo et al., ) introduces high frequency signals in graph convolution to capture local information to address heterophily. Similarly, ACMGCN (Luan et al., ) and FB-GNNs (Luan et al., ) proposes filterbanks to fuse identify, low-pass, and high-pass filter signals. To mitigate the limitation of local heterophilous neighbors, Mixhop (Abu-El-Haija et al., ) and H2GCN Zhu et al. () introduces multi-hop neighbors during the graph convolution. To extends the local neighbors in heterophilous graphs to global neighbors, GloGNN (Li et al., ) proposes signed and learnable coefficient matrix and Geom-GCN (Pei et al., ) introduces new neighbors from the space of geometric embeddings.

For the node level, SnoH (Wang et al., ) modifies the message propagation for each node with varying receptive field in graph convolution. Similarly, CO-GNN (Finkelshtein et al., ) assigns each node with different types of directed message propagation in a cooperative manner, NoSAF (Wang et al., ) provides each node with a node-specific layer aggregation with varying filter weights, and Node-MOE (Han et al., ) identifies heterophilous nodes with a feature inconsistency measurement, then treats these nodes differently with different filters. Different from these approaches, DisamGCL (Zhao et al., ) adopts a contrastive learning objective, by identifying ambiguous nodes with historical label predictions and treating these nodes as negative samples.

To our best knowledge, only DMP (Yang et al., ) proposes to treat each feature differently in heterophily-oriented GNNs. DMP first defines the attribute homophily as feature consistency across the graph topology, then use a learnable weight to automatic learn the layer-wised weights for each features. However, this kind of attribute homophily suffers from ”good heterophily” issue (Ma et al., ) and these homophily values haven’t been used to guide the feature selection in graphs. Furthermore, these learnable weights in GNNs for feature selection still inferior than our statistic-based metrics, TFI.

## Appendix B Proof of Theorem 1

- Theorem1.
Given a graph $\mathcal{G}=\{\mathcal{V},\mathcal{E}\}$ with node features $\mathbf{X_{m}}\in\mathbb{R}^{N}$ in the $m$-th dimension and uniform node labels $\mathbf{Y}\in\mathbb{R}^{N}$ over $\mathcal{V}$, we can obtain aggregated features by applying graph convolution $k$ times, i.e., $\mathbf{\tilde{X}_{m}}=(\mathbf{\hat{A}})^{k}\mathbf{X_{m}}$. For a classifier that predicts label $Y$ using the aggregated features $\tilde{X}_{m}$, its accuracy rate $P_{A}$ is upper bounded by:
$P_{A}\leq\frac{I(Y;\tilde{X}_{m})+\log 2}{\log(C)}$
(17)

Proof. For a node with its aggregated feature $\tilde{X}_{m}$, denote its true label as $Y$ and the predicted label as $\hat{Y}=f(\tilde{X}_{m})$, where $f(\cdot)$ represents the classifier. The occurrence of an error $E$ in the classifier can be expressed as:

$$ $E:=\begin{cases}1&\text{if }\hat{Y}\neq Y,\\ 0&\text{if }\hat{Y}=Y.\end{cases}$ (18) $$

Considering the Markov chain $Y\to\tilde{X}\to\hat{Y}$, we apply Fano’s inequality to obtain:

$$ $H(Y|\tilde{X})\leq H_{b}(P_{E})+P_{E}\log(C-1)$ (19) $$

where $P_{E}$ is the error rate and $H_{b}(\cdot)$ is the binary entropy function.

To express $P_{E}$, we rearrange the inequality:

$$ $P_{E}\geq\frac{H(Y|\tilde{X})-H_{b}(P_{E})}{\log(C-1)}$ (20) $$

Noting that $H(Y|\tilde{X})=H(Y)-I(Y;\tilde{X})=\log(C)-I(Y;\tilde{X})$ and that $H_{b}(P_{E})\leq\log 2$, we can substitute these terms into the equation:

$$ $P_{E}\geq 1-\frac{I(Y;\tilde{X}_{m})+\log 2}{\log(C)}$ (21) $$

Finally, converting the expression to the accuracy rate, we find:

$$ $P_{A}=1-P_{E}\leq\frac{I(Y;\tilde{X}_{m})+\log 2}{\log(C)}$ (22) $$

This concludes the proof.

## Appendix C Implementation Details

In this section, we provide all implementation details in Section 3 and Section 5. We have also made the code publicly available, which can be accessed here: [https://anonymous.4open.science/r/graph-feature-selection-BF28](https://anonymous.4open.science/r/graph-feature-selection-BF28).

### C.1 DATASETS

We use Children, Computers, Fitness, History, Photo, Amazon-Ratings, Minesweeper, Questions, Roman-Empire, Tolokers as mentioned in Table 3 and Squirrel, Chameleon, Actor, Texas, Cornell, Wisconsin, Cora, CiteSeer, PubMed as
mentioned in Table 14 for all experiments. All datasets used in this work are in compliance with the MIT license. Tables 3 and 14 shows the dataset statistics and the values of homophily metrics measured on these datasets. The detailed definition of these homophily measurements are shown in Appendix A. The descriptions of main datasets are given below:

**Table 3: Dataset Statistics**
| Dataset | #Nodes | #Edges | #Features | #Classes | Ave. Degrees | Domain | Feat. Modeling | $\mathbf{h_{node}}$ | $\mathbf{h_{edge}}$ | $\mathbf{h_{class}}$ | $\mathbf{h_{adj}}$ | Avg. TFI |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Children | 76,875 | 1,554,578 | 768 | 24 | 20.22 | E-commerce | PLMs | 0.4579 | 0.4220 | 0.2372 | 0.2913 | 0.0225 |
| Comp. | 87,229 | 721,081 | 768 | 10 | 8.27 | E-commerce | PLMs | 0.8469 | 0.8322 | 0.7601 | 0.7988 | 0.0208 |
| Fitness | 173,055 | 1,773,500 | 768 | 13 | 10.25 | E-commerce | PLMs | 0.8991 | 0.9004 | 0.7940 | 0.8528 | 0.0366 |
| History | 41,551 | 358,574 | 768 | 12 | 8.63 | E-commerce | PLMs | 0.7812 | 0.6626 | 0.2654 | 0.5463 | 0.0296 |
| Photo | 48,362 | 500,939 | 768 | 12 | 10.36 | E-commerce | PLMs | 0.7792 | 0.7491 | 0.7229 | 0.6892 | 0.0234 |
| Amazon. | 24,492 | 93,050 | 300 | 5 | 3.80 | E-commerce | FastText | 0.3793 | 0.3804 | 0.1270 | 0.1357 | 0.0177 |
| Mines. | 10,000 | 39,402 | 7 | 2 | 3.94 | Games | One-hot | 0.6832 | 0.6828 | 0.0094 | 0.0108 | 0.0202 |
| Questions | 48,921 | 153,540 | 301 | 2 | 3.14 | Website | FastText | 0.8963 | 0.8396 | 0.0722 | 0.2759 | 0.0049 |
| Roman. | 22,662 | 32,927 | 300 | 18 | 1.45 | Website | FastText | 0.0415 | 0.0469 | 0.0230 | -0.0778 | 0.4870 |
| tolokers | 11,758 | 519,000 | 10 | 2 | 44.14 | Social | Statistics | 0.6331 | 0.5945 | 0.1867 | 0.0887 | 0.0044 |

- •
Children and History (Yan et al., ) datasets are derived from the Amazon-Books dataset, consisting of items with the second-level label “Children” and “History” , respectively. In both datasets, nodes represent books, and edges indicate frequent co-purchases or co-views between books. Each book’s label corresponds to its third-level category. The node attributes are derived from the book’s title and description using Pre-trained Language Models (PLMs). The task is to classify the books into 24 categories for Children and 12 categories for History. More details can be found on: [https://github.com/sktsherlock/TAG-Benchmark](https://github.com/sktsherlock/TAG-Benchmark).
- •
Computers and Photo (Yan et al., ) datasets are extracted from the Amazon-Electronics dataset, including products with the second-level label “Computers” and “Photo”, respectively. Nodes in these datasets represent electronics products, and edges signify frequent co-purchases or co-views. The labels correspond to the third-level category of the products. User reviews embedded by PLMs were used as the text attributes of the nodes, with the review having the highest number of votes being selected, or, if such a review was unavailable, a random review was chosen instead. The task is to classify the products into 10 categories for Computers and 12 categories for Photo. More details can be found on: [https://huggingface.co/datasets/Sherirto/CSTAG/tree/main](https://huggingface.co/datasets/Sherirto/CSTAG/tree/main).
- •
Fitness dataset (Yan et al., ) is derived from the Amazon-Sports dataset, consisting of fitness-related items with the second-level label “Fitness”. Nodes represent fitness products, and edges indicate frequent co-purchases or co-views between products, encoded by PLM. The labels are based on the third-level category, and the task is to classify items into 13 categories. More details can be found on: [https://huggingface.co/datasets/Sherirto/CSTAG/tree/main/Fitness](https://huggingface.co/datasets/Sherirto/CSTAG/tree/main/Fitness).
- •
Amazon-Ratings (Platonov et al., ) is derived from the Amazon product co-purchasing network metadata, provided by the SNAP (Leskovec & Krevl, ) Datasets, which nodes represent products and edges connect products that are frequently bought together. Node features are created using the mean of fastText (Grave et al., ) embeddings from product descriptions. The task is to predict the average rating of a product, grouped into five classes. More details can be found on: [https://github.com/yandex-research/heterophilous-graphs/tree/main/data](https://github.com/yandex-research/heterophilous-graphs/tree/main/data).
- •
Minesweeper (Platonov et al., ) is inspired by the classic Minesweeper game and is synthetic in nature. It consists of a regular 100x100 grid, where each node represents a cell connected to its eight neighboring cells (except for cells on the edges, which have fewer neighbors). In this setup, $20\%$ of the nodes are randomly designated as mines. Each node has one-hot-encoded features representing the number of neighboring mines, with $50\%$ of the nodes having missing features, indicated by a separate binary attribute. The task to recognize if a nodes is mine or not. More details can be found [https://github.com/yandex-research/heterophilous-graphs/tree/main/data](https://github.com/yandex-research/heterophilous-graphs/tree/main/data).
- •
Questions (Platonov et al., ) is based on user interactions on the Yandex Q question-answering platform. Nodes’ labels represent users, and edges connect users if one answered the other’s question during a one-year period. Node features are the mean of fastText (Grave et al., ) embeddings from user descriptions, with an additional binary feature indicating users without descriptions. The task is to predict which users remained active (i.e., not deleted or blocked) at the end of the period. More details can be found [https://github.com/yandex-research/heterophilous-graphs/tree/main/data](https://github.com/yandex-research/heterophilous-graphs/tree/main/data).
- •
Roman-Empire (Platonov et al., ) is constructed from the “Roman Empire” article in the English Wikipedia. Nodes in the graph represent individual words in the article, and edges connect words that either appear consecutively in the text or are related via syntactic dependencies. The node’s class is syntactic role obtained using spaCy (Honnibal et al., ), and fastText (Grave et al., ) are used for word embeddings of node features. More details can be found [https://github.com/yandex-research/heterophilous-graphs/tree/main/data](https://github.com/yandex-research/heterophilous-graphs/tree/main/data).
- •
Tolokers (Platonov et al., ) is based on data from the Toloka crowdsourcing platform (Likhobaba et al., ). The nodes represent tolokers (workers) who have participated in at least one of 13 projects and edges connect workers who have collaborated on the same task. Node features are based on profile information and task performance statistics. The task is to predict which tolokers have been banned from a project. More details can be found [https://github.com/yandex-research/heterophilous-graphs/tree/main/data](https://github.com/yandex-research/heterophilous-graphs/tree/main/data).

For all datasets, we randomly split the train, validation, and test set as $50\%:25\%:25\%$ for $10$ runs. Specifically, to investigatethe impact of supervision percentage on the performance of GFS, in Figure 6, we combine the train, validation and test sets for TFI calculation with a given ratio $r$.

### C.2 Model

The details of all the GNN methods used in our experiments are introduced as follows:

- •
GCN (Kipf & Welling, ) performs a layer-wise propagation of node features and aggregates features from neighboring nodes to capture local graph structures. Each layer of the network updates the node embeddings by applying a convolution operation over the graph, which combines the node’s own features with the features of its neighbors. The authors’ implementation is available at [https://github.com/tkipf/gcn](https://github.com/tkipf/gcn).
- •
GAT (Velicković et al., ) leverages self-attention mechanisms to perform node classification on graph-structured data. The innovation of GAT lies in its ability to learn different attention coefficients for each neighboring node dynamically. Specifically, the attention coefficient between node $i$ and its neighbor $j$ is computed as:
$e_{ij}=\text{LeakyReLU}(\mathbf{a}^{T}[W\mathbf{h}_{i}\|W\mathbf{h}_{j}]),$
where $\mathbf{h}_{i}$ and $\mathbf{h}_{j}$ are the feature vectors of nodes $i$ and $j$, $W$ is a shared weight matrix, and $\mathbf{a}$ is a learnable attention vector. The coefficients are normalized using the softmax function:
$\alpha_{ij}=\frac{\exp(e_{ij})}{\sum_{k\in\mathcal{N}_{i}}\exp(e_{ik})},$
where $\mathcal{N}_{i}$ denotes the set of neighbors of node $i$. The final node representation is computed as a weighted sum of its neighbors’ features:
$\mathbf{h}_{i}^{\prime}=\sigma\left(\sum_{j\in\mathcal{N}_{i}}\alpha_{ij}W
\mathbf{h}_{j}\right).$
The multi-head attention mechanism improves stability and expressiveness by concatenating or averaging the outputs of multiple independent attention heads, which is set with $num_heads=8$ in our experiments. This allows GAT to assign different importances to each neighboring node while maintaining computational efficiency. The authors’ implementation is available at [https://github.com/PetarV-/GAT](https://github.com/PetarV-/GAT).
- •
SAGE (Hamilton et al., ) introduces an inductive framework for node embeddings by aggregating features from a node’s local neighborhood rather than requiring all nodes to be available during training. Its key innovation lies in the aggregation functions that can efficiently generate embeddings for unseen nodes. The aggregation process involves sampling and aggregating feature information from a node’s neighborhood at each layer of the network. The general form of feature aggregation is:
$h_{v}^{(k)}=\sigma\left(W^{(k)}\cdot\text{AGGREGATE}\left(h_{u}^{(k-1)},
\forall u\in\mathcal{N}(v)\right)\right),$
where $\mathcal{N}(v)$ represents the set of neighbors of node $v$, and AGGREGATE is set to mean operation in our experiment. This allows the model to generalize across evolving graphs and unseen nodes. The authors’ implementation is available at [https://github.com/williamleif/GraphSAGE](https://github.com/williamleif/GraphSAGE).
- •
GT (Shi et al., ) is a transformer-based architecture designed for graph learning. It adapts the traditional Transformer model to graph data by incorporating node and edge features into the attention mechanism. The key innovation of GT is the use of multi-head attention to propagate node features across graph edges while also considering edge information. In our experiments, we employ the Graph Transformer with $8$ attention heads. The attention coefficients are calculated as:
$\alpha_{ij}=\frac{\exp\left(\text{LeakyReLU}\left(\mathbf{q}_{i}^{\top}\mathbf
{k}_{j}+\mathbf{e}_{ij}\right)\right)}{\sum_{k\in\mathcal{N}(i)}\exp\left(
\text{LeakyReLU}\left(\mathbf{q}_{i}^{\top}\mathbf{k}_{k}+\mathbf{e}_{ik}
\right)\right)},$
where $\mathbf{q}_{i}$ and $\mathbf{k}_{j}$ are the query and key vectors for nodes $i$ and $j$, and $\mathbf{e}_{ij}$ represents the edge feature between nodes $i$ and $j$. This allows the model to aggregate both node and edge information. The author’s implementation is available at [https://github.com/PaddlePaddle/PGL/tree/main/ogb_examples/nodeproppred/unimp](https://github.com/PaddlePaddle/PGL/tree/main/ogb_examples/nodeproppred/unimp)
- •
SGC (Simplifying Graph Convolution) (Wu et al., ) is a simplified variant of GCN designed to reduce computational complexity while maintaining similar performance. SGC removes the non-linear activation functions between layers of a traditional GCN, collapsing multiple layers into a single linear transformation. This reduces the overall complexity of the model and results in faster training and inference times. The propagation of node features in SGC can be expressed as:
$\mathbf{H}=\mathbf{S}^{K}\mathbf{X}\mathbf{W},$
where $\mathbf{S}$ is the normalized adjacency matrix, $\mathbf{X}$ is the input feature matrix, $\mathbf{W}$ is the weight matrix, and $K$ is the number of propagation steps (or layers). By precomputing $\mathbf{S}^{K}\mathbf{X}$, the model reduces to simple logistic regression on the preprocessed features, significantly speeding up the training process. In our experiments, we use $K=1$ for optimal performance. The authors’ implementation is available at [https://github.com/Tiiiger/SGC](https://github.com/Tiiiger/SGC).
- •
APPNP (Gasteiger et al., ) builds upon Graph Convolutional Networks (GCNs) by utilizing personalized PageRank for improved propagation of node features while avoiding oversmoothing. The model separates the neural network prediction from the propagation process, allowing it to handle larger neighborhood sizes efficiently. The propagation is controlled by the following iterative equation:
$Z^{(k+1)}=(1-\alpha)\hat{A}Z^{(k)}+\alpha H,$
where $\hat{A}$ is the normalized adjacency matrix, $\alpha$ is the teleport probability (set as $\alpha=0.1$ in our experiments), and $H$ is the initial node feature set. After $k$ (set default $k=2$) propagation steps, the output node features are computed as:
$Z=\text{softmax}(Z^{(k)}).$
This structure permits the model to aggregate information from both local and distant nodes without increasing the depth of the neural network. The authors’ implementation is available at [https://github.com/klicperajo/ppnp](https://github.com/klicperajo/ppnp).
- •
ACMGCN (Luan et al., ) addresses the heterophily problem in graph neural networks (GNNs) by introducing an Adaptive Channel Mixing (ACM) framework. This model adapts to both homophilic and heterophilic graphs by dynamically learning to balance information from three channels: aggregation, diversification, and identity. The key innovation lies in its ability to adaptively learn different weights for each node, allowing the model to exploit local graph structure and node feature similarities. The three channels are combined using learned weights, allowing the model to emphasize different types of information for different nodes:
$H^{(l)}=\alpha_{L}H_{L}^{(l)}+\alpha_{H}H_{H}^{(l)}+\alpha_{I}H_{I}^{(l)},$
where $H_{L}^{(l)}$, $H_{H}^{(l)}$, and $H_{I}^{(l)}$ represent the low-pass (aggregation), high-pass (diversification), and identity channels, respectively, and $\alpha_{L}$, $\alpha_{H}$, $\alpha_{I}$ are learned weights that balance these channels. This flexible channel mixing enables ACMGCN to significantly outperform standard GNN models on heterophilic graphs while maintaining strong performance on homophilic graphs. The authors’ implementation is available at [https://github.com/SitaoLuan/ACM-GNN](https://github.com/SitaoLuan/ACM-GNN).
- •
FAGCN (Bo et al., ) tackles the limitation of traditional GNNs that primarily focus on low-frequency signals. FAGCN introduces a self-gating mechanism to adaptively combine both low-frequency and high-frequency signals, allowing it to handle both assortative and disassortative networks effectively. The key innovation is its ability to dynamically adjust the contribution of each frequency type in the message-passing process. The aggregation of node features is expressed as:
$h_{i}^{\prime}=\epsilon h_{i}+\sum_{j\in\mathcal{N}(i)}\alpha_{Lij}(F_{L}h_{j}
)+\alpha_{Hij}(F_{H}h_{j}),$
where $F_{L}$ and $F_{H}$ are low-pass and high-pass filters, respectively, and $\alpha_{Lij}$ and $\alpha_{Hij}$ are the learned attention coefficients for each type of signal. This enables FAGCN to adapt to different graph structures and alleviates the over-smoothing problem common in deep GNNs. The authors’ implementation is available at [https://github.com/bdy9527/FAGCN](https://github.com/bdy9527/FAGCN).

### C.3 Training Details

All models are implemented using the PyTorch (Ansel et al., ) framework and the DGL (Wang et al., ) library. We run experiments on a machine with $4$ NVIDIA RTX A6000 GPUs, each with 48GB of memory. For most of the tables and figures illustrated in Section 5, we perform a grid search on all dataset and the hyperparameters used for search are list as follow:

- •
Number of layers: $\{2,3\}$
- •
Hidden dimension: $\{128,256,512\}$
- •
Learning rate: $\{3\times 10^{-5},10^{-4},3\times 10^{-4},10^{-3},3\times 10^{-3},10^{-2}\}$
- •
Weight decay: $\{0,10^{-5},10^{-3}\}$
- •
Dropout rate: $\{0.1,0.2,0.4,0.6,0.8\}$

Specially, for response of GCN+GFS, GCN, and MLP to number of layers and hidden dimension, as shown in Figure 5 and Figure 11, the hyperparameters range of set are list as follow:

- •
Number of layers: $\{1,2,3,4,5,6,7,8,9,10\}$
- •
Hidden dimension: $\{16,32,64,128,256,512,1024\}$
- •
Learning rate: $\{10^{-5},3\times 10^{-5},10^{-4},3\times 10^{-4},10^{-3},3\times 10^{-3},10^{
-2},3\times 10^{-2}\}$
- •
Weight decay: $\{0,10^{-5},3\times 10^{-5},10^{-4},3\times 10^{-4},10^{-3}\}$
- •
Dropout rate: $\{0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9\}$

To enhance the performance of all GNN models, we use skip connections (He et al., ) and layer normalization (Ba et al., ) across all methods.

**Table 4: Hyperparameters for GNN baselines and GNN+GFS on Children.**
| Dataset | Model | Num of Layers | Hidden Dim | Learning Rate | Weight Decay | Dropout |
| --- | --- | --- | --- | --- | --- | --- |
| Children | GCN | 2 | 512 | 3e-5 | 0 | 0.2 |
| GCN+GFS | 2 | 256 | 3e-4 | 1e-4 | 0.2 |  |
| GAT | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| GAT+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| SAGE | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| SAGE+GFS | 3 | 512 | 3e-4 | 1e-3 | 0.2 |  |
| GT | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| GT+GFS | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| SGC | 2 | 512 | 3e-3 | 0 | 0.2 |  |
| SGC+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| APPNP | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| APPNP+GFS | 2 | 512 | 3e-4 | 1e-5 | 0.2 |  |
| ACMGCN | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| ACMGCN+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| FAGCN | 2 | 512 | 3e-3 | 0 | 0.6 |  |
| FAGCN+GFS | 2 | 512 | 3e-4 | 1e-3 | 0.2 |  |

**Table 5: Hyperparameters for GNN baselines and GNN+GFS on Computers.**
| Dataset | Model | Num of Layers | Hidden Dim | Learning Rate | Weight Decay | Dropout |
| --- | --- | --- | --- | --- | --- | --- |
| Computers | GCN | 2 | 512 | 3e-5 | 0 | 0.2 |
| GCN+GFS | 3 | 512 | 1e-4 | 3e-5 | 0.2 |  |
| GAT | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| GAT+GFS | 3 | 512 | 3e-4 | 1e-3 | 0.2 |  |
| SAGE | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| SAGE+GFS | 3 | 512 | 3e-4 | 1e-3 | 0.2 |  |
| GT | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| GT+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| SGC | 2 | 512 | 3e-3 | 0 | 0.2 |  |
| SGC+GFS | 3 | 512 | 3e-4 | 1e-5 | 0.2 |  |
| APPNP | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| APPNP+GFS | 3 | 512 | 3e-4 | 1e-3 | 0.2 |  |
| ACMGCN | 2 | 512 | 3e-3 | 0 | 0.6 |  |
| ACMGCN+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| FAGCN | 2 | 512 | 3e-3 | 0 | 0.2 |  |
| FAGCN+GFS | 2 | 512 | 3e-4 | 1e-3 | 0.2 |  |

**Table 6: Hyperparameters for GNN baselines and GNN+GFS on Fitness.**
| Dataset | Model | Num of Layers | Hidden Dim | Learning Rate | Weight Decay | Dropout |
| --- | --- | --- | --- | --- | --- | --- |
| Fitness | GCN | 2 | 512 | 3e-3 | 0 | 0.2 |
| GCN+GFS | 3 | 512 | 3e-5 | 0 | 0.8 |  |
| GAT | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| GAT+GFS | 3 | 512 | 3e-4 | 1e-3 | 0.2 |  |
| SAGE | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| SAGE+GFS | 3 | 512 | 3e-4 | 1e-5 | 0.2 |  |
| GT | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| GT+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| SGC | 2 | 512 | 3e-3 | 0 | 0.2 |  |
| SGC+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| APPNP | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| APPNP+GFS | 3 | 512 | 3e-4 | 1e-5 | 0.2 |  |
| ACMGCN | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| ACMGCN+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| FAGCN | 3 | 512 | 3e-5 | 0 | 0.2 |  |
| FAGCN+GFS | 3 | 512 | 3e-3 | 0 | 0.4 |  |

**Table 7: Hyperparameters for GNN baselines and GNN+GFS on History.**
| Dataset | Model | Num of Layers | Hidden Dim | Learning Rate | Weight Decay | Dropout |
| --- | --- | --- | --- | --- | --- | --- |
| History | GCN | 2 | 512 | 3e-4 | 0 | 0.4 |
| GCN+GFS | 2 | 128 | 1e-3 | 3e-4 | 0.4 |  |
| GAT | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| GAT+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| SAGE | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| SAGE+GFS | 3 | 512 | 3e-4 | 1e-5 | 0.2 |  |
| GT | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| GT+GFS | 3 | 256 | 3e-4 | 0 | 0.2 |  |
| SGC | 2 | 512 | 3e-3 | 0 | 0.2 |  |
| SGC+GFS | 3 | 512 | 3e-3 | 0 | 0.2 |  |
| APPNP | 2 | 512 | 3e-3 | 0 | 0.6 |  |
| APPNP+GFS | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| ACMGCN | 2 | 512 | 3e-3 | 0 | 0.6 |  |
| ACMGCN+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| FAGCN | 2 | 512 | 3e-3 | 0 | 0.2 |  |
| FAGCN+GFS | 3 | 256 | 3e-3 | 0 | 0.2 |  |

**Table 8: Hyperparameters for GNN baselines and GNN+GFS on Photo.**
| Dataset | Model | Num of Layers | Hidden Dim | Learning Rate | Weight Decay | Dropout |
| --- | --- | --- | --- | --- | --- | --- |
| Photo | GCN | 2 | 512 | 3e-5 | 0 | 0.2 |
| GCN+GFS | 3 | 256 | 1e-3 | 3e-4 | 0.1 |  |
| GAT | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| GAT+GFS | 3 | 512 | 3e-4 | 1e-5 | 0.2 |  |
| SAGE | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| SAGE+GFS | 3 | 512 | 3e-4 | 1e-5 | 0.2 |  |
| GT | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| GT+GFS | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| SGC | 2 | 512 | 3e-3 | 0 | 0.2 |  |
| SGC+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| APPNP | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| APPNP+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| ACMGCN | 2 | 512 | 3e-3 | 0 | 0.4 |  |
| ACMGCN+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| FAGCN | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| FAGCN+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |

**Table 9: Hyperparameters for GNN baselines and GNN+GFS on Amazon-Ratings.**
| Dataset | Model | Num of Layers | Hidden Dim | Learning Rate | Weight Decay | Dropout |
| --- | --- | --- | --- | --- | --- | --- |
| Amazon-Ratings | GCN | 2 | 512 | 3e-4 | 1e-3 | 0.4 |
| GCN+GFS | 2 | 512 | 1e-3 | 1e-3 | 0.4 |  |
| GAT | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| GAT+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| SAGE | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| SAGE+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| GT | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| GT+GFS | 3 | 512 | 3e-4 | 1e-5 | 0.2 |  |
| SGC | 2 | 512 | 3e-3 | 0 | 0.4 |  |
| SGC+GFS | 3 | 128 | 3e-4 | 0 | 0.2 |  |
| APPNP | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| APPNP+GFS | 3 | 512 | 3e-4 | 1e-5 | 0.2 |  |
| ACMGCN | 2 | 512 | 3e-4 | 0 | 0.4 |  |
| ACMGCN+GFS | 3 | 512 | 3e-4 | 1e-3 | 0.2 |  |
| FAGCN | 2 | 512 | 3e-3 | 0 | 0.6 |  |
| FAGCN+GFS | 3 | 512 | 3e-4 | 1e-5 | 0.2 |  |

**Table 10: Hyperparameters for GNN baselines and GNN+GFS on Minesweeper.**
| Dataset | Model | Num of Layers | Hidden Dim | Learning Rate | Weight Decay | Dropout |
| --- | --- | --- | --- | --- | --- | --- |
| Minesweeper | GCN | 2 | 512 | 3e-5 | 0 | 0.2 |
| GCN+GFS | 2 | 128 | 1e-3 | 3e-4 | 0.2 |  |
| GAT | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| GAT+GFS | 3 | 512 | 3e-4 | 1e-3 | 0.2 |  |
| SAGE | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| SAGE+GFS | 3 | 512 | 3e-4 | 1e-3 | 0.2 |  |
| GT | 3 | 512 | 3e-5 | 0 | 0.2 |  |
| GT+GFS | 3 | 512 | 3e-5 | 1e-3 | 0.2 |  |
| SGC | 2 | 512 | 3e-3 | 0 | 0.2 |  |
| SGC+GFS | 2 | 256 | 3e-3 | 0 | 0.2 |  |
| APPNP | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| APPNP+GFS | 2 | 512 | 3e-4 | 1e-5 | 0.2 |  |
| ACMGCN | 2 | 512 | 3e-4 | 0 | 0.6 |  |
| ACMGCN+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| FAGCN | 2 | 512 | 3e-3 | 0 | 0.2 |  |
| FAGCN+GFS | 2 | 512 | 3e-4 | 0 | 0.2 |  |

**Table 11: Hyperparameters for GNN baselines and GNN+GFS on Questions.**
| Dataset | Model | Num of Layers | Hidden Dim | Learning Rate | Weight Decay | Dropout |
| --- | --- | --- | --- | --- | --- | --- |
| Questions | GCN | 2 | 512 | 3e-5 | 0 | 0.2 |
| GCN+GFS | 3 | 128 | 1e-4 | 1e-3 | 0.2 |  |
| GAT | 2 | 512 | 3e-4 | 0 | 0.4 |  |
| GAT+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| SAGE | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| SAGE+GFS | 3 | 512 | 3e-4 | 1e-3 | 0.2 |  |
| GT | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| GT+GFS | 3 | 512 | 3e-4 | 1e-3 | 0.2 |  |
| SGC | 2 | 512 | 3e-3 | 0 | 0.8 |  |
| SGC+GFS | 3 | 128 | 3e-3 | 1e-3 | 0.2 |  |
| APPNP | 2 | 512 | 3e-4 | 0 | 0.4 |  |
| APPNP+GFS | 2 | 512 | 3e-5 | 1e-5 | 0.2 |  |
| ACMGCN | 3 | 512 | 3e-5 | 0 | 0.2 |  |
| ACMGCN+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| FAGCN | 2 | 512 | 3e-3 | 0 | 0.2 |  |
| FAGCN+GFS | 2 | 256 | 3e-3 | 0 | 0.2 |  |

**Table 12: Hyperparameters for GNN baselines and GNN+GFS on Roman-Empire.**
| Dataset | Model | Num of Layers | Hidden Dim | Learning Rate | Weight Decay | Dropout |
| --- | --- | --- | --- | --- | --- | --- |
| Roman-Empire | GCN | 2 | 512 | 3e-5 | 0 | 0.2 |
| GCN+GFS | 3 | 128 | 1e-2 | 1e-4 | 0.3 |  |
| GAT | 2 | 512 | 3e-3 | 0 | 0.6 |  |
| GAT+GFS | 3 | 512 | 3e-4 | 1e-5 | 0.2 |  |
| SAGE | 2 | 512 | 3e-3 | 0 | 0.6 |  |
| SAGE+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| GT | 2 | 512 | 3e-3 | 0 | 0.8 |  |
| GT+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| SGC | 2 | 512 | 3e-3 | 0 | 0.2 |  |
| SGC+GFS | 2 | 512 | 3e-3 | 0 | 0.2 |  |
| APPNP | 2 | 512 | 3e-3 | 0 | 0.8 |  |
| APPNP+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| ACMGCN | 2 | 512 | 3e-3 | 0 | 0.8 |  |
| ACMGCN+GFS | 3 | 512 | 3e-3 | 0 | 0.2 |  |
| FAGCN | 2 | 512 | 3e-3 | 0 | 0.4 |  |
| FAGCN+GFS | 2 | 512 | 3e-3 | 0 | 0.2 |  |

**Table 13: Hyperparameters for GNN baselines and GNN+GFS on Tolokers.**
| Dataset | Model | Num of Layers | Hidden Dim | Learning Rate | Weight Decay | Dropout |
| --- | --- | --- | --- | --- | --- | --- |
| Tolokers | GCN | 2 | 512 | 3e-4 | 1e-5 | 0.2 |
| GCN+GFS | 3 | 128 | 1e-3 | 3e-5 | 0.2 |  |
| GAT | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| GAT+GFS | 3 | 512 | 3e-4 | 1e-5 | 0.2 |  |
| SAGE | 2 | 512 | 3e-5 | 0 | 0.2 |  |
| SAGE+GFS | 2 | 512 | 3e-4 | 1e-5 | 0.2 |  |
| GT | 3 | 512 | 3e-5 | 0 | 0.2 |  |
| GT+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| SGC | 2 | 512 | 3e-3 | 0 | 0.2 |  |
| SGC+GFS | 3 | 512 | 3e-3 | 0 | 0.2 |  |
| APPNP | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| APPNP+GFS | 3 | 512 | 3e-4 | 1e-5 | 0.2 |  |
| ACMGCN | 2 | 512 | 3e-4 | 0 | 0.2 |  |
| ACMGCN+GFS | 3 | 512 | 3e-4 | 0 | 0.2 |  |
| FAGCN | 2 | 512 | 3e-3 | 0 | 0.2 |  |
| FAGCN+GFS | 3 | 512 | 3e-4 | 1e-5 | 0.2 |  |

### C.4 Optimization-based Graph Feature Selection

We implement optimization-based metrics $\theta_{\text{Soft}}$ and $\theta_{\text{Hard}}$ mentioned in Section 5.

- •
$\theta_{\text{Soft}}$: We use two learnable weight matrices, $\textbf{W}_{\text{GNN}}\in\mathbb{R}^{M}$ and $\textbf{W}_{\text{MLP}}\in\mathbb{R}^{M}$, to guide feature selection for GNN and MLP. For the input node features $\textbf{X}\in\mathbb{R}^{N\times M}$, we divide them into a GNN-favored part $\textbf{X}_{\mathcal{G}}$ and a GNN-disfavored part $\textbf{X}_{\neg\mathcal{G}}$, computed as follows:
$\displaystyle\textbf{X}_{\mathcal{G}}$
$\displaystyle=\frac{\textbf{W}_{\text{GNN}}^{2}}{\textbf{W}_{\text{GNN}}^{2}+
\textbf{W}_{\text{MLP}}^{2}+\epsilon}\cdot\textbf{X}$
(23)
$\displaystyle\textbf{X}_{\neg\mathcal{G}}$
$\displaystyle=\frac{\textbf{W}_{\text{MLP}}^{2}}{\textbf{W}_{\text{GNN}}^{2}+
\textbf{W}_{\text{MLP}}^{2}+\epsilon}\cdot\textbf{X}$
(24)
where $\epsilon$ is set to $10^{-7}$ to prevent division by zero. In this formulation, $\textbf{W}_{\text{GNN}}$ represents the proportion of each feature that is GNN-favored, while $\textbf{W}_{\text{MLP}}$ represents the proportion that is MLP-favored. The sum of $\textbf{X}_{\mathcal{G}}$ and $\textbf{X}_{\neg\mathcal{G}}$ equals the original input X, indicating that the features are divided based on the relative influence of GNN and MLP preferences. These weight parameters are updated during the training process.
- •
$\theta_{\text{Hard}}$: We use a learnable weight matrix, $\textbf{W}_{\text{Hard}}\in\mathbb{R}^{M\times 2}$, to guide feature selection for GNN and MLP. For the input node features $\textbf{X}\in\mathbb{R}^{N\times M}$, we divide them into a GNN-favored part $\textbf{X}_{\mathcal{G}}$ and a GNN-disfavored part $\textbf{X}_{\neg\mathcal{G}}$, computed as follows:
$\displaystyle\textbf{M}_{\text{Hard}}$
$\displaystyle=\text{Gumbel-Softmax}(\textbf{W}_{\text{Hard}})$
(25)
$\displaystyle\textbf{X}_{\mathcal{G}}$
$\displaystyle=\textbf{X}\cdot\textbf{M}_{\text{Hard}}[:,0]$
(26)
$\displaystyle\textbf{X}_{\neg\mathcal{G}}$
$\displaystyle=\textbf{X}\cdot\textbf{M}_{\text{Hard}}[:,1]$
(27)
In this case, $\textbf{M}_{\text{Hard}}\in\mathbb{R}^{M\times 2}$ is the output of the Gumbel-Softmax applied to $\textbf{W}_{\text{Hard}}$. For each row in $\textbf{M}_{\text{Hard}}$, one value will be 1 and the other will be 0. A value of 1 indicates the selection of that feature, meaning that either the GNN-favored part or the MLP-favored part is chosen for each feature. This binary selection mechanism ensures that each feature is exclusively assigned to either $\textbf{X}_{\mathcal{G}}$ or $\textbf{X}_{\neg\mathcal{G}}$, depending on which component is favored. The use of $\text{Gumbel-Softmax}(\cdot)$ (Jang et al., ) allows for differentiable sampling, enabling the weight parameters to be updated during the training process.

### C.5 Details of Pretrained Node Embeddings

We implement experiments using both GCN and MLP pretrained features to evaluate their effect when applying GCN+GFS on either the original node features, $X$, or on the pretrained node embeddings, MLP($X$) and GCN($X$, $A$). The results of this evaluation are presented in Figure 8 in the main text.

- •
MLP($X$): We pretrain a Multi-Layer Perceptron (MLP) model with the following setup: $2$ layers, a hidden dimension of $128$, trained for $1000$ steps using the ELU activation function. For each of the $10$ data splits, the MLP is trained separately with the Adam optimizer, using a learning rate of $1e-2$ and a weight decay of $5e-4$. The pretrained MLP is then used to extract MLP($X$) corresponding to the training splits, which serves as the node embeddings for the experiment.
- •
GCN($X$, $A$): We pretrain a Graph Convolutional Network (GCN) with $2$ layers, a hidden dimension of $128$, and trained for $1000$ steps. The activation function used is ReLU, with a dropout rate of $0.5$. Similarly, GCN is trained separately on each of the 10 data splits using the Adam optimizer, with a learning rate of $1e-2$ and a weight decay of $5e-4$. The pretrained GCN is then used to extract GCN($X$, $A$), corresponding to the training splits, as the node embeddings.

## Appendix D More Experimental Results

### D.1 GCN+GFS on additional datasets

We report the performance of GCN+GFS on additional datasets including Squirrel, Chameleon, Actor, Texas, Cornell, Wisconsin, Cora, CiteSeer, and PubMed. The dataset statistics and descriptions are shown in Figure 14.

**Table 14: Addtional Dataset Statistics**
| Dataset | #Nodes | #Edges | #Features | #Classes | Ave. Degrees | $\mathbf{h_{node}}$ | $\mathbf{h_{edge}}$ | $\mathbf{h_{class}}$ | $\mathbf{h_{adj}}$ | Avg. TFI |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Squirrel | 2223 | 46998 | 2089 | 5 | 21.14 | 0.1759 | 0.2072 | 0.0725 | -0.0076 | 0.0173 |
| Chameleon | 890 | 8854 | 2325 | 5 | 9.95 | 0.2273 | 0.2361 | 0.0601 | 0.0360 | 0.0177 |
| Actor | 7600 | 26659 | 932 | 5 | 3.51 | 0.2197 | 0.2167 | 0.0074 | 0.0015 | 0.0029 |
| Texas | 183 | 279 | 1703 | 5 | 1.52 | 0.0748 | 0.0609 | 0.0017 | -1.4628 | 0.0348 |
| Cornell | 183 | 277 | 1703 | 5 | 1.51 | 0.1246 | 0.1227 | 0.0482 | -1.1984 | 0.0202 |
| Wisconsin | 251 | 450 | 1703 | 5 | 1.79 | 0.1934 | 0.1778 | 0.0447 | -0.3947 | 0.0182 |
| Cora | 2708 | 10556 | 1433 | 7 | 3.90 | 0.8252 | 0.8100 | 0.7657 | 0.7717 | 0.0188 |
| CiteSeer | 3327 | 9228 | 3703 | 6 | 2.77 | 0.7166 | 0.7391 | 0.6267 | 0.6673 | 0.0103 |
| PubMed | 19717 | 88651 | 500 | 3 | 4.50 | 0.7924 | 0.8024 | 0.6641 | 0.6836 | 0.0440 |

As shown in Figure 9, the performance of the GCN+GFS model surpasses the standard GCN across additional datasets. The selective feature processing approach provided by GFS allows GNNs to focus on beneficial features, leading to improved performance across diverse datasets.

Figure: Figure 9: Performance on additional datasets.
Refer to caption: x9.png

### D.2 Sensitivity of GFS

Figure 10 shows the response of GCN+GFS to $r$ on $6$ other datasets, where GFS collapses to a GNN when $r=1.0$ or an MLP when $r=0.0$, as all features are sent to GNN or MLP, respectively. Although in some datasets like Minesweeper and Questions, where all the features may favor GNNs, GFS doesn’t significantly improve GCN performance, it doesn’t diminish GFS’s overall effectiveness on most datasets.

Figure: Figure 10: The performance of GCN+GFS is shown as the ratio $r$ of GNN-favored features in TFI increases. The point representing the best performance is highlighted as $\bigstar$.
Refer to caption: x10.png

Figure 11 shows how GCN, MLP, and GCN+GFS respond to changes in number of layers, dimension of hidden embeddings, learning rate, weight decay, and dropout rate on a homophilous graph (Computers) and a heterophilous graph (Amazon-Ratings) on other $6$ datasets. In most cases, GFS outperforms GCN, but for certain hyperparameter settings on the Questions dataset, GFS does not consistently outperform GCN. This is likely because the node features in these datasets are already highly graph-favored, making the additional feature selection offered by GFS less impactful. These results highlight the robustness of GFS under different hyperparameter settings.

Figure: Figure 11: Response of GCN+GFS, GCN, and MLP to $5$ hyperparameters on more datasets.
Refer to caption: x11.png

### D.3 Percentage of Supervision in TFI

In Figure 12, the influence of the supervision percentage in TFI on the performance of GCN+GFS across other $6$ datasets is shown. The results highlight that as the supervision percentage increases, the performance of GCN+GFS generally improves or stabilizes. Even with only $10\%$ supervision, GCN+GFS outperforms the standard GCN, demonstrating the robustness of the TFI approach. For datasets like Fitness and Amazon, there is a noticeable improvement in performance as the supervision percentage increases, but it stabilizes at around $30\%$. In datasets such as History, Photos, and Mines, the model performance is relatively stable, indicating that a small percentage of supervision is sufficient to achieve optimal performance. The performance on Questions shows more variability with increasing supervision, possibly due to the graph-favored dataset’s features. This figure supports the claim that TFI requires minimal supervision to improve GCN+GFS performance, which is particularly useful in semi-supervised learning settings.

Figure: Figure 12: Influence of the percentage of the supervision in TFI on the model performance of GCN+GFS.
Refer to caption: x12.png

### D.4 The number of k-hop neighbors in TFI

We further investigate the influence of the number of neighbor hops in TFI on the performance of GCN+GFS. We set $k$ from $\{1,2,\dots,8\}$ as shown in Eq. (5). As illustrated in Figure 13, increasing the number of neighbor hops does not significantly affect model performance and results in only minor deviations in most datasets. This indicates that 1-hop neighbors are sufficient for TFI to select GNN-favored or GNN-disfavored features.

Figure: Figure 13: Influence of the number of neighbor hops in TFI on the performance of GCN+GFS.
Refer to caption: x13.png

## Peer Reviews

confidence: 4
contribution: 2
initial_rating: 3
presentation: 3
questions: See Weaknesses.
soundness: 3
strengths: 1 This paper is well-organized and easy to follow. 

2 The figures regarding the design motivation and the proposed framework (Figures 1 and 3) are clear.

3 The theoretical proofs are detailed.
summary: This paper investigates the problem of feature selection for graph Convolution Networks (GCNs). It begins by introducing TFI, a metric designed to guide the selection of relevant features. Following that, it introduces GFS, a plug-in method that distinguishes between features that benefit graph convolution and those that do not contribute positively or may even have a negative impact. Then, the two sets of features are processed separately using GCN and Multi-Layer Perceptrons (MLP), respectively. Evaluations on node classification tasks demonstrate the effectiveness of the proposed TFI and GFS.
weaknesses: 1 The proposed feature selection metric TFI, which leverages mutual information between features and labels, is not novel. This is a conventional approach for feature selection, as outlined in [1]. Although the TFI utilizes the features derived from the neighborhood average (AX), the approach is incremental. 

2 This paper lacks a comparative analysis of classic feature selection methods, such as [2]. 

3 The GPS exhibits limited robustness to the selection of the hyperparameter r.

4 It is unclear how the high-pass filters of FAGCN and ACMGNN, as presented in Table 1, align with the TFI metric with low-pass AX.

5 A significant concern is the applicability of the proposed GFS. The paper asserts that TFI is computed on training nodes. Thus, I am concerned that the unusual dataset division of 50/25/25 for training/validation/testing employed in this paper is crucial for the effectiveness of GFS. The question then becomes: how would the results be affected if a public splitting, such as 20 per class for training in GCN, SGC, GAT, and APPNP, were applied?

[1] Feature Selection: A Data Perspective. ACM Comput. Surv. 50(6).

[2] Multi-label Feature Selection via Global Relevance and Redundancy Optimization. IJCAI 2020.

---

confidence: 3
contribution: 3
initial_rating: 5
presentation: 3
questions: 1. As shown in Figure 2, the performance gap for the Roman dataset between GCN and MLP in regions with low TFI is minimal. How is the improvement over GCN in Figure 4 so significant regardless of $r$?
2. Since features are updated in each GNN/MLP layer, is the TFI computed at each layer to get (dis)favored features, or is it only computed at the beginning? This point does not seem clearly explained in the paper.
soundness: 2
strengths: 1. The paper is well-motivated and presents a novel approach to distinguish between GNN-favored and GNN-disfavored features, treating them separately to learn with different methods.
2. The paper is clearly presented and includes solid theoretical guarantees, making it easy to follow.
3. The authors provide sufficient empirical analysis, including ablation studies and comparisons with state-of-the-art methods.
summary: The paper introduces a new method to identify features that are favored and disfavored by GNNs. It uses topological feature selection to integrate these features into GNNs, leading to significant improvements in their performance.
weaknesses: The evaluation method is not fair enough. The performance results in Table 1 should be presented with only the difference of with or without GPS, while keeping the model architecture the same (e.g., number of layers and hidden dimension).

---

confidence: 4
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. How is the decomposition of node features in the graph executed? 
2. What impact does the dimension of the initial node features have?
3. Can this method only be applied to node features, or can it also be used on edge features, and what would be the effect?
4. Is it possible to measure what a "good heterophily" feature looks like? Can other metrics be used for supplementary description, such as Label Homophily, Feature Homophily, Mutual Information, and so on?
soundness: 3
strengths: 1. The writing of the paper is clear and easy to understand, with a well-defined theme.
2. The paper provides a theoretical foundation for using TFI to guide feature selection through simple proofs.
3. The experimental results on node classification tasks are impressively good.
summary: GNNs have strong graph learning capabilities. However, GNNs are not effective at learning all graph data features. In previous work, many metrics based on graph topology and features have been proposed to describe feature homogeneity in order to assess the effectiveness of GNNs. However, there has been no metric that can directly and effectively guide which features GNNs should learn. To fill this gap, this paper proposes an evaluation metric, Topological Feature Informative (TFI), to compute the learnability of features for GNNs. Subsequently, a dual-channel embedding architecture combining GNNs and MLPs is used to embed these features separately.
weaknesses: The paper does not provide any statistical analysis of the selected features, which raises some questions.

---

confidence: 5
contribution: 3
initial_rating: 5
presentation: 3
questions: Q1: I am still unclear about how TFI fundamentally differs from other mutual information-based metrics for feature selection in graph learning. Could the authors provide a clearer explanation of its unique contributions and advantages?

Q2: How does the proposed method handle extremely sparse features, which are common in real-world graphs? Are there specific challenges or limitations in this context?

Q3: It appears that the proposed method offers limited advantages on homophilous graphs compared to its performance on heterophilous graphs. Could the authors provide insights to clarify this discrepancy and explain why the method may be less effective in homophilous settings?

Q4: I am curious about the robustness of the proposed method when applied to noisy graphs, where certain features may contain varying levels of noise. Could the authors provide insights or results on its performance in these scenarios?"
soundness: 3
strengths: S1: The concept of distinguishing between GNN-favored and GNN-disfavored features is interesting and novel. 

S2: The paper is well-written and structured, easy for readers to follow the methodology and findings. 

S3:  The experimental results consistently demonstrate the effectiveness of the proposed GFS trick.
summary: The paper explores the varying impact of graph convolution across different feature dimensions in GNNs. It introduces a novel metric called Topological Feature Informativeness (TFI), designed to identify features that are either favored or disfavored by GNNs. The authors propose a Graph Feature Selection (GFS) method that leverages TFI to process GNN-favored features with GNNs and GNN-disfavored features with MLPs, enhancing overall performance. The experimental results across multiple GNN architectures and datasets demonstrate that GFS improves accuracy with minimal computational overhead. The work emphasizes the importance of feature-aware processing in GNNs, showing that graph convolution is not uniformly beneficial for all feature dimensions.
weaknesses: W1: As far as I am concerned, the theoretical analysis of the proposed strategy is incomplete. The paper does not clearly explain the specific conditions under which distinguishing between GNN-favored and GNN-disfavored features is most effective. 

W2: The proposed TFI metric, while beneficial, appears similar to existing mutual information-based methods. The authors need to provide more clarification on how TFI uniquely contributes to feature selection in graph learning, emphasizing its distinct advantages over prior approaches.

W3: Although the experimental results demonstrate strong performance of the GFS-augmented models, I remain somewhat unconvinced. To strengthen the validity of the findings, it would be helpful to include experiments on synthetic datasets, such as those generated by the SBM model with varying levels of heterophily and homophily. Such experiments could offer more reliable insights into the potential advantages of the proposed methodology.

Meta-review: This paper proposes a novel technique called e Graph Feature Selection (GFS) to enhance the performance of GCNs. The method leverages a Topological Feature Informativeness (TFI) to distinguish GNN-favored features and GNN-disfavored features. The proposed method is justified by theories and experiments.   

The paper is well written. The motivation is clearly presented and the results nicely supports the hypothesis that the authors made. The theoretical analysis is solid and detailed. The numerical experiments well support that the proposed method enhances performances of GCNs. Thus, I recommend acceptance of the paper.

Decision: Accept (Poster)