Joint Graph Rewiring and Feature Denoising via Spectral Resonance

## Abstract

Abstract In graph learning the graph and the node features both contain noisy information about the node labels. In this paper we propose joint denoising and rewiring (JDR)—an algorithm to jointly rewire the graph and denoise the features, which improves the performance of downstream node classification graph neural nets (GNNs). JDR improves the alignment between the leading eigenspaces of graph and feature matrices. To approximately solve the associated non-convex optimization problem, we propose a heuristic that efficiently handles real-world graph datasets with multiple classes and different levels of homophily or heterophily. We theoretically justify JDR in a stylized setting and verify the effectiveness of our approach through extensive experiments on synthetic and real-world graph datasets. The results show that JDR consistently outperforms existing rewiring methods on node classification using GNNs as downstream models.

## 1 Introduction

Graph neural networks (GNNs) are a powerful deep learning tool for graph-structured data, with applications in physics (Mandal et al., ; Linkerhägner et al., ), chemistry (Gilmer et al., ), biology (Gligorijević et al., ) and beyond (Zhou et al., ).
Typical tasks across disciplines include graph classification (Duvenaud et al., ; Xu et al., ), node classification (Kipf and Welling, ; Li et al., ) and link prediction (Pan et al., ).

Graph datasets contains two distinct types of information: the graph structure and the node features.
The graph encodes interactions between entities and thus the classes or communities they belong to, similarly to the features.
Since many real graphs inherently contain missing and spurious links and the features are noisy, we ask a natural question:
Is there a simple way to jointly rewire the graph and denoise the features which improves performance of a downstream GNN?

Indeed, recent work demonstrates that rewiring the graph by judiciously adding and removing edges may improve GNN performance by facilitating information flow.
In a GNN, the graph serves not only to encode interactions but also to organize computations (Battaglia et al., ).
So even when it correctly encodes interactions it may not be an effective computational graph and rewiring it may help.
Preprocessing methods rewire the graph using its geometric and spectral properties (Topping et al., ; Nguyen et al., ; Karhadkar et al., ).
End-to-end methods (Giraldo et al., ; Gutteridge et al., ; Qian et al., ) dynamically rewire the graph during training leveraging both the graph and the node features.
However, unlike preprocessing methods, they do not output an improved graph which restricts their interpretability and reusability. Our focus is on the preprocessing methods.

Figure: Figure 1: Schematic overview of joint denoising and rewiring (JDR). In this example, we consider a noisy graph as it occurs in many different real-world scenarios, in the sense that it contains edges between and within classes and its node features are not fully aligned with the labels. The graph’s adjacency matrix $\bm{A}$ and binary node features $\bm{X}$ can be decomposed via spectral decomposition and SVD. The rewiring of $\bm{A}$ is performed by combining the information of its own eigenvectors $\bm{V}$ and the singular vectors $\bm{U}$ from $\bm{X}$. The same applies vice versa for denoising, and both are performed iteratively $K$ times. We synthesise the rewired graph $\tilde{\bm{A}}$ and the denoised features $\tilde{\bm{X}}$ by multiplying back with the final $\bm{V}_{(K)}$ and $\bm{U}_{(K)}$. To get specific properties like sparsity or binarity we can perform an Update step, e.g. by thresholding (as done here). The resulting denoised and rewired graph is displayed on the right. Its structure now better represents the communities and the first entry of the features indicates the class assignment.
Refer to caption: x1.png

In this work, we propose a new rewiring scheme that also uses node features (which contain information about node labels) to produce an enhanced graph.
That both the graph and the features are correlated with labels is implicit in high-quality stylized models of graphs with features, including community models such as the contextual stochastic block model (cSBM) (Deshpande et al., ) and neighborhood graphs on points from low-dimensional manifolds.
This motivates various spectral clustering and non-linear dimensionality reduction methods (Shi and Malik, ; Ng et al., ).
In the cSBM, seminal theoretical work shows that jointly leveraging the graph (stochastic block model (SBM)) and the features (a Gaussian mixture model (GMM)) improves over unsupervised clustering using either piece of information alone. However, the associated efficient inference algorithms based on belief propagation (Deshpande et al., ; Duranthon and Zdeborová, ) rely on perfect knowledge of the distribution of the cSBM and cannot be applied to arbitrary real-world data.

Our contributions are as follows:

- 1.
We take inspiration from work on the cSBM to design a practical algorithm for joint graph rewiring and feature denoising, which can improve the node classification performance of any downstream GNN on real-world data. We achieve this by adapting the graph and the features so as to maximize alignment between their leading eigenspaces.
If these spaces are well-aligned we say that the graph and the features are in resonance.
- 2.
We design an alternating optimization algorithm, joint denoising and rewiring (JDR), which approximates alignment maximization on spectrally-complex real-world graph data with multiple classes, possibly homophilic or heterophilic. We prove that JDR improves alignment between the graph and the features, but also with the labels, on a stylized generative model with noise from the Gaussian orthogonal ensemble (GOE); a recent conjecture in the literature suggests that this generalizes to cSBM.
- 3.
We run extensive experiments to show that JDR outperforms existing preprocessing rewiring strategies while being guided solely by denoising.

This last point suggests that although there exist graphs with topological and geometrical characteristics which make existing rewiring schemes beneficial, a greater issue in real-world graphs is noise in the sense of missing and spurious links.
This is true even when the graphs correctly reflect the ground truth information.
In a citation network, for example, citations that should exist may be missing because of incomplete scholarship.
Conversely, citations that should not exist may be present because the authors engaged in bibliographic ornamentation.
Our method is outlined in Figure 1 and the code repository is available online(^1^11[https://github.com/jlinki/JDR](https://github.com/jlinki/JDR)).

## 2 Joint Denoising and Rewiring

### 2.1 Preliminaries

We let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be an undirected graph with $|\mathcal{V}|=N$ nodes and an adjacency matrix $\bm{A}$. To each node we associate an $F$-dimensional feature vector and collect these vectors in the rows of matrix $\bm{X}\in\mathbb{R}^{N\times F}$. We make extensive use of the graph and feature spectra, namely the eigendecomposition $\bm{A}=\bm{V}\bm{\Lambda}\bm{V}^{T}$ and the SVD $\bm{X}=\bm{U}\bm{\Sigma}\bm{W}^{T}$, with eigen- and singular values ordered from largest to smallest. (As discussed below, in heterophilic graphs we order the eigenvalues of $\bm{A}$ according to their absolute value.)
The graph Laplacian is $\bm{L}=\bm{D}-\bm{A}$, where $\bm{D}$ is the diagonal node degree matrix.
For $k>2$ node classes, we use one-hot labels $\bm{y}\in\{0,1\}^{N\times k}$. We write $[L]$ for the set $\{1,2,\ldots,L\}$.
In the balanced two-class case, we consider nodes to be ordered so that the first half has label $\bm{y}_{i}=-1$ and the second half $\bm{y}_{i}=1$.
In semi-supervised node classification, the task is to label the nodes based on the graph ($\bm{A}$ and $\bm{X}$) and a subset of the labels $\bm{y}$.
Homophilic graphs are those where nodes are more likely to connect with nodes with similar features or labels (e.g., friendship networks (McPherson et al., )); heterophilic graphs are those where nodes more likely to connect with dissimilar nodes (e.g., protein interaction networks (Zhu et al., )).

### 2.2 Motivation via the contextual stochastic block model

For simplicity, we first explain our method for $k=2$ classes and graphs generated from the cSBM. We then extend it to real-world graphs with multiple classes and describe the full practical algorithm.

Contextual Stochastic Block Model.
CSBMs (Deshpande et al., ) extend SBMs (Abbe, ), a community graph model, by high-dimensional node features.
They have become a key generative model for studying GNNs (Baranwal et al., ; Wu et al., ; Kothapalli et al., ); for further pointers see Appendix A.2.
We use cSBMs to build intuition about the graph rewiring and denoising problem.
In a balanced 2-class SBM, the nodes are divided into two equal-sized communities with node labels $\bm{y}_{i}\in\left\{\pm 1\right\}$(^2^22This can be naturally extended to several classes of different size (Duranthon and Zdeborová, ).).
Pairs of nodes connect independently at random, with probability $c_{\text{in}}/N$ inside communities and $c_{\text{out}}/N$ across communities.

In the sparse regime (Abbe, ), with average node degree $d=\mathcal{O}(1)$, it is common to parameterize probabilities as $c_{\text{in}}=d+\lambda\sqrt{d}$ and $c_{\text{out}}=d-\lambda\sqrt{d}$, where $|\lambda|$ can be seen as the signal-to-noise ratio (SNR) of the graph.
The signal $\bm{X}_{i}\in\mathbb{R}^{F}$ at node $i$ comes from a GMM,

$$ $\bm{X}_{i}=\sqrt{\frac{\mu}{N}}\bm{y}_{i}\bm{u}+\frac{\bm{z}_{i}}{\sqrt{F}},$ (1) $$

where $\bm{u}\sim\mathcal{N}\left(0,\bm{I}_{F}/F\right)$ is the randomly drawn mean and $\bm{z}_{i}\sim\mathcal{N}\left(0,\bm{I}_{F}\right)$ is i.i.d. Gaussian standard noise.
We set $\gamma=\frac{N}{F}$ and, following Chien et al. (), parameterize the graphs generated from the cSBM using $\phi=\frac{2}{\pi}\arctan(\lambda\sqrt{\gamma}/\mu)$.
For $\phi\to 1$ we get homophilic behavior; for $\phi\to-1$ we get heterophilic behavior.
Close to either extreme the node features contain little information.
For $\phi\to 0$ the graph is Erdős–Rényi and only the features contain information.

Denoising and Rewiring the cSBM. In the cSBM, $\bm{A}$ and $\bm{X}$ offer different noisy views on the labels.
One can show that up to a scaling and a shift, the adjacency matrix is approximately $\pm\bm{y}\bm{y}^{T}+\bm{Z}_{ER},$ which means that it is approximately a rank-one matrix with labels in the range, corrupted with “Erdős–Rényi-like noise” $\bm{Z}_{ER}$ (Erdös and Rényi, ).
It similarly follows directly from the definition that the feature matrix $\bm{X}$ is approximately (again up to a scaling and a shift) $\bm{y}\bm{u}^{T}+\bm{Z}_{G}$ where $\bm{Z}_{G}$ is white Gaussian noise.
It thus makes sense to use the information from $\bm{X}$ to enhance $\bm{A}$ and vice versa.
Deshpande et al. () show that solving

$$ $\displaystyle\operatorname*{maximize}_{\bm{v}\in\mathbb{R}^{N},\bm{u}\in \mathbb{R}^{F}}\leavevmode\nobreak\ \langle\bm{v},\bm{Av}\rangle+b\langle\bm{v },\bm{Xu}\rangle$ (2) $\displaystyle\mathrm{subject\leavevmode\nobreak\ to}\leavevmode\nobreak\ \|\bm {v}\|_{2}=\|\bm{u}\|_{2}=1,\langle\bm{v},\textbf{1}\rangle\leq\delta$ $$

for some carefully chosen value of $b$ allows one to analyze detection bounds in unsupervised community detection with $k=2$.
It is easy to see that in the high-SNR case ($\lambda$ and $\mu$ far away from the detection threshold), when the second leading eigenvector of $\bm{A}$ and the leading left singular vectors $\bm{X}$ approximately coincide with the labels, the optimal $\bm{v}^{*}$ is related to those vectors and aligned with the labels.
This suggests the following rationale for denoising: (1) We can interpret the value of (2) as a measure of alignment. Since $\bm{v}^{*}$ corresponds to the labels, we can relate this measure to the quality of the label estimation.
(2) We may leverage this alignment to rewire the graph and denoise the features. Namely, we could perturb $\bm{A}$ and $\bm{X}$ in a way that improves the alignment.

In real datasets, however, the optimal value of $b$ is unknown, the scaling of $\bm{X}$ is arbitrary, and things are further complicated by having (many) more than 2 classes.
Moreover, (2) is computationally hard.
We thus define a simple related measure of alignment which alleviates these issues.

Figure: (a)
Refer to caption: x2.png

###### Definition 1 .

Recall the decompositions $\bm{A}=\bm{V\Lambda V}^{T}$, $\bm{X}=\bm{U\Sigma W}^{T}$, and let $\bm{V}_{L}$, $\bm{U}_{L}$ denote the first $L$ columns of $\bm{V}$ and $\bm{U}$ and $\|.\|_{\mathrm{sp}}$ the spectral norm. We define graph–feature alignment as

$$ $\displaystyle\mathrm{Alignment}_{L}(\bm{A},\bm{X})=\|\bm{V}_{L}^{T}\bm{U}_{L} \|_{\mathrm{sp}}.$ (3) $$

Remark: The logic of this definition is that for a cSBM with high SNR and $k$ classes, the information about labels is indeed contained in the leading $L=k$ vectors of $\bm{V}$ and $\bm{U}$.
The quantity $\mathrm{Alignment}_{L}(\bm{A},\bm{X})$ is the cosine of the angle between the subspaces spanned by the columns of $\bm{V}_{L}$ and $\bm{U}_{L}$.
To denoise the features and rewire the graph, we seek to improve the alignment.

Given $\mathrm{Alignment}_{L}(\bm{A},\bm{X})$ and a graph with $\bm{A}_{0}$ and $\bm{X}_{0}$, the jointly denoised graph and features are the solution to

$$ $\displaystyle\operatorname*{maximize}_{\bm{A},\bm{X}}\mathrm{Alignment}_{L}( \bm{A},\bm{X})$ (4) $\displaystyle\mathrm{subject\leavevmode\nobreak\ to}\leavevmode\nobreak\ \|\bm {A}-\bm{A}_{0}\|\leq\delta_{A},\|\bm{X}-\bm{X}_{0}\|\leq\delta_{X}.$ $$

The parameters $\delta_{A},\delta_{X}>0$ modulate the strength of alignment.
We will show empirically that a stronger alignment indicates a better representation of the labels by $\bm{A}$ and $\bm{X}$ and thus a better graph.
Figure 2 visualizes this connection. It shows that the response of the graph to features is maximized when the spectra of the graph and the features are aligned. We refer to the condition where the alignment is high as spectral resonance; see Appendix A.1.1 for further discussion.

###### Definition 1 .

### 2.3 Joint Denoising and Rewiring Algorithm

Maximizing the alignment (4) directly, e.g., using gradient descent, is computationally challenging.
Here we propose a heuristic which alternates between spectral interpolation and graph synthesis. We later prove that the resulting algorithm indeed improves alignment, both with the labels and between the graph and the features, under a stylized noise model.
The algorithm, illustrated in Figure 1, comprises three steps. In Step 1, we compute the spectral decompositions of $\bm{A}$ and $\bm{X}$.
To improve the alignment, we interpolate between the $L$ largest eigenvectors in Step 2.
Based on the new eigenvectors, we synthesize a new graph in Step 3.
The three steps are iterated until a stopping criterion is met.
As is standard in the rewiring literature, the hyperparameters of the algorithm are tuned on a validation set.
Formalizing this results in the JDR algorithm:

- Step 1:
Decomposition
$\displaystyle\bm{A}=\bm{V}\boldsymbol{\Lambda}\bm{V}^{T}\text{ with }\bm{V}=(
\bm{v}_{1},\bm{v}_{2},\dots,\bm{v}_{N})\text{ and }\bm{X}=\bm{U}\boldsymbol{
\Sigma}\bm{W}^{T}\text{ with }\bm{U}=(\bm{u}_{1},\bm{u}_{2},\dots,\bm{u}_{N})$
- Step 2:
Interpolation:
For every $i\in[L]$,
$\displaystyle\tilde{\bm{v}}_{i}$
$\displaystyle=(1-\eta_{A})\bm{v}_{i}+\eta_{A}\leavevmode\nobreak\ \mathrm{sign
}(\langle\bm{v}_{i},\bm{u}_{j}\rangle)\bm{u}_{j}$
$\displaystyle\tilde{\bm{u}}_{i}$
$\displaystyle=(1-\eta_{X})\bm{u}_{i}+\eta_{X}\leavevmode\nobreak\ \mathrm{sign
}(\langle\bm{u}_{i},\bm{v}_{j}\rangle)\bm{v}_{j}$
where $j$ is chosen as $\operatorname*{argmax}_{j\in[L]}|\langle\bm{v}_{i},\bm{u}_{j}\rangle|$ when updating $\bm{v}_{i}$ and as $\operatorname*{argmax}_{j\in[L]}|\langle\bm{u}_{i},\bm{v}_{j}\rangle|$ when updating $\bm{u}_{i}$. $\eta_{A}$ and $\eta_{X}$ are hyperparameters that are tuned with a downstream algorithm on a validation set. We use $\mathrm{sign}()$ to handle sign ambiguities in decompositions.
- Step 3:
Graph Synthesis
$\displaystyle\tilde{\bm{A}}=\tilde{\bm{V}}\bm{\Lambda}\tilde{\bm{V}}^{T}\text{
and }\tilde{\bm{X}}=\tilde{\bm{U}}\bm{\Sigma}\bm{W}^{T}$
- Step 4:
Iterate steps $K$ times with
$\displaystyle\bm{A}\leftarrow\tilde{\bm{A}}\text{ and }\bm{X}\leftarrow\tilde{
\bm{X}}.$

Following (3), we consider the $L$ leading eigenvectors of $\bm{A}$ and $\bm{X}$ for interpolation. Since these bases may be rotated with respect to each other (we note that (3) is insensitive to relative rotations, see Appendix A.1.2), when updating an eigenvector of $\bm{A}$, we interpolate it with the most similar eigenvector of $\bm{X}$.
We show empirically that this heuristic yields strong results, but also prove that it improves alignment with labels with a stylized noise model.
We emphasize that the interpolation rates $\eta_{A}$ and $\eta_{X}$ are the same across different eigenvectors and iterations $K$.
After $K$ steps, we synthesize the final weighted dense graph $\tilde{\bm{A}}=\bm{V}_{(K)}\bm{\Lambda}\bm{V}_{(K)}^{T}$.
To efficiently apply GNNs, we can enforce sparsity, e.g., via thresholding or selecting the top-$k$ entries per node. A detailed pseudocode is given in Appendix A.1.

An illustration. A simple edge case to illustrate how the algorithm works is when either only $\bm{A}$ or $\bm{X}$ contains information.
In a cSBM with $\phi=0$, $\bm{X}$ contains all the information, so the best hyperparameter choice is $\eta_{X}=0$ and (4) simplifies to a maximization over $\bm{A}$.
Since there are only two classes, it is sufficient to consider $L=1$.
From (2) we know that the leading left singular vector $\bm{u}_{1}$ of $\bm{X}$ is well-aligned with the labels.
We thus replace the second leading eigenvector $\bm{v}_{2}$ in $\bm{A}$ by $\bm{u}_{1}$ by choosing $\eta_{A}=1.0$.
After graph synthesis, the new $\bm{v}_{2}$ of $\tilde{\bm{A}}$ is not yet equal to $\bm{u}_{1}$, since $\bm{u}_{1}$ was not orthogonal to the other $\bm{v}_{i}$. We thus repeat the three steps $K$ times.
For $\phi=\pm 1$ all information is contained in the graph; a similar argument can then be constructed mutatis mutandis.

JDR improves alignment.
We can prove that JDR improves alignment under a stylized cSBM-like model.
In fact, we can show something even stronger: the algorithm improves alignment with true labels.
However, since we do not know the labels in practice, we focus on improving the alignment as defined in (4).
To do this, we model the graph and features as rank-1 “label” matrices $\bm{A}=\lambda\bm{yy}^{T}+\bm{O}_{A}$ and $\bm{B}=\lambda\bm{yy}^{T}+\bm{O}_{B}$, which are corrupted by independent GOE noise $\bm{O}_{A}$ and $\bm{O}_{B}$ and $\left\|\bm{y}\right\|=1$.
This is a common model in the analysis of cSBM (Deshpande et al., ). The Gaussian adjacency equivalence conjecture (Shi et al., ) suggests a generalization of the findings to the true binary adjacency case.
One iteration of JDR with $L=1$ first interpolates between the leading eigenvectors $\bm{v}_{1}\left(\bm{A}\right)=\bm{v}_{A}$ and $\bm{v}_{1}\left(\bm{B}\right)=\bm{v}_{B}$; graph synthesis then yields $\bm{A}_{\eta}=\bm{A}+\lambda_{1}\left(\bm{A}\right)\left(-\bm{v}_{A}\bm{v}_{A}
^{T}+\tilde{\bm{v}}_{A}\tilde{\bm{v}}_{A}^{T}\right)$ with $\tilde{\bm{v}}_{A}=(1-\eta)\bm{\bm{v}}_{A}+\mathrm{sign}(\langle\bm{\bm{v}}_{A
},\bm{v}_{B}\rangle)\eta\bm{v}_{B}$.

###### Proposition 1 .

Let $\lambda>\sqrt{2}$. There exists an $\eta^{0}\in(0,1)$ such that
for all $\eta\in(0,\eta^{0})$, we have

$$ $\displaystyle\mathbb{P}\big{[}\left\langle\bm{v}_{1}\left(\bm{A}\right),\bm{y} \right\rangle^{2}<\langle\bm{v}_{1}(\bm{A}_{\eta}),\bm{y}\rangle^{2}\big{]}\to 1 \text{ as }N\to\infty.$ $$

In words, interpolation improves alignment of the largest eigenvector with the labels $\bm{y}$ for sufficiently large graphs. By applying the same argument to $\bm{B}$, we obtain that our method indeed improves alignment and multiple iterations approximate solutions to (4). The proof can be found in Appendix A.3.

###### Proposition 1 .

## 3 Experiments

**Table 1: Comparison of state-of-the-art preprocessing rewiring approaches. Note that we refer to the complexity per iteration. $N$ denotes the number of nodes, $m$ the number of edges and $d_{\text{max}}$ is the maximum node degree.**
| Method | Add edge | Remove edge | Use Features | Heterophilic? | Complexity |
| --- | --- | --- | --- | --- | --- |
| DIGL (Gasteiger et al., ) | ✓ | ✗ | ✗ | ✗ | $\mathcal{O}(N)$ |
| FoSR (Karhadkar et al., ) | ✓ | ✗ | ✗ | ✓ | $\mathcal{O}(N^{2})$ |
| BORF (Nguyen et al., ) | ✓ | ✓ | ✗ | ✓ | $\mathcal{O}(md_{\text{max}}^{3})$ |
| JDR (Ours) | ✓ | ✓ | ✓ | ✓ | $\mathcal{O}(N)$ |

We extensively evaluate JDR on both synthetic data generated from the cSBM and real-world benchmark datasets.
We follow experimental setting from Chien et al. () and evaluate JDR for semi-supervised node classification with different downstream GNNs.
We also adopt their data splits, namely the sparse splitting $2.5\%/2.5\%/95\%$ for training, validation and testing, respectively, or the dense splitting $60\%/20\%/20\%$.
For the general experiments, we perform $100$ runs with different random splits.
For the scalability experiments, we use the experimental settings of the respective works (Lim et al., ; Platonov et al., ).
We report the average accuracy and the $95\%$-confidence interval calculated via bootstrapping with 1000 samples.
All experiments are reproducible using the code provided.

Baselines.
Following recent works on rewiring, we use graph convolution network (GCN) (Kipf and Welling, ) as our downstream GNN.
To obtain a more comprehensive picture, we additionally evaluate the performance on the more recent generalized PageRank graph neural network (GPRGNN) (Chien et al., ).
We compare our algorithm with the state-of-the-art rewiring methods first-order spectral rewiring (FoSR) (Karhadkar et al., ), batch Ollivier-Ricci flow (BORF) (Nguyen et al., ) and diffusion improves graph learning (DIGL) (Gasteiger et al., ).
FoSR approximates which edges should be added to maximize the spectral gap to reduce oversquashing.
BORF adds edges in regions of negative curvature in the graph, which indicate bottlenecks that can lead to an oversquashing of the messages passed along these edges.
A positive curvature indicates that there are so many edges in this area that messages could be oversmoothed, which is why edges are removed here.
We compare computational and implementation aspects of JDR and baselines in Table 1.
On the cSBM, we compare to an optimal baseline, namely the approximate message passing-belief propagation (AMP-BP) algorithm (Duranthon and Zdeborová, ).
AMP-BP is asymptotically optimal (in the large dimension limit) for unsupervised or semi-supervised community detection in the cSBM.
It relies on knowing the distribution of the cSBM and is thus not applicable to real-world graphs with unknown characteristics and complex features.

Hyperparameters.
Unless stated otherwise, we use the hyperparameters from Chien et al. () for the GNNs and optimize the hyperparameters of JDR using a mixture of grid and random search on the validation set.
We use the top-$64$ values of $\tilde{\bm{A}}$ to enforce sparsity and interpolation to update the features.
For DIGL, FoSR and BORF, we tune their hyperparameters using a grid search, closely following the given parameter range from the original papers.
For all hyperparameter searches we use GCN and GPRGNN as the downstream models on $10$ runs with different random splits.
A detailed list of all hyperparameters can be found in Appendix A.7 or in the code repository.

### 3.1 Results on Synthetic Data

We first test JDR on data generated from the cSBM, as we can easily vary the SNR of the graph and the features to verify its denoising and rewiring capabilities.
We focus on the sparse splitting, since for the dense splitting GPRGNN already matches the performance of AMP-BP.

Figure: Figure 3: Alignment of the leading eigenspaces according to (3) for graphs from the cSBM with different $\phi$.
Refer to caption: x4.png

Does JDR maximize Alignment?
Before discussing Figure 4, which shows the results of baselines and JDR for different values of $\phi$, we verify empirically that our alternating optimization algorithm indeed approximates solutions to (4).
As shown in Figure 3, the quantity $\mathrm{Alignment}_{L}(\bm{A},\bm{X})$ improves significantly after running JDR, across all $\phi$. As we show next, this happens simultaneously with improvements in downstream performance, which lends credence to the intuitive reasoning that motivates our algorithm.
For additional alignment results on real-world data and baselines, refer to Appendix A.5.4.

Heterophilic Regime.
For $\phi<-0.25$, the predictions of GCN are only slightly better than random.
GPRGNN performs much better, since it can learn higher order polynomial filters to deal with heterophily.
GCN +JDR outperforms the baseline by a very large margin; it handles heterophilic data well.
Using JDR for GPRGNN further improves its already strong performance in this regime.
Both GNNs benefit less from the denoising in the weakly heterophilic setting where they exhibit the worst performance across all $\phi$.
The difference between $\phi=0$ and the weakly heterophilic regime is that “optimal denoising” for $\phi=0$ is straightforward, since all the information is contained in $\bm{X}$. We show similar findings for spectral clustering on the cSBM in Appendix A.5.5.

Figure: Figure 4: Test accuracy on graphs from the cSBM across different $\phi$. The error bars indicate the $95\%$ confidence interval. JDR improves the performance for both GNNs across all $\phi$.
Refer to caption: x5.png

Weak Graph Regime.
For $|\phi|\leq 0.25$, where the SNR of the graph is very low, both GNNs perform poorly.
Intuitively, when the graph is very noisy, a GNN is a suboptimal model, since it leverages the graph structure.
A simple MLP baseline, using only the node features, outperforms GNNs in this setting, with all three approaches lagging far behind AMP-BP.
Using JDR, we see significant improvements for both GNNs, which almost catch up with AMP-BP for $\phi=0$.
Although all information was available in the node features, the GNN with JDR now clearly outperform the MLP by a very large margin.
We argue that this is because in the semi-supervised setting with few labels available, the GNN generalizes much better.

Homophilic Regime.
For $\phi>0.25$, GCN and GPRGNN perform similarly well, with GPRGNN achieving better results for $\phi\to 1.0$.
With JDR, they become much more comparable to each other and closer to AMP-BP.
Even though the hyperparameters of JDR were tuned using only GCN as a downstream model, it also improves the performance of GPRGNN for all $\phi$.
The general robustness to hyperparameter changes is also analyzed in detail in Appendix A.6.

### 3.2 Results on Real-World Data

We evaluate JDR on five common homophilic benchmarks datasets, namely the citation graphs Cora, CiteSeer, PubMed (Sen et al., )
and the Amazon co-purchase graphs Computers and Photo (McAuley et al., ).
For heterophilic datasets, we rely on the Wikipedia graphs Chameleon and Squirrel (Rozemberczki et al., ), the WebKB datasets Texas and Cornell used in Pei et al. () and the actor co-occurence network Actor (Tang et al., ).
To show the scalability of JDR on larger heterophilic datasets, we further report the results for the Yandex Q user network Questions (Platonov et al., ) and the social networks Penn94 and Twitch-Gamers (Lim et al., ).
Further details about all datasets are in Appendix A.4.
Following Chien et al. (), we evaluate the homophilic datasets in the sparse splitting, staying close to the original setting of Kipf and Welling () and the heterophilic datasets in dense splitting (Pei et al., ).
The remaining larger graphs are evaluated using their original splits.
For further results and splits, see Appendix A.5.

**Table 2: Results on real-world homophilic datasets in the sparse splitting (2.5%/2.5%/95%): Mean accuracy across runs (%) $\pm\ 95\%$ confidence interval. Best average accuracy in bold.**
| Method | Cora | CiteSeer | PubMed | Computers | Photo |
| --- | --- | --- | --- | --- | --- |
| GCN | $77.26$$\pm 0.35$ | $67.16$$\pm 0.37$ | $84.22$$\pm 0.09$ | $84.42$$\pm 0.31$ | $91.33$$\pm 0.29$ |
| GCN+DIGL | $79.27$$\pm 0.26$ | $68.03$$\pm 0.33$ | $84.60$$\pm 0.09$ | $\bm{86.00}$$\pm\bm{0.24}$ | $92.00$$\pm 0.23$ |
| GCN+FoSR | $77.23$$\pm 0.34$ | $67.03$$\pm 0.34$ | $84.21$$\pm 0.09$ | $84.34$$\pm 0.27$ | $91.36$$\pm 0.28$ |
| GCN+BORF | $77.23$$\pm 0.35$ | $66.96$$\pm 0.38$ | $84.22$$\pm 0.09$ | $84.46$$\pm 0.30$ | $91.26$$\pm 0.30$ |
| GCN+JDR | $\bm{79.96}$$\pm\bm{0.26}$ | $\bm{69.35}$$\pm\bm{0.28}$ | $\bm{84.79}$$\pm\bm{0.08}$ | $85.66$$\pm 0.36$ | $\bm{92.52}$$\pm\bm{0.23}$ |
| GPRGNN | $79.65$$\pm 0.33$ | $67.50$$\pm 0.35$ | $84.33$$\pm 0.10$ | $84.06$$\pm 0.48$ | $92.01$$\pm 0.41$ |
| GPRGNN+DIGL | $79.77$$\pm 0.30$ | $67.50$$\pm 0.35$ | $84.72$$\pm 0.10$ | $\bm{86.25}$$\pm\bm{0.28}$ | $92.31$$\pm 0.25$ |
| GPRGNN+FoSR | $79.22$$\pm 0.31$ | $67.30$$\pm 0.38$ | $84.32$$\pm 0.09$ | $84.21$$\pm 0.46$ | $92.07$$\pm 0.37$ |
| GPRGNN+BORF | $79.43$$\pm 0.30$ | $67.48$$\pm 0.36$ | $84.36$$\pm 0.10$ | $84.08$$\pm 0.43$ | $92.11$$\pm 0.38$ |
| GPRGNN+JDR | $\bm{80.77}$$\pm\bm{0.29}$ | $\bm{69.17}$$\pm\bm{0.30}$ | $\bm{85.05}$$\pm\bm{0.08}$ | $84.77$$\pm 0.35$ | $\bm{92.68}$$\pm\bm{0.25}$ |

**Table 3: Results on real-world heterophilic dataset in the dense splitting (60%/20%/20%): Mean accuracy across runs (%) $\pm\ 95\%$ confidence interval. Best average accuracy in bold.**
| Method | Chameleon | Squirrel | Actor | Texas | Cornell |
| --- | --- | --- | --- | --- | --- |
| GCN | $67.65$$\pm 0.42$ | $57.94$$\pm 0.31$ | $34.00$$\pm 0.31$ | $75.62$$\pm 1.12$ | $64.68$$\pm 1.25$ |
| GCN+DIGL | $58.04$$\pm 0.48$ | $39.64$$\pm 0.34$ | $39.57$$\pm 0.29$ | $\bm{91.05}$$\pm\bm{0.73}$ | $\bm{88.49}$$\pm\bm{0.74}$ |
| GCN+FoSR | $67.67$$\pm 0.39$ | $58.12$$\pm 0.35$ | $33.98$$\pm 0.30$ | $78.31$$\pm 1.07$ | $65.64$$\pm 1.06$ |
| GCN+BORF | $67.78$$\pm 0.43$ | OOM | $33.95$$\pm 0.31$ | $76.66$$\pm 1.10$ | $68.72$$\pm 1.11$ |
| GCN+JDR | $\bm{69.76}$$\pm\bm{0.50}$ | $\bm{61.76}$$\pm\bm{0.39}$ | $\bm{40.47}$$\pm\bm{0.31}$ | $85.12$$\pm 0.74$ | $84.51$$\pm 1.06$ |
| GPRGNN | $69.15$$\pm 0.51$ | $53.44$$\pm 0.37$ | $39.52$$\pm 0.22$ | $92.82$$\pm 0.67$ | $87.79$$\pm 0.89$ |
| GPRGNN+DIGL | $66.57$$\pm 0.46$ | $42.98$$\pm 0.37$ | $39.61$$\pm 0.21$ | $91.11$$\pm 0.72$ | $88.06$$\pm 0.81$ |
| GPRGNN+FoSR | $68.96$$\pm 0.45$ | $52.34$$\pm 0.37$ | $39.47$$\pm 0.21$ | $93.16$$\pm 0.66$ | $87.51$$\pm 1.04$ |
| GPRGNN+BORF | $69.44$$\pm 0.56$ | OOM | $39.55$$\pm 0.20$ | $93.53$$\pm 0.68$ | $88.83$$\pm 1.06$ |
| GPRGNN+JDR | $\bm{71.00}$$\pm\bm{0.50}$ | $\bm{60.62}$$\pm\bm{0.38}$ | $\bm{41.89}$$\pm\bm{0.24}$ | $\bm{93.85}$$\pm\bm{0.54}$ | $\bm{89.45}$$\pm\bm{0.84}$ |

Homophilic Datasets. Table 2 shows the results of JDR compared to the baselines.
For both GNNs, JDR achieves the best results on four out of five datasets.
GCN and GPRGNN with JDR achieve similar performance here, which is consistent with the findings for the homophilic cSBM.
DIGL also performs strongly on the datasets and ranks first on Computers.
However, with GPRGNN as a downstream model, the improvements are quite small.
FoSR and BORF only marginally improve the performance of the GNNs in this setting.

Heterophilic Datasets. The results in Table 3 show that GCN +JDR can catch up significantly compared to GPRGNN, but GPRGNN +JDR generally performs better.
This is in line with the findings for the heterophilic cSBM.
DIGL performs well on Actor, Texas and Cornell despite its inherent homophily assumption.
The reason for this is the chosen smoothing kernel, which results in a graph that is evenly connected everywhere with small weights.
GCN then largely ignores the graph and thus performs very similarly to an MLP, which performs already quite well on these datasets (Chien et al., ).
However, this fails for GPRGNN, which can make better use of the weak, complex graph structures.
FoSR and BORF also improve performance here in most cases, but they are outperformed by JDR in all cases, often by a large margin.
The out-of-memory error on Squirrel for BORF results from its computational complexity of $\mathcal{O}(md_{\text{max}}^{3})$, because the dataset has a large number of edges $m$ and a high maximum node degree $d_{\text{max}}$.

**Table 4: Results on large datasets. Mean accuracy (ROC AUC for imbalanced Questions) across runs (%) $\pm\ 95\%$ confidence interval. Best results in bold.**
| Method | Questions | Penn94 | Twitch-gamers |
| --- | --- | --- | --- |
| # nodes | $48,921$ | $41,554$ | $168,114$ |
| # edges | $0.15$M | $1.36$M | $6.8$M |
| GCN | $75.31\pm 0.81$ | $80.40\pm 0.18$ | $64.56\pm 0.19$ |
| GCN+DIGL | $73.35\pm 0.64$ | $74.70\pm 0.32$ | $61.64\pm 0.14$ |
| GCN+FoSR | $75.51\pm 0.73$ | $80.54\pm 0.31$ | $64.65\pm 0.15$ |
| GCN+JDR | $\bm{77.52}\pm\bm{0.63}$ | $\bm{82.30}\pm\bm{0.61}$ | $\bm{65.14}\pm\bm{0.19}$ |

Larger Graphs.
Scalability is a problem for preprocessing rewiring methods because applying them to large graphs requires significant amounts of memory and compute (see complexity in Table 1).
Since the decompositions needed for JDR can be truncated to the largest $L$ vectors, it is still applicable to larger graphs.
The experimental results on larger heterophilic datasets in Table 4 verify this.
They show that JDR can significantly improve performance for these larger graphs, while FoSR only achieves marginal improvements.
BORF ran out of memory on all of these datasets and DIGL is unable to improve due to its inherent homophily assumption.
While scaling JDR to even larger graphs with millions of nodes is possible in principle, it requires more optimized and efficient implementations and is therefore left for future work.

## 4 Relation to Prior Work

JDR is most related to preprocessing rewiring methods which we thus use as baselines. To provide a more thorough overview, we also place it within the extended literature.

Graph Rewiring.
Recent work show that even when a graph correctly encodes interactions it may not be an effective computational graph for a GNN due to conditions such as oversquashing (Alon and Yahav, ; Di Giovanni et al., ) and oversmoothing (Chen et al., ).
Recently many methods have been proposed to address this, notably graph rewiring methods. They can be divided into preprocessing and end-to-end methods.
Preprocessing methods rewire the graph using geometric and spectral properties, including curvature (Topping et al., ; Nguyen et al., ; Fesser and Weber, ; Bober et al., ), expansion (Deac et al., ; Banerjee et al., ), effective resistance (Black et al., ; Shen et al., ), and spectral gap (Karhadkar et al., ).
Conceptually related is diffusion-based rewiring (Gasteiger et al., ) that smooths the graph with a diffusion kernel.
This can be interpreted as graph denoising, but is only suitable for homophilic graphs.
Our approach is related to rewiring but with several key differences (see Table 1). Our rewiring strategy aims to denoise the graph (rather than control some geometric property) with the goal to improve downstream performance, while the classical rewiring literature focuses on optimizing the graph for message passing computations.

Early end-to-end methods randomly drop edges during training to reduce oversmoothing (Rong et al., ). Subsequent work (Gutteridge et al., ; Qian et al., ) incorporates latent features to dynamically rewire the graph.
Ji et al. () use the estimated labels of a GNN to rewire the graph during training of the same GNN.
Giraldo et al. () use curvature information for dynamic rewiring.
Graph Transformers (Dwivedi and Bresson, ; Rampasek et al., ) aim to overcome oversquashing in GNNs via global attention.
In order to handle large graphs, these works still need to revert to sparse, non-global attention (Gabrielsson et al., ; Shirzad et al., ). Unlike preprocessing methods, end-to-end methods cannot output an improved graph which restricts their interpretability and reusability.

Graph Denoising. There is extensive literature on denoising signals on graphs using graph filters (Chen et al., ; Ma et al., ; Liu et al., ). However, we are interested in modifying the structure of the graph itself (rewiring), in a way that can benefit any downstream algorithm.

Graph Structure Learning. The aim of graph structure learning (GSL) (Zhu et al., ) is to make GNNs more robust against adversarial perturbations of the graph or to learn a graph for data where there is no graph to start with (Jin et al., ; Chen et al., ; Wang et al., ; Zhu et al., ).
Lv et al. () build a neighborhood graph over features and interpolate between it and the input graph, which is a form of alignment.
Unlike our method, they do not use spectral information, are unable to deal with noisy features and are only suitable for homophilic graphs where similarity-based connection rules apply.
Even though both our work and GSL consider noisy graph settings, they are conceptually very different.
We do not add noise to graph datasets which corresponds to a perturbation rate of 0 0 (“clean data”) in GSL nomenclature.
Instead, we acknowledge that in every real world dataset, there is noise in the graph structure and the node features, and one manifestation of this noise is a misalignment of their leading eigenspaces.
We then use this to rewire the graph (and denoise features) so as to improve the overall node classification performance of downstream GNNs.
Naturally, GSL methods have difficulties to improve over baselines in this setting (Jin et al., ; Dong and Kluger, ).
Our method, on the other hand, is not designed to handle strong perturbations and therefore cannot compete with GSL methods developed for specifically this purpose.

Graph Regularization.
Laplacian regularization (Ando and Zhang, ), originally stemming from semi-supervised representation learning, has been adapted by recent methods (Yang et al., ; Ma et al., ) to also improve the performance of GNNs.
An extra loss term is added during the GNNs training, which contains additional information about the graph structure to reduce oversmoothing.
Their main limiting factor is the underlying homophiliy assumption: It is assumed that connected nodes are more likely to share the same label.

## 5 Conclusion and Limitations

Our experimental results clearly show that spectral resonance is a powerful principle on which to build graph rewiring (and feature denoising) algorithms. JDR consistently outperforms existing rewiring methods DIGL, FoSR and BORF on both synthetic and real-world graph datasets. The smaller performance gains of GPRGNN suggest that this more powerful GNN is already able to leverage the complementary spectra of graphs and features to some extent.

Still, we should emphasize that preprocessing rewiring methods only use the graph, not the features. A direct comparison is thus not fair, as JDR cannot be used at all without features. Nonetheless, we have proposed the first generally applicable algorithm for cases where such information is available.

On a meta level, our results suggest that noise in real-world graphs is an important limiting factor for the performance of GNNs. It would therefore also be interesting to see whether feature-agnostic rewiring from a denoising perspective, for example using link prediction, could be used to improve the downstream performance. A related idea that we tried, but could not get to work well, is combining existing geometric rewiring algorithms with JDR. Intuitively, there should be a way to benefit from both removing noise and facilitating computation, but we have to leave that for future work.

We also note that most current rewiring methods can be applied to graph level tasks, while JDR is limited to node classification tasks, as it is derived via cSBM, which only considers community structures. It is therefore a fundamental research question how to extend the cSBM idea to graph-level problems. Given the already ubiquitous impact of cSBM on GNN research, this could potentially have a major impact on the field.

## Appendix A Appendix

### A.1 The JDR algorithm

Figure: Algorithm 1 Joint Denoising and Rewiring

#### A.1.1 Low-dimensional Graphs and Relation to Resonance

Figure: Figure 5: Visualization of the first six eigenmodes of $\bm{L}$ of the $8\times 8$ grid graph.
Refer to caption: x6.png

low-dimensional coordinates,
We finally mention that although our algorithm is motivated by the cSBM, it could have equivalently been motivated by ubiquitous low-dimensional graphs.
In such graphs, node labels are related to the which are in turn given by the eigenvectors of the graph Laplacian; this is illustrated in Figure 5.
If, for example, the labels are given by the sign of the first non-constant eigenfunction (the slowest-changing normal mode), our notion of alignment with $L=1$ clearly remains meaningful.

This also further motivates our terminology of resonance. In a purist sense, resonance is a dynamical phenomenon where driving a system with a frequency corresponding to an eigenvalue of the Laplacian yields a diverging response. Importantly, the shape of the response is then an eigenfunction. In a broad sense, resonance signifies alignment with Laplacian eigenfunctions, which are the natural modes. For graphs, this is closely related to alignment with eigenvectors of the adjacency matrix (it is equivalent for $d$-regular graphs). As Figure 2(b) shows, maximizing alignment between feature and graph spectra indeed leads to the largest response of the graph to the features.

#### A.1.2 Rotational invariance of Alignment

Here, we show that the alignment measure (3) is invariant to rotations of the subspaces for non-unique eigenvalues.
Let $A\in\mathbb{R}^{N\times N}$ be the adjacency matrix and $X\in\mathbb{R}^{N\times F}$ the feature matrix. Let the eigendecomposition of $A$ be

$$ $\displaystyle A=\lambda_{1}U_{1}U_{1}^{T}+\cdots+\lambda_{p}U_{p}U_{p}^{T}$ $$

where $p\leq N$, $U_{i}\in\mathbb{R}^{N\times s_{i}}$ with $s_{i}$ being the multiplicity of the eigenvalue $\lambda_{i}$, $\sum_{i}s_{i}=N$, $U_{i}^{T}U_{i}=I$, and $U_{j}^{T}U_{i}=0$ for $i\neq j$. Let similarly the SVD of $X$ be

$$ $\displaystyle X=\sigma_{1}V_{1}W_{1}^{T}.+\cdots+\sigma_{q}V_{q}W_{q}^{T},$ $$

where $V_{i}\in\mathbb{R}^{N\times t_{i}}$, $W_{i}\in\mathbb{R}^{F\times t_{i}}$, $\sum_{i}t_{i}=F$, with analogous orthogonality conditions. Assume both $(\lambda_{i})$ and $(\sigma_{i})$ are sorted from largest to smallest.
Assume also for simplicity that $L=\sum_{i=1}^{p^{\prime}}s_{i}=\sum_{i=1}^{q^{\prime}}t_{i}$ so that the leading $L$-dimensional subspace of the graph $A$ is spanned by the columns of the block matrix $U_{L}=[U_{1}\cdots U_{p^{\prime}}]$. Of course it is also spanned by the columns of $\widetilde{U}_{L}=[U_{1}Q_{1}\cdots U_{p^{\prime}}Q_{p^{\prime}}]$ where invertible matrices $Q_{i}\in\mathbb{R}^{s_{i}\times s_{i}}$ reflect the fact that the eigensolver may return any of the infinitely many (when $s_{i}>1$) orthogonal bases for the subspaces spanned by the columns of $U_{i}$. The $Q_{i}$ are orthogonal since $U_{i}Q_{i}$ are orthogonal. Similarly the leading $L$-dimensional subspace of the features is spanned by the columns of $V_{L}=[V_{1}\cdots V_{q^{\prime}}]$ but also of $\widetilde{V}_{L}=[V_{1}R_{1}\cdots V_{q^{\prime}}R_{q^{\prime}}]$ for any orthogonal $(R_{i})$. Now

$$ $\displaystyle\|\widetilde{U}_{L}^{T}\widetilde{V}_{L}\|_{\mathrm{sp}}$ $\displaystyle=\|\mathrm{blockdiag}(Q_{1}^{T},\ldots,Q_{p^{\prime}}^{T})\ U_{L} ^{T}V_{L}\ \mathrm{blockdiag}(R_{1},\cdots,R_{q^{\prime}})\|_{\mathrm{sp}}$ $\displaystyle=\|U_{L}^{T}V_{L}\|_{\mathrm{sp}}$ $$

for any choice of $Q$s and $R$s since both block-diagonal matrices are orthogonal and the spectral norm is unitarily-invariant.

#### A.1.3 Computational Complexity

The complexity of JDR results mainly from the SVD and eigendecomposition, which is of order $\mathcal{O}(FN\text{min}(F,N))$ for SVD and $\mathcal{O}(N^{3})$ for the eigendecomposition ($F=N$). Since we only need the leading $k$ eigenvectors this reduces to $\mathcal{O}(FNk)$. If the matrix is additionally sparse as is often the case for real-world graphs with binary node features this reduces further to $\mathcal{O}(z(A)k)$ where $z(A)$ is the number of non-zero elements in $A$. If we assume that neither the average degree $d$ nor $k$ is scaled by $N$ (or $F$), the complexity becomes $\mathcal{O}(N)$.

### A.2 Contextual Stochastic Block Models

SBMs and GMMs are landmark theoretical models for studying clustering, classification problems and developing algorithmic tools. The cSBM (Deshpande et al., ), a combination of the two, has become a key model for studying node classification problems on graphs, inspiring numerous designs of GNNs like GPRGNN (Chien et al., ), GIANT (Chien et al., ) or ASGC (Chanpuriya and Musco, ).
Many theoretical studies of node-level GNN problems are based on the cSBM, e.g. on double descent (Shi et al., ), neural collapse (Kothapalli et al., ), OOD generalization (Baranwal et al., ), or oversmoothing (Wu et al., ).
Beyond being a standard synthetic benchmark, the cSBM is also used to to verify hypotheses about GNNs (Ma et al., ; Luan et al., ).

As for any model, the cSBM also comes with limitations. One possible limitation of our work is that cSBM assumes that the features are linear as in a GMM, which makes a linear classifier optimal. If the class boundaries are highly nonlinear, this is no longer true, and the spectrum of $\bm{X}$ may need to be “linearized”, e.g. via Laplacian eigenmaps or diffusion maps.
Still, the results on real-world data show that the cSBM model is already highly transferable, suggesting that the high-dimensional features in real-world graph datasets are often quite linear.

### A.3 Proof of Proposition 1

Notation.
We order eigenvalues from largest to smallest and denote the eigenvector associated with the eigenvalue $\lambda_{j}$ by $\bm{v}_{j}\left(\bm{A}\right)$.
For the leading eigenvectors of $\bm{A}$ and $\bm{B}$ we write $\bm{v}_{1}\left(\bm{A}\right)=\bm{v}_{A}$ and $\bm{v}_{1}\left(\bm{B}\right)=\bm{v}_{B}$.

###### Proof.

Recall that in the BBP transition (Baik et al., ), with the random matrices defined in the main text

$$ $\displaystyle\bm{\bm{A}}$ $\displaystyle=\lambda uu^{T}+\bm{O}_{A}$ $\displaystyle\bm{\bm{B}}$ $\displaystyle=\lambda uu^{T}+\bm{O}_{B}$ $$

when $\lambda>1$, we have as $N\to\infty$ that

$$ $\displaystyle\lambda(A)$ $\displaystyle\overset{a.s.}{\rightarrow}\lambda+\frac{1}{\lambda}$ (5) $\displaystyle\left\langle v_{1}\left(\bm{A}\right),u\right\rangle^{2}$ $\displaystyle\overset{a.s.}{\rightarrow}1-\frac{1}{\lambda^{2}}.$ $$

To prove the proposition, we need to show that when $u$ is replaced by a vector correlated with $\bm{O}_{A}$, the alignment between the spike and the leading eigenvector increases.
Based on (5), and if we choose the same direction of the leading eigenvector as of $u$, i.e. $\left\langle u,v_{A}\right\rangle,\left\langle u,v_{B}\right\rangle>0$ (which we can do due to the sign ambiguity in the spectral decomposition), we can write the leading eigenvector of $\bm{A}$ and $\bm{B}$ as

$$ $\displaystyle v_{A}$ $\displaystyle=\sqrt{1-\frac{1}{\lambda^{2}}}u+\frac{1}{\lambda}w_{A}$ (6) $\displaystyle v_{B}$ $\displaystyle=\sqrt{1-\frac{1}{\lambda^{2}}}u+\frac{1}{\lambda}w_{B},$ $$

where $w_{A},w_{B}$ are two norm-1 vectors orthogonal to
$u$, and when $N\to\infty$ we also have $\langle w_{A},w_{B}\rangle\overset{a.s.}{\to}0$. In this case, we can represent the denoised matrix as

$$ $\displaystyle\bm{A}_{\eta}=$ $\displaystyle\lambda uu^{T}+\lambda_{1}(\bm{A})\left(-v_{A}v_{A}^{T}+\tilde{v} _{A}\tilde{v}_{A}^{T}\right)+\bm{O}_{A}$ (7) $\displaystyle=$ $\displaystyle\lambda uu^{T}+\lambda_{1}(\bm{A})\left(-v_{A}v_{A}^{T}+\left( \left(1-\eta\right)v_{A}+\eta v_{B}\right)\left(\left(1-\eta\right)v_{A}+\eta v _{B}\right)^{T}\right)+\bm{O}_{A}$ $\displaystyle=$ $\displaystyle\lambda_{1}(\bm{S})v_{1}(\bm{S})v_{1}(\bm{S})^{T}+\lambda_{2}(\bm {S})v_{2}(\bm{S})v_{2}(\bm{S})^{T}+\lambda_{3}(\bm{S})v_{3}(\bm{S})v_{3}(\bm{S })^{T}+\bm{O}_{A}$ $$

where $\bm{S}=\lambda uu^{T}+\lambda_{1}(\bm{A})\left(-v_{A}v_{A}^{T}+\left(\left(1-
\eta\right)v_{A}+\eta v_{B}\right)\left(\left(1-\eta\right)v_{A}+\eta v_{B}
\right)^{T}\right)=\sum_{i=1}^{3}\lambda_{i}(\bm{B})v_{i}(\bm{S})v_{i}(\bm{S})
^{T}$ is a rank-3 symmetric matrix, with eigenvalues as functions of $\lambda$ and $\eta$.
The eigenvectors are linear combinations of $u$, $w_{A}$ and $w_{B}$ whose weights are also functions of $\lambda$ and $\eta$.
In particular, by the eigendecomposition of $\bm{S}$, we get

$$ $\displaystyle\lambda_{1}(\bm{S})$ $\displaystyle=\lambda+c_{\lambda_{1}}\eta^{2}+\mathcal{O}\left(\eta^{3}\right)$ (8) $\displaystyle\lambda_{2}(\bm{S})$ $\displaystyle=c_{\lambda_{2}}\eta+\mathcal{O}\left(\eta^{2}\right)$ $\displaystyle\lambda_{3}(\bm{S})$ $\displaystyle=c_{\lambda_{3}}\eta+\mathcal{O}\left(\eta^{2}\right)$ $\displaystyle v_{1}(\bm{S})$ $\displaystyle=u+\eta(a_{1}w_{a}+b_{1}w_{b}+c_{1}u)+\mathcal{O}(\eta^{2})=u+ \eta\delta u+\mathcal{O}(\eta^{2})$ $\displaystyle v_{2}(\bm{S})$ $\displaystyle=a_{2}w_{a}+b_{2}w_{b}+c_{2}u+\mathcal{O}(\eta)$ $\displaystyle v_{3}(\bm{S})$ $\displaystyle=a_{3}w_{a}+b_{3}w_{b}+c_{3}u+\mathcal{O}(\eta)$ $$

where $c_{\lambda_{1}}$, $c_{\lambda_{2}}$, $c_{\lambda_{3}}$ and $\{a,b,c\}_{\{1,2,3\}}$ are constants that depend only on $\lambda$. In this case, with a small $\eta$, only the leading eigenvalue $\bar{\lambda}:=\lambda_{1}(\bm{S})$ would be outside of the bulk and the corresponding eigenvector is the leading eigenvalue we are looking for. We also note that $c_{\lambda_{1}}>0$, and $c_{\lambda_{2}}+c_{\lambda_{3}}<0$, $a_{2}=-a_{3}$, $b_{1}+a_{1}=0$ and $c_{1}=c_{2}=c_{3}=0$.
Using the first-order eigenvector perturbation method compared to the spike-1 matrix $\bar{\bm{A}}=\bar{\lambda}uu^{T}+\bm{O}_{A}$, we have

$$ $\displaystyle\delta v_{1}(\bm{A}_{\eta})$ $\displaystyle:=v_{1}({\bm{A}}_{\eta})-v_{1}(\bar{\bm{A}})$ $\displaystyle=\eta\sum_{j\neq 1}\frac{\lambda_{1}(\bm{S})v_{j}(\bar{\bm{A}})^{ T}\left(u\delta u^{T}+\delta uu^{T}\right)v_{1}(\bar{\bm{A}})}{\lambda_{1} \left(\bar{\bm{A}}\right)-\lambda_{j}\left(\bar{\bm{A}}\right)}v_{j}(\bar{\bm{ A}})$ $\displaystyle+\eta\sum_{j\neq 1}\frac{v_{j}(\bar{\bm{A}})^{T}\left(c_{\lambda_ {2}}v_{2}(\bm{S})v_{2}(\bm{S})^{T}+c_{\lambda_{3}}v_{3}(\bm{S})v_{3}(\bm{S})^{ T}\right)v_{1}(\bar{A})}{\lambda_{1}\left(\bar{\bm{A}}\right)-\lambda_{j}\left (\bar{\bm{A}}\right)}v_{j}(\bar{\bm{A}})+\mathcal{O}\left(\eta^{2}\right).$ $$

Recalling (6) and (8), we get

$$ $\displaystyle\left\langle\delta v_{1}({\bm{A}_{\eta}}),u\right\rangle$ $\displaystyle=\lambda\eta\left(\lambda a_{1}+2\left(\lambda-\frac{\lambda}{ \lambda^{2}}\right)b_{1}\right)\sum_{j\neq 1}\frac{\left(u^{T}v_{j}\right)^{2} }{\lambda_{1}\left(\bar{\bm{A}}\right)-\lambda_{j}\left(\bar{\bm{A}}\right)}$ $\displaystyle-\eta c_{\lambda_{2}}\sqrt{1-\frac{1}{\lambda^{2}}}a_{2}^{2}\sum_ {j\neq 1}\frac{\left(u^{T}v_{j}\right)^{2}}{\lambda_{1}\left(\bar{\bm{A}} \right)-\lambda_{j}\left(\bar{\bm{A}}\right)}$ $\displaystyle-\eta c_{\lambda_{3}}\sqrt{1-\frac{1}{\lambda^{2}}}a_{3}^{2}\sum_ {j\neq 1}\frac{\left(u^{T}v_{j}\right)^{2}}{\lambda_{1}\left(\bar{\bm{A}} \right)-\lambda_{j}\left(\bar{\bm{A}}\right)}+\mathcal{O}\left(\eta^{2}\right)$ $\displaystyle=\eta\left(-\lambda a_{1}\left(\lambda-\frac{2}{\lambda}\right)-( c_{\lambda_{2}}+c_{\lambda_{3}})a_{2}^{2}\sqrt{1-\frac{1}{\lambda^{2}}}\right) \sum_{j\neq 1}\frac{\left(u^{T}v_{j}\right)^{2}}{\lambda_{1}\left(\bar{\bm{A}} \right)-\lambda_{j}\left(\bar{\bm{A}}\right)}+\mathcal{O}\left(\eta^{2}\right)$ $$

Noting that $a_{1}<0$ and $c_{\lambda_{2}}+c_{\lambda_{3}}<0$, for any $\lambda>\sqrt{2}$, we can always find a $\eta=\eta^{0}$, which makes $\langle\delta v_{1}({\bm{A}}_{\eta}),u\rangle$ positive. Since $\bar{\bm{A}}$ follows the standard BBP transition, we finally have

$$ $\left\langle v_{1}({\bm{A}}_{\eta}),u\right\rangle=\left\langle v_{1}(\bar{\bm {A}}),u\right\rangle+\left\langle\delta v_{1}({\bm{A}_{\eta}}),u\right\rangle> \sqrt{1-\frac{1}{\bar{\lambda}^{2}}}>\sqrt{1-\frac{1}{{\lambda}^{2}}}.$ $$

∎

The above results are primarily based on the eigendecomposition of $\bm{B}$, for which analytical expressions can be derived, but they are quite complex. We can easily verify the results for specific values of $\lambda$. For $\lambda=2$ we have

$$ $\displaystyle c_{\lambda_{1}}=\frac{75}{64},c_{2}=-\frac{5\left(\sqrt{2}+1 \right)}{8},c_{3}=\frac{5\left(\sqrt{2}-1\right)}{8},a_{1}=-\frac{5\sqrt{3}}{1 6},a_{2}=\frac{8\sqrt{\frac{2}{3}}}{5}$ $$

and for $\lambda=4$ we have

$$ $\displaystyle c_{\lambda_{1}}=\frac{4335}{8192},c_{2}=-\frac{17\left(\sqrt{2}+ 1\right)}{64},c_{3}=\frac{17\left(\sqrt{2}-1\right)}{64},a_{1}=-\frac{17\sqrt{ 15}}{256},a_{2}=\frac{128\sqrt{\frac{2}{15}}}{17}.$ $$

**Table 5: Properties of the real-world benchmark datasets. For directed graphs we transform the graph to undirected in all experiments. $\mathcal{H}(\mathcal{G})$ indicates the homophily measure.**
| Dataset | Classes | Features | Nodes | Edges | Directed | $\mathcal{H}(\mathcal{G})$ |
| --- | --- | --- | --- | --- | --- | --- |
| Cora | $7$ | $1,433$ | $2,708$ | $5,278$ | False | $0.810$ |
| Citeseer | $6$ | $3,703$ | $3,327$ | $4,552$ | False | $0.736$ |
| PubMed | $3$ | $500$ | $19,717$ | $44,324$ | False | $0.802$ |
| Computers | $10$ | $767$ | $13,752$ | $245,861$ | False | $0.777$ |
| Photo | $8$ | $745$ | $7,650$ | $119,081$ | False | $0.827$ |
| Chameleon | $6$ | $2,325$ | $2,277$ | $31,371$ | True | $0.231$ |
| Squirrel | $5$ | $2,089$ | $5,201$ | $198,353$ | True | $0.222$ |
| Actor | $5$ | $932$ | $7,600$ | $26,659$ | True | $0.219$ |
| Texas | $5$ | $1,703$ | $183$ | $279$ | True | $0.087$ |
| Cornell | $5$ | $1,703$ | $183$ | $277$ | True | $0.127$ |
| Roman-empire | $18$ | $300$ | $22,662$ | $32,927$ | False | $0.047$ |
| Amazon-ratings | $5$ | $300$ | $24,492$ | $93,050$ | False | $0.380$ |
| Minesweeper | $2$ | $7$ | $10,000$ | $39,402$ | False | $0.683$ |
| Tolokers | $2$ | $10$ | $11,758$ | $519,000$ | False | $0.595$ |
| Questions | $2$ | $301$ | $48,921$ | $153,540$ | False | $0.840$ |
| Penn94 | $2$ | $4,814$ | $41,554$ | $1,362,229$ | False | $0.470$ |
| Twitch-gamers | $2$ | $7$ | $168,114$ | $6,797,557$ | False | $0.545$ |

###### Proof.

### A.4 Datasets

Table 5 shows the properties of the real-world datasets used.
We also provide the homophily measure $\mathcal{H}(\mathcal{G})$ proposed in Pei et al. (), which we compute using the build-in function of Pytorch Geometric (Fey and Lenssen, ).
For the cSBM, following (Chien et al., ), we choose $N=5000$, $F=2000$ and thus have $\gamma=\frac{N}{F}=2.5$.
Since the threshold to recover communities in cSBM is $\lambda^{2}+\mu^{2}/\gamma>1$ (Deshpande et al., ), we use a margin such that $\lambda^{2}+\mu^{2}/\gamma=1+\epsilon$.
We choose the same $\epsilon=3.25$ as Chien et al. () in all our experiments to be above the detection threshold and $d=5$ to obtain a sparse graph to be close to the properties of real-world graphs.
From the recovery threshold, we can parameterize the resulting arc of an ellipse with $\lambda\geq 0$ and $\mu\geq 0$ using $\phi=\arctan(\lambda\sqrt{\gamma}/\mu)$.
Table 6 shows the parameters $\mu^{2}$ and $\lambda$ and the homophily measure $\mathcal{H}(\mathcal{G})$for the different values of $\phi$.

**Table 6: Properties of the synthetic datasets generated from the cSBM with $\epsilon=3.25$. $\mathcal{H}(\mathcal{G})$ indicates the homophily measure.**
| $\boldsymbol{\phi}$ | $\mu^{2}$ | $\lambda$ | $\mathcal{H}(\mathcal{G})$ |
| --- | --- | --- | --- |
| $-1.0$ | $0.0$ | $-2.06$ | $0.039$ |
| $-0.875$ | $0.40$ | $-2.02$ | $0.049$ |
| $-0.75$ | $1.56$ | $-1.90$ | $0.076$ |
| $-0.625$ | $3.28$ | $-1.71$ | $0.119$ |
| $-0.5$ | $5.31$ | $-1.46$ | $0.170$ |
| $-0.375$ | $7.35$ | $-1.15$ | $0.241$ |
| $-0.25$ | $9.07$ | $-0.79$ | $0.325$ |
| $-0.125$ | $10.22$ | $-0.40$ | $0.408$ |
| $0.0$ | $10.63$ | $0.0$ | $0.496$ |
| $0.125$ | $10.22$ | $0.40$ | $0.583$ |
| $0.25$ | $9.07$ | $0.79$ | $0.671$ |
| $0.375$ | $7.35$ | $1.15$ | $0.751$ |
| $0.5$ | $5.31$ | $1.46$ | $0.837$ |
| $0.625$ | $3.28$ | $1.71$ | $0.879$ |
| $0.75$ | $1.56$ | $1.90$ | $0.925$ |
| $0.875$ | $0.40$ | $2.02$ | $0.955$ |
| $1.0$ | $0.0$ | $2.06$ | $0.963$ |

### A.5 Additional Results

We provide a number of additional experiments which did not fit in the main text. These include more experiments on additional heterophilic datasets from Platonov et al. (), results for the homophilic datasets in the dense splitting, more experiments with DIGL (Gasteiger et al., ), more alignment results and results for synthetic and real-world data using spectral clustering with and w/o JDR. The clustering experiments in particular allow an interpretation of how JDR works: Applying it to a graph increases its “spectral clusterability”.

#### A.5.1 Additional Heterophilic Datasets

In order to get a more comprehensive picture of the performance of JDR, we also test JDR on all the datasets proposed there by Platonov et al. ().
Table 7 shows the results on these datasets using their original splits and comparing DIGL, FoSR, BORF and JDR.
In general, the performance increases are relatively small for all methods, but overall JDR still performs best. It achieves significant performance increases on Tolokers and Questions.
For Minesweeper none of the methods is really able to improve performance.
The reason for this is in the synthetic design of its graph and the features: The graph does not contain any information about the labels, as it only connects neighboring cells (it is solely a computational graph). The same is partially true for node features which indeed contain information about neighboring mines, but only for $50\%$ of the nodes. This renders JDR unsuitable, which is also reflected in an interesting way in the experiments: We found that the choice of hyperparameters has hardly any influence on performance. But also the results of the other rewiring methods indicate that they cannot be applied here. The graph is a standard grid-graph, which should not exhibit any interesting geometric properties and not contain any insights about the labels. Overall, this is a typical error case for any rewiring method.
But it could also be discussed to what extent this dataset is an interesting graph dataset for node classification at all, since the connectivity does not contain any information about the labels.

**Table 7: Comparison of DIGL, FoSR, BORF and JDR on real-world heterophilic datasets from Platonov et al. (): Mean accuracy (%) and ROC AUC for imbalanced Minesweeper, Tolokers and Questions$\pm\ 95\%$ confidence interval. Best average accuracy in bold. OOM indicates an out-of-memory error.**
| Method | Roman-empire | Amazon-ratings | Minesweeper | Tolokers | Questions |
| --- | --- | --- | --- | --- | --- |
| GCN | $78.64$$\pm 0.42$ | $46.19$$\pm 0.58$ | $\bm{90.08}$$\pm\bm{0.31}$ | $84.61$$\pm 0.59$ | $75.31$$\pm 0.81$ |
| GCN+DIGL | $75.32$$\pm 0.61$ | $45.92$$\pm 0.41$ | $88.16$$\pm 0.57$ | $81.62$$\pm 0.59$ | $73.35$$\pm 0.64$ |
| GCN+FoSR | $78.58$$\pm 0.43$ | $46.30$$\pm 0.44$ | $90.07$$\pm 0.51$ | $84.50$$\pm 0.47$ | $75.51$$\pm 0.73$ |
| GCN+BORF | $78.66$$\pm 0.42$ | $46.44$$\pm 0.54$ | $90.06$$\pm 0.38$ | OOM | OOM |
| GCN+JDR | $\bm{78.86}$$\pm\bm{0.48}$ | $\bm{46.47}$$\pm\bm{0.67}$ | $90.01$$\pm 0.32$ | $\bm{84.73}$$\pm\bm{0.45}$ | $\bm{77.52}$$\pm\bm{0.63}$ |
| GPRGNN | $71.46$$\pm 0.29$ | $45.84$$\pm 0.21$ | $87.80$$\pm 0.51$ | $72.01$$\pm 0.65$ | $65.30$$\pm 1.01$ |
| GPRGNN+DIGL | $71.59$$\pm 0.37$ | $\bm{46.43}$$\pm\bm{0.34}$ | $\bm{87.96}$$\pm\bm{0.50}$ | $73.09$$\pm 1.16$ | $69.98$$\pm 0.49$ |
| GPRGNN+FoSR | $71.44$$\pm 0.30$ | $45.94$$\pm 0.36$ | $87.83$$\pm 0.58$ | $72.72$$\pm 0.69$ | $65.45$$\pm 0.68$ |
| GPRGNN+BORF | $71.46$$\pm 0.26$ | $45.79$$\pm 0.33$ | $87.81$$\pm 0.51$ | OOM | OOM |
| GPRGNN+JDR | $\bm{71.85}$$\pm\bm{0.31}$ | $46.19$$\pm 0.24$ | $87.91$$\pm 0.49$ | $\bm{75.54}$$\pm\bm{0.73}$ | $\bm{73.60}$$\pm\bm{0.86}$ |

#### A.5.2 Homophilic Datasets in the Dense Splitting

**Table 8: Comparison of DIGL, BORF and JDR on real-world homophilic datasets using the dense splitting: Mean accuracy (%) $\pm\ 95\%$ confidence interval. Best average accuracy in bold.**
| Method | Cora | CiteSeer | PubMed | Computers | Photo |
| --- | --- | --- | --- | --- | --- |
| GCN | $88.14$$\pm 0.27$ | $79.02$$\pm 0.25$ | $86.14$$\pm 0.10$ | $89.03$$\pm 0.12$ | $94.07$$\pm 0.10$ |
| GCN+DIGL | $88.74$$\pm 0.28$ | $79.13$$\pm 0.27$ | $\bm{87.81}$$\pm\bm{0.09}$ | $\bm{90.34}$$\pm\bm{0.12}$ | $\bm{94.87}$$\pm\bm{0.10}$ |
| GCN+FoSR | $88.09$$\pm 0.28$ | $79.23$$\pm 0.25$ | $86.14$$\pm 0.10$ | $88.98$$\pm 0.12$ | $94.04$$\pm 0.09$ |
| GCN+BORF | $88.18$$\pm 0.24$ | $79.17$$\pm 0.24$ | $86.14$$\pm 0.10$ | $89.14$$\pm 0.11$ | $94.00$$\pm 0.10$ |
| GCN+JDR | $\bm{88.76}$$\pm\bm{0.25}$ | $\bm{80.25}$$\pm\bm{0.27}$ | $86.20$$\pm 0.10$ | $88.93$$\pm 0.13$ | $94.20$$\pm 0.08$ |
| GPRGNN | $88.57$$\pm 0.0.25$ | $79.42$$\pm 0.30$ | $89.16$$\pm 0.15$ | $88.95$$\pm 0.18$ | $94.49$$\pm 0.11$ |
| GPRGNN+DIGL | $88.49$$\pm 0.24$ | $79.62$$\pm 0.29$ | $88.89$$\pm 0.16$ | $\bm{90.15}$$\pm\bm{0.14}$ | $94.27$$\pm 0.10$ |
| GPRGNN+FoSR | $88.37$$\pm 0.25$ | $79.75$$\pm 0.31$ | $\bm{89.28}$$\pm\bm{0.17}$ | $88.85$$\pm 0.19$ | $94.50$$\pm 0.10$ |
| GPRGNN+BORF | $88.56$$\pm 0.27$ | $79.39$$\pm 0.31$ | $89.04$$\pm 0.18$ | $88.90$$\pm 0.19$ | $94.52$$\pm 0.10$ |
| GPRGNN+JDR | $\bm{89.33}$$\pm\bm{0.25}$ | $\bm{81.00}$$\pm\bm{0.28}$ | $89.24$$\pm 0.15$ | $87.35$$\pm 0.32$ | $\bm{94.78}$$\pm\bm{0.08}$ |

Table 8 shows the results of DIGL, FoSR, BORF and JDR on real-world homophilic datasets in the dense splitting.
The improvements of rewiring are smaller overall compared to the sparse splitting, but all four methods are able to improve it in most cases.
With GCN as the downstream model, DIGL now performs best.
JDR can still achieve the best result on two out of five data sets.
When using GPRGNN as downstream model, JDR performs best on three out of five datasets.
DIGL and FoSR are still able to achieve small performance improvements on most datasets and both rank first place on one dataset.
BORF, on the other hand, is not able to improve the performance in most cases.
This suggests that a more powerful GNN architecture benefits less from DIGL, FoSR or BORF, while JDR can still improve it even further.
The computer dataset is an exception for both downstream GNNs, JDR is not really able to improve the performance at all, while DIGL can clearly improve it.

#### A.5.3 Combining JDR and DIGL

We compare our method to DIGL (Gasteiger et al., ) in the main text.
We use the personalized PageRank (PPR) diffusion kernel and the same top-$64$ values sparsening method as in JDR in all experiments.
Figure 6 shows the additional results for DIGL on the synthetic datasets from the cSBM.
Table 9 shows the results on the real-world homophilic datasets in the sparse splitting and Table 10 on the heterophilic datasets in the dense splitting.
Here, in addition to the individual results for JDR and DIGL, the results for a combination of the two methods are also shown.
For this purpose, the graph was first denoised with JDR and then diffused with DIGL.
To do this, we fixed the hyperparameters of JDR and then tuned the parameter $\alpha$ of DIGL.
We think this is interesting as both methods enhance the graph in different ways and thus should be combinable. In principle, this should also be possible for a combination of JDR with BORF or FoSR, but so far we have not been able to get this to work.

Homophilic datasets. For the homophilic datasets, both DIGL and JDR can improve the results when GCN is used as a downstream model.
Still, DIGL is outperformed by JDR on four out the five datasets.
The two methods can be combined on three of the five data sets to achieve even better results.
This gives empirical support for the assumption that the two methods use a distinct way of performing rewiring in this case and a combination therefore can further increase accuracy.
The picture is somewhat different for GPRGNN as a downstream model.
The improvements for DIGL are significantly smaller here, whereas JDR shows clear improvements across all datasets.
This suggests that a more powerful GNN architecture benefits less from DIGL, while JDR can still improve it even further.
A combination of the two methods does not lead to an increase in performance here.
Although the performance is still significantly better compared to no rewiring or just DIGL, JDR alone usually performs better.

**Table 9: Comparison of DIGL and JDR on real-world homophilic dataset using the sparse splitting: Mean accuracy (%) $\pm\ 95\%$ confidence interval. Best average accuracy in bold.**
| Method | Cora | CiteSeer | PubMed | Computers | Photo |
| --- | --- | --- | --- | --- | --- |
| GCN | $77.26$$\pm 0.35$ | $67.16$$\pm 0.37$ | $84.22$$\pm 0.09$ | $84.42$$\pm 0.31$ | $91.33$$\pm 0.29$ |
| GCN+DIGL | $79.27$$\pm 0.26$ | $68.03$$\pm 0.33$ | $84.60$$\pm 0.09$ | $\bm{86.00}$$\pm\bm{0.24}$ | $92.00$$\pm 0.23$ |
| GCN+JDR | $79.96$$\pm 0.26$ | $\bm{69.35}$$\pm\bm{0.28}$ | $84.79$$\pm 0.08$ | $85.66$$\pm 0.36$ | $92.52$$\pm 0.23$ |
| GCN+JDR+DIGL | $\bm{80.48}$$\pm\bm{0.26}$ | $69.19$$\pm 0.29$ | $\bm{84.83}$$\pm\bm{0.10}$ | $84.78$$\pm 0.34$ | $\bm{92.69}$$\pm\bm{0.22}$ |
| GPRGNN | $79.65$$\pm 0.33$ | $67.50$$\pm 0.35$ | $84.33$$\pm 0.10$ | $84.06$$\pm 0.48$ | $92.01$$\pm 0.41$ |
| GPRGNN+DIGL | $79.77$$\pm 0.30$ | $67.50$$\pm 0.35$ | $84.72$$\pm 0.10$ | $\bm{86.25}$$\pm\bm{0.28}$ | $92.31$$\pm 0.25$ |
| GPRGNN+JDR | $\bm{80.77}$$\pm\bm{0.29}$ | $69.17$$\pm 0.30$ | $\bm{85.05}$$\pm\bm{0.08}$ | $84.77$$\pm 0.35$ | $\bm{92.68}$$\pm\bm{0.25}$ |
| GPRGNN+JDR+DIGL | $80.55$$\pm 0.27$ | $\bm{69.47}$$\pm\bm{0.27}$ | $84.87$$\pm 0.10$ | $85.98$$\pm 0.21$ | $92.67$$\pm 0.27$ |

**Table 10: Comparison of DIGL and JDR on real-world heterophilic dataset using the dense splitting: Mean accuracy (%) $\pm\ 95\%$ confidence interval. Best average accuracy in bold.**
| Method | Chameleon | Squirrel | Actor | Texas | Cornell |
| --- | --- | --- | --- | --- | --- |
| GCN | $67.65$$\pm 0.42$ | $57.94$$\pm 0.31$ | $34.00$$\pm 0.31$ | $75.62$$\pm 1.12$ | $64.68$$\pm 1.25$ |
| GCN+DIGL | $58.04$$\pm 0.48$ | $39.64$$\pm 0.34$ | $39.57$$\pm 0.29$ | $\bm{91.05}$$\pm\bm{0.73}$ | $\bm{88.49}$$\pm\bm{0.74}$ |
| GCN+JDR | $\bm{69.76}$$\pm\bm{0.50}$ | $\bm{61.76}$$\pm\bm{0.39}$ | $\bm{40.47}$$\pm\bm{0.31}$ | $85.12$$\pm 0.74$ | $84.51$$\pm 1.06$ |
| GCN+JDR+DIGL | $66.06$$\pm 0.43$ | $36.62$$\pm 0.29$ | $40.30$$\pm 0.27$ | $88.90$$\pm 0.73$ | $88.06$$\pm 0.77$ |
| GPRGNN | $69.15$$\pm 0.51$ | $53.44$$\pm 0.37$ | $39.52$$\pm 0.22$ | $92.82$$\pm 0.67$ | $87.79$$\pm 0.89$ |
| GPRGNN+DIGL | $66.57$$\pm 0.46$ | $42.98$$\pm 0.37$ | $39.61$$\pm 0.21$ | $91.11$$\pm 0.72$ | $88.06$$\pm 0.81$ |
| GPRGNN+JDR | $\bm{71.00}$$\pm\bm{0.50}$ | $\bm{60.62}$$\pm\bm{0.38}$ | $\bm{41.89}$$\pm\bm{0.24}$ | $\bm{93.85}$$\pm\bm{0.54}$ | $\bm{89.45}$$\pm\bm{0.84}$ |
| GPRGNN+JDR+DIGL | $70.07$$\pm 0.44$ | $59.37$$\pm 0.35$ | $41.57$$\pm 0.20$ | $91.52$$\pm 0.70$ | $87.77$$\pm 1.81$ |

Heterophilic datasets. Since DIGL rewires the graph by adding edges between nodes with short diffusion distance, it is expected to perform poorly on the heterophilic datasets.
The results using GCN show that this is only true for Chameleon and Squirrel, while for Actor, Texas and Cornell there are still considerable improvements.
For the datasets Texas and Cornell, DIGL even achieve the best results.
JDR, on the other hand, improves performance across datasets and GNNs.
This is also in line with the finding on the cSBM in Figure 6(a).
However, we can also see that DIGL can not really improve performance of GPRGNN.
JDR, on the other hand, can still achieve an improvement across all datasets.
A combination of DIGL and JDR is generally not particularly useful in this scenario, likely because DIGL has difficulties on the heterophilic datasets anyway.

Figure: (a)
Refer to caption: x7.png

#### A.5.4 Alignment

Here, we give a more detailed view on how much JDR actually increases alignment on cSBM and real-world datasets compared to the baseline methods.
For cSBM, we can see in Figure 7 that DIGL only increases alignment in the homophilic regime.
In the heterophilic regime it clearly decreases alignment. We expect this because it promotes connections among nodes at short diffusion distance. Also the random teleport probability found on these datasets is $1.0$, which results in a random uniformly connected graph. Similar to the results on the real-world datasets Cornell and Texas from the main text, we can see this in the classification performance in Figure 6. In the heterophilic regime, the performance of DIGL matches exactly the MLP, while in the homophilic regime, we can see some performance increases. With GPRGNN, DIGL is not really able to improve performance at all (except for $\phi=0$).
In Figure 8, we can see that JDR increases alignment across all settings and more strongly than the baseline methods on the real-world graphs (except for Citeseer).
DIGL also increases alignment on many homophilic graphs and on heteophilic Texas.
We would like to note that when only rewiring a graph, increasing the alignment might not always be the best thing to do: If the graph is very good, it would be the features that should be made more aligned to the graph.
But only JDR is able to do this, as it also denoises the features as well.
It is also interesting that DIGL decreases the alignment for Cora and PubMed, but still achieves a good performance. This indicates that DIGL in this case improves the graph in a different way than JDR. So here, it should be possible to combine both methods to achieve even better performance. And indeed, A.5.3 shows that this is exactly the case for Cora, PubMed and Photo on GCN.
FoSR and BORF do not visibly change the alignment, since their modifications to the graph are usually too small.

Figure: Figure 7: Alignment of the leading eigenspaces according to (3) for graphs from the cSBM with different $\phi$. We compare JDR to DIGL and no rewiring.
Refer to caption: x9.png

Figure: (a) Homophilic Datasets
Refer to caption: x10.png

#### A.5.5 Spectral Clustering

In addition to the GNNs as a downstream algorithm, we also experimente with spectral clustering (SC).
Spectral clustering either works with an existing graph, or a k-nearest neighbor graph is created from given (high-dimensional) node features.
Then the $k$ largest eigenvectors of the graph are calculated (the first one is usually omitted as it is a constant vector) and their entries are then used as coordinates for a k-means clustering of the nodes into $k$ classes.
We show that JDR using the hyperparameters found with GCN as a downstream model, improves the performance of a spectral clustering algorithm acting directly on $\bm{A}$ or $\bm{X}$.
This indicates a close connection between GCNs and spectral clustering such that a good denoised graph for GCN is also a good graph for spectral clustering.
Intuitively, since spectral clustering is related to the graph cut, this means that in this case the classes are connected with fewer edges, making them easier to cluster based on the cut.

Figure: (a)
Refer to caption: x12.png

**Table 11: Results on real homophilic datasets using spectral clustering: Mean accuracy (%) and best result in bold. Here, all methods use the hyperparameters found using GCN as downstream algorithm.**
| Method | Cora | CiteSeer | Pubmed | Computers | Photo |
| --- | --- | --- | --- | --- | --- |
| SC(A) | $33.83$ | $24.16$ | $58.94$ | $37.35$ | $30.58$ |
| SC(A)+DIGL | $29.54$ | $22.18$ | $59.65$ | $61.55$ | $25.41$ |
| SC(A)+FoSR | $33.83$ | $24.61$ | $58.72$ | $36.97$ | $30.54$ |
| SC(A)+BORF | $35.01$ | $25.22$ | $58.90$ | $37.35$ | $33.37$ |
| SC(A)+JDR | 67.76 | 63.36 | $\bm{72.90}$ | $\bm{62.29}$ | $\bm{65.67}$ |
| SC(X) | $29.76$ | $45.57$ | $60.45$ | $28.53$ | $48.46$ |
| SC(X)+JDR | 34.68 | 45.90 | $\bm{60.47}$ | $\bm{28.55}$ | $\bm{48.58}$ |

**Table 12: Results on real heterophilic datasets using spectral clustering: Mean accuracy (%) and best result in bold. Here, all methods use the hyperparameters found using GCN as downstream algorithm.**
|  | Chameleon | Squirrel | Actor | Texas | Cornell |
| --- | --- | --- | --- | --- | --- |
| SC(A) | $31.71$ | $22.40$ | $25.92$ | $48.09$ | $39.89$ |
| SC(A)+DIGL | $\bm{32.06}$ | $22.69$ | $25.91$ | $43.72$ | $40.44$ |
| SC(A)+FoSR | $31.44$ | $\bm{24.46}$ | $25.93$ | $55.19$ | $38.80$ |
| SC(A)+BORF | $31.97$ | OOM | $25.97$ | $\bm{56.83}$ | $43.17$ |
| SC(A)+JDR | $31.36$ | $22.15$ | $\bm{28.63}$ | $52.46$ | $\bm{44.26}$ |
| SC(X) | $23.54$ | $20.17$ | $\bm{31.01}$ | $49.18$ | $45.36$ |
| SC(X)+JDR | 24.59 | 21.03 | $23.99$ | $\bm{55.74}$ | $\bm{49.73}$ |

cSBM. Figure 9 displays the results of spectral clustering with and w/o JDR.
Figure 9(a) indicates the expected behavior that spectral clustering using $\bm{A}$ performs particularly poorly in the weak graph regime, since in this case there is hardly any information about the labels in $\bm{A}$.
By using JDR, this limitation is completely removed and the performance is close to AMP-BP across all $\phi$.
The rewired graph now contains more information about the labels, which was previously only available in $\bm{X}$.
For spectral clustering of $\bm{X}$ in Figure 9(b), the relation is exactly the other way around.
In the strong heterophilic or homophilic regime the performance is poor since most information is contained in the graph structure.
Using JDR this limitation is removed and the performance becomes closer to AMP-BP across all $\phi$.
Although a slight denoising of $\bm{X}$ by $\bm{A}$ would be possible for $\phi=\pm 0.375$, there is no performance advantage here and these settings now show the weakest performance across all $\phi$.

Real-world Datasets.
For the real world datasets, we compare the spectral clustering of $\bm{A}$ using the different rewiring methods DIGL, FoSR, BORF and JDR.
For the spectral clustering of $\bm{X}$ we can only evaluate JDR.
Again we use the hyperparameters found using GCN as downstream model.
The results in Table 11 on homophilic datasets show a significant benefit of using JDR across all datasets.
FoSR, BORF and DIGL are also able to improve the performance in some settings but not very consistently.
There are also performance improvements across all datasets for the spectral clustering of $\bm{X}$ with JDR, but these are significantly smaller.
This indicates that the rewiring of the graph has a significantly greater influence on performance here than the denoising of the features.
It also gives an indication of how JDR works on real-world data: It increases “spectral clusterability” of the graph.
Table 12 shows the results for the heterophlic datasets.
The results here are much more inconsistent.
It is striking that DIGL improves Chameleon and Squirrel, while it has actually worsened performance for GCN.
BORF can improve the performance on Texas and Cornell by a large margin, although DIGL and JDR perform better with the GCN.
FoSR improves the performance of Squirrel.
For the results of JDR, it is worth looking at them together with the spectral clustering of $\bm{X}$.
On Chameleon and Squirrel the performance decreases for $\bm{A}$ but clearly increases for $\bm{X}$.
On Texas and Cornell it is improved in all cases, but on $\bm{A}$ not as strongly as for BORF.
On Actor, the performance for $\bm{X}$ has dropped, while JDR is also the only method that really improves the result for $\bm{A}$.
To summarize, the improvements for JDR can be assigned to one of the two sources of information, either $\bm{A}$ or $\bm{X}$, for each dataset.

Figure: (a)
Refer to caption: x14.png

### A.6 Ablations

We perform several ablations of our method to investigate what happens in different scenarios and what effects changes in parameters have.
First, we present our ablations of the JDR method.
We show separate results for denoising only the graph JDR (A) or the features JDR (X) using the GNNs on the cSBM and real-world data.
Also, we show several ablations of the hyperparameters of JDR.
We therefore use a dataset created from cSBM, the homophilic dataset Cora and the heterophilic dataset Chameleon.
Ablations on the real-world datasets are performed for all hyperparameters of JDR and show its robustness to change in these parameters.

JDR. The main motivation for these ablations is to show how much impact the denoising of $\bm{A}$ and $\bm{X}$ respectively have on the results for a dataset and how big the additional benefit is to do this jointly.
Therefore we look at the results if we denoise only the graph JDR (A) or the features JDR (X).
Doing this for the cSBM in Figure 10, we can observe the expected behavior, which is particularly pronounced for GCN in Figure 10(a).
In the weak graph regime, the performance increase results purely from denoising $\bm{A}$, so JDR (A) achieves the same performance as JDR.
The same holds for JDR (X) in the strong homophilic regime and for $\phi=-1.0$.
In the remaining intermediate regimes, we can often observe a performance benefit of both JDR (A) and JDR (X), which becomes much stronger when we combine both.
This benefit of combining both is particularly pronounced for $\phi=-0.375$, where JDR (X) alone even reduces performance, while JDR clearly improves performance.
In Figure 10(b), we can basically observe the same behavior, but less strongly pronounced.
Moreover, it happens in several cases here, again especially in the intermediate regime, that the performance is reduced by JDR (X), but improved for the joint denoising.

Figure: Figure 11: Average accuracy of GCN on all real-world datasets tested for denoising only the features JDR (X), rewiring only the graph JDR (A) and joint denoising and rewiring JDR.
Refer to caption: x16.png

Figure: Figure 12: Average accuracy of GPRGNN on all real-world datasets tested for denoising only the features JDR (X), rewiring only the graph JDR (A) and joint denoising and rewiring JDR. It can be observed that for most datasets, the major improvement is achieved by JDR (A). Only for Squirrel and Chameleon it is JDR (X). In most cases using JDR on both $\bm{X}$ and $\bm{A}$ achieves the best performance.
Refer to caption: x17.png

**Table 13: Comparison for JDR using hyperparameters tuned on different downstream models. The "*" indicates that the hyperparameters of JDR where tuned using the same GNN as downstream model, no symbol mean that the respective other GNN model was used. Results on real-world homophilic datasets using sparse splitting (2.5%/2.5%/95%): Mean accuracy (%) $\pm\ 95\%$ confidence interval. Best average accuracy in bold.**
| Method | Cora | CiteSeer | PubMed | Computers | Photo | $\uparrow$Gain |
| --- | --- | --- | --- | --- | --- | --- |
| GCN | $77.26$$\pm 0.35$ | $67.16$$\pm 0.37$ | $84.22$$\pm 0.09$ | $84.42$$\pm 0.31$ | $91.33$$\pm 0.29$ | - |
| GCN+JDR* | $\bm{79.96}$$\pm\bm{0.26}$ | $\bm{69.35}$$\pm\bm{0.28}$ | $\bm{84.79}$$\pm\bm{0.08}$ | $\bm{85.66}$$\pm\bm{0.36}$ | $\bm{92.52}$$\pm\bm{0.23}$ | $1.59$ |
| GCN+JDR | $78.85$$\pm 0.29$ | $69.11$$\pm 0.28$ | $84.20$$\pm 0.09$ | $85.61$$\pm 0.21$ | $92.25$$\pm 0.25$ | $1.13$ |
| GPRGNN | $79.65$$\pm 0.33$ | $67.50$$\pm 0.35$ | $84.33$$\pm 0.10$ | $84.06$$\pm 0.48$ | $92.01$$\pm 0.41$ | - |
| GPRGNN+JDR | $80.47$$\pm 0.33$ | $68.94$$\pm 0.29$ | $\bm{85.17}$$\pm\bm{0.09}$ | $84.64$$\pm 0.25$ | $92.64$$\pm 0.21$ | $0.86$ |
| GPRGNN+JDR* | $\bm{80.77}$ $\pm\bm{0.29}$ | $\bm{69.17}$$\pm\bm{0.30}$ | $85.05$$\pm 0.08$ | $\bm{84.77}$$\pm\bm{0.35}$ | $\bm{92.68}$$\pm\bm{0.25}$ | $0.98$ |

Figure 11 and Figure 12 show the same investigation for the real-world datasets using GCN and GPRGNN, respectively.
In most cases, the greater performance gain results from JDR (A) and the joint denoising performs best.
Only for the datasets Chameleon for both GNNs and Squirrel for GPRGNN, the denoising of $\bm{X}$ has the greater influence.
Also the case where the denoising of $\bm{X}$ reduces the performance, but a joint denoising performs best, occurs here, e.g. for Citeseer or Cornell.
Overall, this confirms that our method indeed performs joint denoising, especially when both graph and node contain relevant information both benefit from denoising.

Hyperparameters. In Table 13 we show how the downstream GNN performs if JDR was tuned on a different downstream GNN.
We use GCN and GPRGNN for this.
The results show that the hyperparameters of JDR are quite robust to different GNN downstream models as it achieves similar gains using the respective other hyperparameters.
Another way to show the robustness of JDR is to perform ablations of the actual hyperparameters.
To do this, we first look at a data set generated from the cSBM and examine the influence of the number of denoising iterations $K$ and the number of entries of the adjacency matrix to be retained $A_{k}$.
Figure 13 show the results of this study.
As expected increase both results in better performance but will also increase the computational complexity.
Based on this, we choose $A_{k}=64$ for all experiments as a good trade-off between computational and memory cost and accuracy over different numbers of denoising iterations.
We also investigate this effect together with the rest of the hyperparameters for the real-world datasets Cora in Figure 14 and Chameleon in Figure 15.
We again examine the number of denoising iterations $K$ and the number of entries of the adjacency matrix to be retained $A_{k}$.
Additionally, we study the interpolation ratios $\eta_{X}$ and $\eta_{A}$ and the number of eigenvectors for the denoising $L_{X}$ and $L_{A}$.
Both are analyzed relative to the value found by random search and for both $\bm{A}$ and $\bm{X}$ at the same time.
For the interpolation ratios $\eta_{X}$ and $\eta_{A}$, we show the influence of using only a reduced number of digits of the best found value (0 corresponds to no denoising) and for the number of eigenvectors $L_{X}$ and $L_{A}$ we test different offsets (0 corresponding to the best value found using random search).
Overall, we con observe a strong robustness to changes in the hyperparameters.
Only the number of denoising iterations $K$ should not be too high for the heterophilic data set Chameleon.

Figure: Figure 13: Average accuracy of GCN on cSBM with $\phi=0.0$ for different numbers of denoise iterations and different numbers of entries $A_{k}$ to keep for each node in the rewired adjacency matrix. Error bars indicating the $95\%$ confidence interval over $100$ runs.
Refer to caption: x18.png

Figure: (a)
Refer to caption: x19.png

Figure: (a)
Refer to caption: x23.png

### A.7 Hyperparameters

In this section we list all the hyperparameters used for the experiments to ensure the reproducibility of the results.
They are also included in the code.
In all experiments we use the Adam optimizer and the standard early stopping after 200 epochs from (Chien et al., ).
Whenever we use a GCN, it uses two layers, a hidden dimension of $64$ and dropout
with $0.5$.
Whenever we use GPRGNN, we use a polynomial filter of order $10$ (corresponding to $10$ hops) and a hidden dimension of $64$.
For JDR, we always keep the $64$ largest entries of the rewired adjacency matrix $\tilde{\bm{A}}$ per node.
We justify this choice by the ablation in Figure 13.

cSBM. For synthetic data from the cSBM, we generally follow the hyperparameters from (Chien et al., ).
GCN uses a learning rate of $0.01$ and weight decay with $\lambda=0.0005$.
GPRGNN also uses a $\lambda=0.0005$ and both use ReLU non-linearity.
On homophilic graphs ($\phi\geq 0$), GPRGNN uses a learning rate of $0.01$, a weight initialization $\alpha=0.1$ and dropout with $0.5$.
For heterophilic graphs, it uses a learning rate of $0.05$, $\alpha=1.0$ and dropout $0.7$.
The hyperparameters for JDR on the cSBM are shown in Table 17.
We only tuned them using GCN as a downstream model, so for GPRGNN +JDR we use the same ones.

Real-world Datasets. For the real-world datasets, the remaining hyperparameters for GCN are displayed in Table 14 and for GPRGNN in Table 16.
The hyperparameters for JDR can be found in Table 18 and Table 19.
For the rewiring method BORF, we list its hyperparameters in Table 20 and Table 21.
For DIGL, we always use the PPR kernel and sparsify the result by keeping the top-$64$ values for a weighted adjacency matrix.
The values for the random-teleport probabililty $\alpha$ and number of iterations for FoSR are listed in Table 22 and Table 23.

**Table 14: Hyperparameters of GCN. All models use $2$ layers, a hidden dimension of $64$ and dropout with $0.5$. Different type of weight decay and early stopping from (Gasteiger et al., ) was used, if these provided a better performance then using the standard setting in Chien et al. (). The same holds for feature normalization, which was used by default in Chien et al. () for GPRGNN.**
| Dataset | Lr | Normalize $\bm{X}$ | $\lambda_{1}$ | $\lambda_{1}$ layer | Early stopping |
| --- | --- | --- | --- | --- | --- |
| Cora | $0.01$ | False | $0.05$ | First | GPRGNN |
| Citeseer | $0.01$ | True | $0.0005$ | All | GPRGNN |
| PubMed | $0.01$ | True | $0.0005$ | All | GPRGNN |
| Computers | $0.01$ | False | $0.0005$ | All | GPRGNN |
| Photo | $0.01$ | False | $0.0005$ | All | GPRGNN |
| Chameleon | $0.05$ | True | $0.0$ | All | DIGL |
| Squirrel | $0.05$ | True | $0.0$ | All | DIGL |
| Actor | $0.01$ | False | $0.0005$ | All | DIGL |
| Texas | $0.05$ | True | $0.0005$ | All | GPRGNN |
| Cornell | $0.05$ | True | $0.0005$ | All | GPRGNN |

**Table 15: Hyperparameters of GCN for the larger heterophilic datasets. All models use a hidden dimension of $64$, batch norm, no weight decay and the early stopping from Chien et al. ().**
| Dataset | Lr | # layers | Dropout | Residuals |
| --- | --- | --- | --- | --- |
| Questions | $0.005$ | $5$ | $0.2$ | True |
| Penn94 | $0.001$ | $2$ | $0.5$ | False |
| Twitch-gamers | $0.01$ | $4$ | $0.5$ | False |

**Table 16: Hyperparameters of GPRGNN. All models use $10$ hops and a hidden dimension of $64$.**
| Dataset | Lr | Normalize $\bm{X}$ | $\alpha$ | $\lambda_{1}$ | Dropout | Early stopping |
| --- | --- | --- | --- | --- | --- | --- |
| Cora | $0.01$ | True | $0.1$ | $0.0005$ | $0.5$ | GPRGNN |
| Citeseer | $0.01$ | True | $0.1$ | $0.0005$ | $0.5$ | GPRGNN |
| PubMed | $0.05$ | True | $0.2$ | $0.0005$ | $0.5$ | GPRGNN |
| Computers | $0.01$ | False | $0.1$ | $0.0005$ | $0.5$ | GPRGNN |
| Photo | $0.01$ | False | $0.5$ | $0.0$ | $0.5$ | GPRGNN |
| Chameleon | $0.05$ | False | $1.0$ | $0.0$ | $0.7$ | DIGL |
| Squirrel | $0.05$ | True | $0.0$ | $0.0$ | $0.7$ | GPRGNN |
| Actor | $0.01$ | True | $0.9$ | $0.0$ | $0.5$ | GPRGNN |
| Texas | $0.05$ | True | $1.0$ | $0.0005$ | $0.5$ | GPRGNN |
| Cornell | $0.05$ | True | $0.9$ | $0.0005$ | $0.5$ | GPRGNN |
| Questions | $0.005$ | False | $1.0$ | $0.0$ | $0.2$ | GPRGNN |
| Penn94 | $0.01$ | False | $0.1$ | $0.0001$ | $0.2$ | GPRGNN |
| Twitch-gamers | $0.001$ | False | $0.5$ | $0.0001$ | $0.2$ | GPRGNN |

**Table 17: Hyperparameters for GCN on the cSBM in the sparse splitting. For all homophilic datasets the eigenvalues are ordered by value and for all heterophilic datasets they are ordered by absolute value. In all setting we keep the $64$ largest entries of the rewired adjacency matrix $\tilde{\bm{A}}$ per node. Interpolation ratios $\eta$ are rounded to three digits from the best values found by the random search.**
| $\boldsymbol{\phi}$ | JDR | DIGL |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| $K$ | $L_{A}$ | $L_{X}$ | $\eta_{A}$ | $\eta_{X_{1}}$ | $\eta_{X_{2}}$ | $\alpha$ |  |
| $-1.0$ | $28$ | $-$ | $10$ | $-$ | $0.482$ | $0.916$ | $1.0$ |
| $-0.875$ | $41$ | $5$ | $8$ | $0.101$ | $0.479$ | $0.858$ | $1.0$ |
| $-0.75$ | $40$ | $6$ | $9$ | $0.042$ | $0.498$ | $0.846$ | $1.0$ |
| $-0.625$ | $48$ | $6$ | $8$ | $0.036$ | $0.453$ | $0.862$ | $1.0$ |
| $-0.5$ | $50$ | $9$ | $10$ | $0.189$ | $0.412$ | $0.991$ | $1.0$ |
| $-0.375$ | $48$ | $8$ | $10$ | $0.879$ | $0.973$ | $0.773$ | $1.0$ |
| $-0.25$ | $80$ | $1$ | $1$ | $1.000$ | $-$ | $-$ | $1.0$ |
| $-0.125$ | $80$ | $1$ | $1$ | $1.000$ | $-$ | $-$ | $1.0$ |
| $0.0$ | $80$ | $1$ | $1$ | $1.000$ | $-$ | $-$ | $0.95$ |
| $0.125$ | $76$ | $1$ | $-$ | $0.650$ | $-$ | $-$ | $1.0$ |
| $0.25$ | $33$ | $1$ | $-$ | $0.951$ | $-$ | $-$ | $0.5$ |
| $0.375$ | $18$ | $10$ | $10$ | $0.856$ | $0.023$ | $0.228$ | $0.05$ |
| $0.5$ | $18$ | $10$ | $9$ | $0.415$ | $0.263$ | $0.880$ | $0.05$ |
| $0.625$ | $22$ | $8$ | $7$ | $0.264$ | $0.340$ | $0.807$ | $0.05$ |
| $0.75$ | $15$ | $7$ | $9$ | $0.056$ | $0.474$ | $0.778$ | $0.05$ |
| $0.875$ | $16$ | $10$ | $8$ | $0.035$ | $0.228$ | $0.981$ | $0.05$ |
| $1.0$ | $80$ | $-$ | $1$ | $-$ | $1.000$ | $1.000$ | $0.05$ |

**Table 18: Hyperparameters of JDR for all real-world datasets in the dense splitting. Following the findings from cSBM for all homophilic datasets the eigenvalues are ordered by value and for all heterophilic datasets they are ordered by absolute value. In all setting we keep the $64$ largest entries of the rewired adjacency matrix $\tilde{\bm{A}}$ per node. Interpolation ratios $\eta$ are rounded to three digits from the best values found by the random search.**
| Dataset | GNN | GPRGNN |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| $K$ | $L_{A}$ | $L_{X}$ | $\eta_{A}$ | $\eta_{X_{1}}$ | $\eta_{X_{2}}$ | K | $L_{A}$ | $L_{X}$ | $\eta_{A}$ | $\eta_{X_{1}}$ | $\eta_{X_{2}}$ |  |
| Cora | $10$ | $1853$ | $38$ | $0.066$ | $0.173$ | $0.071$ | $10$ | $772$ | $76$ | $0.027$ | $0.434$ | $0.005$ |
| Citeseer | $15$ | $578$ | $1330$ | $0.460$ | $0.173$ | $0.049$ | $4$ | $1390$ | $1169$ | $0.345$ | $0.099$ | $0.585$ |
| PubMed | $12$ | $8$ | $53$ | $0.316$ | $0.004$ | $0.187$ | $1$ | $1772$ | $919$ | $0.197$ | $0.893$ | $0.034$ |
| Computers | $3$ | $718$ | $975$ | $0.398$ | $0.021$ | $0.068$ | $7$ | $583$ | $1533$ | $0.468$ | $0.062$ | $0.127$ |
| Photo | $6$ | $467$ | $1867$ | $0.479$ | $0.071$ | $0.344$ | $4$ | $433$ | $1719$ | $0.413$ | $0.115$ | $0.231$ |
| Chameleon | $7$ | $41$ | $1099$ | $0.066$ | $0.375$ | $0.975$ | $3$ | $31$ | $1331$ | $0.063$ | $0.486$ | $0.755$ |
| Squirrel | $2$ | $4$ | $1941$ | $0.404$ | $0.011$ | $0.022$ | $2$ | $53$ | $1210$ | $0.234$ | $0.495$ | $0.964$ |
| Actor | $29$ | $896$ | $14$ | $0.298$ | $0.235$ | $0.219$ | $11$ | $1171$ | $791$ | $0.476$ | $0.028$ | $0.251$ |
| Texas | $20$ | $21$ | $183$ | $0.514$ | $0.028$ | $0.836$ | $1$ | $109$ | $36$ | $0.182$ | $0.004$ | $0.214$ |
| Cornell | $17$ | $10$ | $125$ | $0.794$ | $0.298$ | $0.113$ | $1$ | $39$ | $67$ | $0.482$ | $0.424$ | $0.068$ |
| Questions | $8$ | $248$ | $284$ | $0.218$ | $0.199$ | $0.841$ | $2$ | $89$ | $2$ | $0.974$ | $0.106$ | $0.311$ |
| Penn94 | $20$ | $60$ | $71$ | $0.445$ | $0.005$ | $0.902$ | $5$ | $172$ | $1851$ | $0.422$ | $0.094$ | $0.138$ |
| Twitch-gamers | $5$ | $7$ | $5$ | $0.235$ | $0.286$ | $0.806$ | $5$ | $2$ | $2$ | $0.165$ | $0.329$ | $0.003$ |

**Table 19: Hyperparameters of JDR for all the homophilic datasets in the sparse splitting. Following the findings from cSBM for all homophilic datasets the eigenvalues are ordered by value and for all heterophilic datasets they are ordered by absolute value. In all setting we keep the $64$ largest entries of the rewired adjacency matrix $\tilde{\bm{A}}$ per node. Interpolation ratios $\eta$ are rounded to three digits from the best values found by the random search.**
| Dataset | GNN | GPRGNN |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| $K$ | $L_{A}$ | $L_{X}$ | $\eta_{A}$ | $\eta_{X_{1}}$ | $\eta_{X_{2}}$ | K | $L_{A}$ | $L_{X}$ | $\eta_{A}$ | $\eta_{X_{1}}$ | $\eta_{X_{2}}$ |  |
| Cora | $10$ | $1853$ | $38$ | $0.066$ | $0.173$ | $0.071$ | $10$ | $772$ | $76$ | $0.027$ | $0.434$ | $0.005$ |
| Citeseer | $15$ | $578$ | $1330$ | $0.460$ | $0.173$ | $0.049$ | $4$ | $1390$ | $1169$ | $0.345$ | $0.099$ | $0.585$ |
| PubMed | $12$ | $8$ | $53$ | $0.316$ | $0.004$ | $0.187$ | $1$ | $1772$ | $919$ | $0.197$ | $0.893$ | $0.034$ |
| Computers | $3$ | $718$ | $975$ | $0.398$ | $0.021$ | $0.068$ | $7$ | $583$ | $1533$ | $0.468$ | $0.062$ | $0.127$ |
| Photo | $6$ | $467$ | $1867$ | $0.479$ | $0.071$ | $0.344$ | $4$ | $433$ | $1719$ | $0.413$ | $0.115$ | $0.231$ |

**Table 20: Hyperparameters for BORF for all real-world datasets in the dense splitting. OOM indicates an out-of-memory error.**
| Dataset | GNN | GPRGNN |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
| # iterations | # added | # removed | # iterations | # added | # removed |  |
| Cora | $2$ | $30$ | $10$ | $1$ | $10$ | $40$ |
| Citeseer | $3$ | $30$ | $40$ | $3$ | $10$ | $50$ |
| PubMed | $2$ | 0 0 | $30$ | $3$ | $20$ | $40$ |
| Computers | $1$ | $20$ | $40$ | $3$ | $20$ | $30$ |
| Photo | $2$ | $40$ | $20$ | $3$ | $50$ | $50$ |
| Chameleon | $2$ | $50$ | $30$ | $1$ | $10$ | $30$ |
| Squirrel | OOM | OOM |  |  |  |  |
| Actor | $2$ | $40$ | $50$ | $2$ | $10$ | $50$ |
| Texas | $1$ | $40$ | $10$ | $2$ | $40$ | $50$ |
| Cornell | 1 | $20$ | $50$ | $1$ | $20$ | $50$ |

**Table 21: Hyperparameters for BORF for the homophilic real-world datasets in the sparse splitting.**
| Dataset | GNN | GPRGNN |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
| # iterations | # added | # removed | # iterations | # added | # removed |  |
| Cora | $2$ | 10 | $40$ | $2$ | $30$ | $50$ |
| Citeseer | $3$ | $50$ | $40$ | $1$ | $20$ | $50$ |
| PubMed | $2$ | 0 | $30$ | $3$ | $20$ | $40$ |
| Computers | $1$ | $20$ | $40$ | $3$ | $20$ | $30$ |
| Photo | $3$ | 0 0 | $50$ | $3$ | $10$ | $20$ |

**Table 22: Values of the hyperparameter $\alpha$ of DIGL and the number of iterations (# iter) of FoSR for the real-world datasets in the dense splitting.**
| Dataset | $\alpha$ DIGL | $\alpha$ DIGL +JDR | #iter FoSR |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
| GCN | GPRGNN | GCN | GPRGNN | GCN | GPRGNN |  |
| Cora | $0.25$ | $0.60$ | $0.20$ | $0.60$ | $150$ | $5$ |
| Citeseer | $0.60$ | $0.50$ | $0.25$ | $0.25$ | $1000$ | $1000$ |
| PubMed | $0.60$ | $0.50$ | $0.60$ | $0.65$ | $10$ | $5$ |
| Computers | $0.05$ | $0.60$ | $0.10$ | $0.65$ | $500$ | $600$ |
| Photo | $0.30$ | $0.70$ | $0.20$ | $0.75$ | $25$ | $250$ |
| Chameleon | $0.15$ | $0.50$ | $0.55$ | $0.40$ | $10$ | $5$ |
| Squirrel | $0.05$ | $0.15$ | $0.10$ | $0.20$ | $5$ | $10$ |
| Actor | $1.00$ | $0.60$ | $0.20$ | $0.05$ | $10$ | $150$ |
| Texas | $1.00$ | $0.00$ | $0.20$ | $0.20$ | $50$ | $75$ |
| Cornell | $1.00$ | $1.00$ | $0.95$ | $0.00$ | $25$ | $5$ |
| Questions | $0.05$ | $1.0$ | $-$ | $-$ | $700$ | $700$ |
| Penn94 | $0.2$ | $0.1$ | $-$ | $-$ | $200$ | $150$ |
| Twitch-gamers | $0.15$ | $0.25$ | $-$ | $-$ | $5$ | $100$ |

**Table 23: Values of the hyperparameter $\alpha$ of DIGL and the number of iterations (# iter) of FoSR for the homophilic real-world datasets in the sparse splitting.**
| Dataset | DIGL | DIGL +JDR | #iter FoSR |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
| GCN | GPRGNN | GCN | GPRGNN | GCN | GPRGNN |  |
| Cora | $0.10$ | $0.30$ | $0.10$ | $0.30$ | $5$ | $75$ |
| Citeseer | $0.30$ | $0.45$ | $0.20$ | $0.45$ | $500$ | $600$ |
| PubMed | $0.35$ | $0.60$ | $0.40$ | $0.60$ | $50$ | $75$ |
| Computers | $0.05$ | $0.65$ | $0.15$ | $0.30$ | $250$ | $800$ |
| Photo | $0.20$ | $0.50$ | $0.10$ | $0.50$ | $5$ | $500$ |

### A.8 Hardware Specifications

Experiments on cSBM, Cora, Citeseer and Photo were conducted on an internal cluster with Nvidia Tesla V100 GPUs with $32$GB of VRAM. The experiments on the remaining datasets (PubMed, Computers, Chameleon, Squirrel, Actor, Cornell and Texas) were performed using Nvidia A100 GPUs with $40$GB or $80$GB of VRAM. The larger VRAM is only necessary for GNN +JDR on PubMed and the larger heterophilic datasets from Lim et al. (); Platonov et al. (), because they have larger numbers of nodes in the graph (and we choose the top-$64$ edges per node after rewiring).
Note that this could be reduced by sacrificing only a little bit of performance as shown in A.6.
One experiment of training and testing on $100$ random splits typically takes about $5\min$.
For the standard benchmark graphs, the longest experiments with GPRGNN +JDR and a different early stopping condition take about $40\min$.
The experiments on the large Twitch-gamers dataset take around $60\min$ (for $10$ splits), but similar to DIGL they require a lot of standard memory (around $500$GB) while performing the decompositions.

### A.9 Random Matrix Theory Insight

Following the derivation from Shi et al. (), we show empirically how the denoising can reduce the empirical risk for a one-layer GCN without non-linearity.
When the number of nodes $N$ goes to infinity and the average node degree satisfies some assumptions, we can apply the Gaussian adjacency equivalence conjecture.
This allows us to replace the binary adjacency in the cSBM with a spiked non-symmetric Gaussian random matrix without changing the training and test loss in the limit.
The equivalent adjacency reads

$$ $\bm{A}=\frac{\lambda}{N}\bm{y}\bm{y}^{T}+\boldsymbol{\Xi}_{gn}$ (9) $$

where with $\bm{\Xi}_{gn}$ has i.i.d. centered normal entries with variance $1/N$. Similarly, we build the features matrix as

$$ $\bm{X}=\frac{\mu}{N}\bm{y}\bm{u}^{T}+\boldsymbol{\Xi}_{x}.$ (10) $$

Compared to the standard cSBM formulation we rescale the variables $\sqrt{\mu\gamma}\to\mu$ and $\sqrt{F}\bm{u}\to\bm{u}$.
Additionally, we define $\alpha=1/\gamma=F/N$ and for simplicity, we consider the case $\bm{I}_{train}=\bm{I}$.
The mean squared error (MSE) loss reads

$$ $L(\boldsymbol{\omega})=\frac{1}{N}\|\bm{AX}\boldsymbol{\omega}-\bm{y}\|^{2}_{F }+\frac{r}{N}\|\boldsymbol{\omega}\|^{2},$ (11) $$

where $r$ is the parameter for the ridge part, $\boldsymbol{\omega}$ are the weights of the GCN and $\|\|_{F}$ indicates the Frobenius norm.
For $N\to\infty$, the MSE concentrates, which means it is only a function of $\mu$, $\lambda$ and $\alpha$.
For denoising $\bm{A}$ we do

$$ $\bm{A}_{\text{den}}=\bm{A}+\eta_{A}\bm{X}\bm{X}^{T}.$ (12) $$

The idea is that although this leads to more noise terms, the signal strength of $\bm{yy}^{T}$ is increased more.
Instead of a weighting of $\frac{\lambda}{N}\bm{yy}^{T}$, we now have $(\frac{\lambda}{N}+\eta_{A}\frac{\mu^{2}F}{N})\bm{yy}^{T}$.
The new MSE also concentrates on a value determined by $\eta_{A}$.
So, numerically, as shown in Figure 16, for any $\mu,|\lambda|>0$ we can always find values of $\eta_{A}$ such that the MSE is decreased.
For denoising $\bm{X}$ we do

$$ $\bm{X}_{\text{den}}=\bm{X}+\eta_{X}\bm{A}\bm{X}$ (13) $$

and show in Figure 17 with the same argumentation as for $\bm{A}$ that an $\eta_{X}$ exists so that the MSE is reduced.
Proof of both cases has yet to be provided and will be the subject of future work.

Figure: (a) $\lambda=0$
Refer to caption: x27.png

Figure: (a) $\mu=0$
Refer to caption: x29.png

## Peer Reviews

confidence: 3
contribution: 4
initial_rating: 8
presentation: 4
questions: could timing comparisons of the proposed methods, more explicitly and beyond just complexity comparison, be provided to observe computational benefits/limitations?
soundness: 4
strengths: The primary novelty of JDR lies in its combined optimization of graph structure and node feature alignment, enhancing data quality by maximizing alignment between the spectral components of the graph and feature matrices. This unified approach addresses both structural and feature-level noise simultaneously, which is rare among existing methods that typically target these types of noise separately. A key concept introduced is “spectral resonance,” where optimal alignment between the graph’s leading eigenvectors and the feature matrix’s singular vectors is achieved, providing a measurable target for denoising and boosting node classification performance. To manage the challenging non-convex optimization, the paper presents an iterative heuristic based on alternating optimization, which simplifies alignment maximization and enables efficient processing of large, real-world graph datasets with multiple classes. Another advantage of JDR is that it outputs a modified graph in the preprocessing stage, enhancing interpretability and reusability for subsequent GNN applications—a contrast to end-to-end methods that alter the graph only during training. Lastly, JDR’s design allows it to adapt effectively to both homophilic and heterophilic graphs, expanding its applicability beyond previous methods, which often work best with specific types of graph structures, such as those with high homophily.

The paper is well-organized, systematically guiding the reader through the challenges, methodology, and outcomes of the proposed Joint Denoising and Rewiring (JDR) algorithm. It begins with an introduction that effectively frames the problem of noisy graph structures and features, establishing the need for an approach that addresses both in unison. The methodology section details the concept of spectral resonance and the iterative optimization heuristic that drives JDR, using clear mathematical definitions and visual aids to support understanding. Following this, a comprehensive experimental section validates the algorithm's effectiveness across synthetic and real-world datasets, offering detailed comparisons with state-of-the-art methods and highlighting the algorithm’s robustness across homophilic and heterophilic graph types. Finally, the paper provides an insightful discussion on related works, situating JDR within the broader landscape of graph preprocessing techniques, before concluding with a summary of contributions and potential directions for future research. Overall, the structure flows logically, with each section building on the previous one to reinforce the practical relevance and theoretical underpinnings of JDR.

Addressing the limitations of the proposed approach is appreciated

The illustrations provided in the work, both in the body and abstract, are informative and well done as well as qualitatively informative.

 The use of the appendix is also well done in providing explicit discussion of the proposed algorithm 

The paper has a robust experimental section with compelling results working in favor of the approaches proposed
summary: The paper presents Joint Denoising and Rewiring (JDR), a novel algorithm designed to simultaneously address noisy graph structures and node features, thereby enhancing node classification performance in graph neural networks (GNNs). The work proposed aims to address the issue of graph datasets that suffer from noise due to missing or spurious edges and misaligned node features. JDR counters this by iteratively refining both the graph structure and features to maximize the alignment between their leading eigenspaces, achieving what the authors call “spectral resonance,” which improves data representation for downstream GNN performance. The algorithm operates in three iterative steps: first, it decomposes the graph’s adjacency matrix and node feature matrix into eigen and singular vectors; then, it aligns the leading eigenvectors of the graph with the primary singular vectors of the feature matrix, creating a more cohesive representation; and finally, it synthesizes the updated graph for subsequent GNN tasks. The authors argue this iterative approach effectively enhances graph quality and feature clarity, especially in datasets exhibiting mixed homophily and heterophily, which are often challenging for GNNs. The proposed work provides extensive experiments show that JDR consistently outperforms existing rewiring methods on synthetic and real-world datasets, yielding better alignment and improved node classification accuracy. Additionally, JDR’s preprocessing method produces an enhanced graph that supports interpretability and reusability, unlike other methods that modify graphs solely during training. The paper highlights JDR’s scalability and versatility across different graph types, underscoring its potential in handling noisy real-world data and establishing it as a valuable advancement in preprocessing techniques for GNNs.
weaknesses: JDR depends on the availability of informative node features for effective rewiring and denoising, which restricts its applicability to settings with substantial node feature information; this reliance could limit its effectiveness in networks that primarily encode structural data. The algorithm’s design is also tailored to node-level tasks, making it less suited for graph-level tasks like graph classification, where global structure matters more than node-specific features

---

confidence: 2
contribution: 3
initial_rating: 5
presentation: 3
questions: My main concern with this work is that although the proposed method looks to be sound, there is really no theorectical ground on its effectiveness in general when applied to a wider range of datasets. Also, the datasets used are very small and I'm not sure how useful and how scalable the method is in practice. It would significantly strengthen the work if the authors can either theoretically show how much improvement can be obtained using their method or empirically demonstration the effectiveness on a wide range of datasets, not just on common datasets that we have already seen too many papers claiming their effectiveness over others. I believe the contributions of such work are rather limited.
soundness: 3
strengths: 1. The proposed method is general and can be applied to a wide range of GNNs for downstream classification tasks. 

2. The proposed method jointly considers graph rewiring and feature denoising.

3. The proposed method improves the performance of a numer of GNNs on a number of popular datasets in the experiments.
summary: The paper proposes to  jointly consider graph rewiring and feature denoising to improve the performance of GNNs for downstream classification tasks.  The proposed method improves the performance of a numer of GNNs on a number of popular datasets in the experiments.
weaknesses: 1. The solution is somewhat incremental and its novelty is low, although it appears to be sound.

2. There is no theoretical guarantee on the degree of improvement using JDR.

3. The experiments were conducted on a small number of datasets that cannot be considered as an evidence that the proposed method is really effective.

---

confidence: 3
contribution: 2
initial_rating: 6
presentation: 3
questions: - As the computation overhead of the proposed method seems large, can it be applied to large graphs, e.g. ogbn-products?
soundness: 3
strengths: - This paper uses cSBMs as a key framework to build intuition about the graph rewiring and denoising problem, providing the theoretical foundation for the alignment target.
- The empirical verification using synthetic data is clear.
- The method is evaluated on both homophilic graphs and heterophilic graphs, showing its generalizability.
summary: This paper considers the graph rewiring problem and feature denoising problem to improve the graph node classification task.
This paper utilizes cSBM modeling to optimize both graph structures and graph features to make them have better alignment, then proposes the JDR method to effectively optimize the alignment function.
weaknesses: - The proposed method involves graph structure matrix and graph feature matrix decomposition, which can be computationally challenging on extremely large real-world graph data, limiting the practicability of the proposed method.
- As the SVD decomposition can have time complexity of $O(N^3)$, it may be not accurate to say the proposed JDR has time complexity of $O(N)$

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. In Figure 2, why do the two subgraphs compare w.r.t. amplitude and normalized amplitude, respectively?
2. Regarding section 2.2, can you explain more about how the structure is revealed parametrically? In line 137, $|\lambda|$ is considered the signal-to-noise ratio, when it can also be considered the degree of homophility?
3. The remark below definition 1, ''the information about the labels is indeed contained in the leading vectors of V and U'', is unclear.Because usually the decomposition of a matrix gives a support space, the weights of each support, and the coefficient of reconstructing the matrix using the support, which leads to the insufficiency of the leading support vectors to be representative.
This should be relevant for W2.
soundness: 3
strengths: 1.The overall structure is well organized and easy to understand, especially the diagram demonstrations are very helpful.
2.The problem formulation is clearly presented.
3.Experiments are comprehensive and convincing.
summary: This work presents a structural rewiring algorithm, with consideration of structural alignment with node features, supported by theoretical investigation on cSBM model.
weaknesses: 1. The theoretical framework seems convincing to me, while how the real data sets fit the parametric model needs further investigation, otherwise the heuristic of denoising features might not be applicable.
2. More explanation is needed on the insights of where the rationale of denoising comes from, e.g. lines 159-190.
3. Since the eigendecomposition is applied on adjacency, the comparison of training and inference time costs between the vanilla GNN and with the add-on of the proposal is needed.

---

confidence: 3
contribution: 2
initial_rating: 5
presentation: 2
questions: See above.
soundness: 3
strengths: 1. The method is well inspired and can achieve better performance than several other graph rewiring methods.
2. An impressive amount of experiments were implemented.
summary: The work proposes a method (JDR) to jointly perform graph rewiring and feature denoising. The method is derived from cSBM to maximize alignment between the eigenspaces of features and the graph. The authors tested the performance of the methods in various datasets and showed the advantages of the proposed JDR method.
weaknesses: 1. The paper should better discuss previous works on GNN versus MLP, and the connection between heterophily and graph noise. [1-4] investigated the phenomena that MLP sometimes perform better than GNN (especially GCN) for heterophilic graphs. In particular, [4] proposed a metric that well correlates with empirical GNN performance, and also discussed the connection between heterophily and graph noise. A better discussion (acknowledgement?) of [4] is needed due to its high relevance to this paper.

2. The comparison between JDR and graph rewiring methods seems not perfectly fair as the authors also mention themselves. Moreover, if JDR indeed achieves optimal denoising, then the graph may no longer be needed in later training. This (JDR(X) + MLP) seems uncovered by the ablation settings. 

3. The authors should check if the rewired graph structures degenerate, which may be a natural consequence of combining matrix factorization scheme and thresholding. 

4. The hyperparameter tuning was not performed for the GNN backbone. This is questionable as the optimal backbone hyperparameter is anticipated to vary across different rewired graphs (and potentially different denoised features). This may matter because the performance gap between JDR and other methods seems small in most cases.

5. Some paragraphs are unclear and not readable. In particular, it is unclear what “findings” in the sentence “The Gaussian adjacency equivalence conjecture (Shi et al., 2024) suggests a generalization of the findings to the true binary adjacency case.” is referring to. The whole section (as well as the proof) needs to be polished.

[1] Ma, Yao, et al. "Is Homophily a Necessity for Graph Neural Networks?." International Conference on Learning Representations.

[2] Gomes, Diana, et al. "When Are Graph Neural Networks Better Than Structure-Agnostic Methods?." I Can't Believe It's Not Better Workshop: Understanding Deep Learning Through Empirical Falsification. 2022.

[3] Luan, Sitao, et al. "Revisiting heterophily for graph neural networks." Advances in neural information processing systems35 (2022): 1362-1375.

[4] Dong, Mingze, and Yuval Kluger. "Towards understanding and reducing graph structural noise for GNNs." International Conference on Machine Learning. PMLR, 2023.

Meta-review: This paper is a clear accept -- the reviewers found the presented method to be novel, interesting, and well-presented. The idea of aligning the singular vector subspaces of node features and the graph adjacency matrix as a way of jointly denoising feature data and rewiring the graph is intuitive and seemingly effective.

The theoretical results, while holding in a highly simplified setting (cSBMs) help build intuition for/justification of the method. The empirical results are strong and comprehensive. During the rebuttal the authors effectively addressed several initial concerns of the reviewers, and the reviewers raised their scores accordingly.

Decision: Accept (Oral)