Are we measuring oversmoothing in graph neural networks correctly?
Authors: Kaicheng Zhang
Piero Deidda
Desmond Higham
Francesco Tudisco
Venue: ICLR 2026 Poster (2026)
Keywords: graph neural networks
oversmoothing
low-rank
Primary area: learning on graphs and other geometries & topologies
TL;DR: We propose the effective rank as a more robust metric for quantifying oversmoothing in graph neural networks.

Abstract: Oversmoothing is a fundamental challenge in graph neural networks (GNNs): as the number of layers increases, node embeddings become increasingly similar, and model performance drops sharply. Traditionally, oversmoothing has been quantified using metrics that measure the similarity of neighbouring node features, such as the Dirichlet energy. We argue that these metrics have critical limitations and fail to reliably capture oversmoothing in realistic scenarios. For instance, they provide meaningful insights only for very deep networks, while typical GNNs show a performance drop already with as few as 10 layers.  As an alternative, we propose measuring oversmoothing by examining the numerical or effective rank of the feature representations. We provide extensive numerical evaluation across diverse graph architectures and datasets to show that rank-based metrics consistently capture oversmoothing, whereas energy-based metrics often fail. Notably, we reveal that drops in the rank align closely with performance degradation, even in scenarios where energy metrics remain unchanged. Along with the experimental evaluation, we provide theoretical support for this approach, clarifying why Dirichlet-like measures may fail to capture performance drop and proving that the numerical rank of feature representations collapses to one for a broad family of GNN architectures.

Published as a conference paper at ICLR 2026
ARE WE MEASURING OVERSMOOTHING IN GRAPH NEURAL NETWORKS CORRECTLY?
Kaicheng Zhang*
Piero Deidda
School Mathematics and Maxwell Institute University of Edinburgh
Gran Sasso Science Institute
Scuola Normale Superiore piero. det dda sns it
K. -.zzaag--af 600 sms ed. ac.uk
Desmond Higham
Francesco Tudisco
School of Mathematics and Maxwell Institute University of Edinburgh
School of Mathematics and Maxwell Institute University of Edinburgh
d.j. J.j.hi hi gham@ed. ac uk
Gran Sasso Science Institute
Miniml.AI f. scuuddc ac.
ABSTRACT
Oversmoothing is a fundamental challenge in graph neural networks (GNNs): as the number of layers increases, node embeddings become increasingly similar, and model performance drops sharply. Traditionally, oversmoothing has been quantified using metries that measure the similarity of neighbouring node features, such as the Dirichlet energy. We argue that these metrics have critical limitations and fail reliably capture oversmoothing realistic scenarios. For instance, they provide meaningful insights only for very deep networks, while typical GNNs show a performance drop already with as few as 10 layers. As an alternative, we propose measuring oversmoothing by examining the numerical or effective rank of the feature representations. We provide extensive numerical evaluation across diverse graph architectures and datasets to show that rank- metrics consistently capture oversmoothing. whereas energy-based metries often fail. Notably, we reveal that drops in the rank align closely with performance degradation, even in scenarios where energy metrics remain unchanged. Along with the experimental evaluation, we provide theoretical support for this approach, clarifying why Dirichlet-like measures may fail to capture performance drop and proving that the numerical rank of feature representations collapses one for a broad family of GNN architectures.
INTRODUCTION
Graph neural networks (GNNs) have emerged as powerful framework for learning representations from graph-structured data, with applications spanning knowledge retrieval and reasoning (Tian)
et al.
2022 Peng et al. 2023), personalised recommendation systems (Peng et al.] 2022 Damianou
al. 2024). social network analysis (Fan al. 2019). and 3D mesh classification (Shi Rajkumar
2020). Central to most GNN architectures is the message-passing paradigm, where node features are iteratively aggregated from their neighbours and transformed using learned functions, such as multi-layer perceptrons or catp-attention mechanisms.
However, the performance message- saases--assnnn-bbaee based GNNs is known to deteriorate after only few layers, essentially placing limit on their depth. This issue, often linked to the increasingly similar learned features as GNNs deepen, is known as oversmoothing (Li al. 2018 Nt Maehara 2019
Wu et al. 2022 Rusch et al.
2023a Zhao et al. 2024 Amdiz-RRdriguez & Errica (025).
In recent years, oversmoothing in GNNs, as well as methods to alleviate it, have been studied based on the decay of some node feature similarity metrics, such as the Dirichlet energy and its variants
OOno & Suzuki 2019
Cai
Wang 2020 Bodnar et al. 2022 Nguyen et al. 2022 Di Giovanni
Eqqual Contribution

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2023). At high level, most of these metrics directly et al., 2023 Wu et al. 2023 Roth & Liebig
measure the norm of the absolute deviation from the dominant eigenspace of the message- message-passing matrix. In linear GNNs without bias terms, this eigenspace is often known and easily computable via e.g. the power method. However, when nonlinear activation functions or biases are used, the dominant eigenspace may change, causing these oversmoothing metrics to fail and give false negative signals about the oversmoothing state of the learned features.
While these metrics are often considered as sufficient but not necessary evidence for oversmoothing
(Rusch et al. 2023a), there is considerable body of literature using these unreliable metrics as
their evidence for non-occurrence of oversmoothing in GNNs (Zhou et al. 2021: Chen al. 2022
Rusch et al. [022]] Wang et al. [022] Maskey et al. | et al. 2023 Rusch et al. 20236
Epping et al. 2024] Scholkemper et al. 2024 Wang & Cho. 2024 Roth 2024 Wang et al. 2025
Arnaiz-Rorriguez & Errical 2025).
However, as we show in Section the performance degradation of GNNs trained on real datasets often happens well before any noticeable decay in these oversmoothing metrics can be observed. The vast majority of empirical studies in the literature that observe the decay of Dirichlet-like energy metrics are conducted over the layers of very deep but untrained (with randomly sampled weights) or effectively untrained" GNNs (Wang et al., 2022 Rusch et al., 2022 2023b Wu et al., 2023 Roth 2024 Wang al.) 2025). where the decay of the metrics is only driven by the small weight initialization. Instead, we show that when GNNs of different depths are trained with proper weight initialization, these metries do not correlate with the model's performance degradation.
Furthermore, we argue that these metrics can only indicate oversmoothing when their values converge exactly to zero, corresponding to either an exact alignment to the dominant eigenspace to the feature representation matrix collapsing to the all-zero matrix. This double implication presents an issue: in realistic settings with a large but not excessively large number of layers, we may observe the decay of the oversmoothing metric by, say, two orders of magnitude while still being far from zero. In such cases, unelear whether the features are aligning with the dominant eigenspace, simply decreasing in magnitude, or exhibiting neither of the two behaviours. As a result, these types of metrics provide little to no explanation for the degradation of GNN performance.
As an alternative to address these shortcomings, we advocate for the use of a continuous approximation of the rank of the network's feature representations to measure oversmoothing. Collapse of the rank of the feature representations was already considered as cause of oversmoothing. e.g. in (Guo) jct al. 2023b, Roth & Licbig 2023), ). however rank rank-based measures were never explicitly studied, Lc. they were never compared with other oversmoothing measures and their capacity in mcasuring oversmoothing was never quantified, theoretically and/or numerically. Our work fills this gap. Indeed, our experimental evaluation across various GNN architectures trained for node classification demonstrates that continuous rank relaxations, such as the numerical rank and the effective rank, correlate strongly with performance degradation in independently trained GNNs-even in settings where popular energy-like metrics show little to no correlation.
Overall, the main contributions of this paper are follows:
We review popular oversmoothing metrics in the current literature and simplify their theoretical analysis from a novel perspective of nonlinear activation eigenvectors.
We notice that the rank can be a better metric for quantifying oversmoothing, and thereby we redefine oversmoothing in GNNs as the convergence towards a low-rank matrix rather than to matrix of exactly rank one.
We provide extensive numerical evidence that continuous rank relaxation functions provide much more compelling measure of oversmoothing than commonly used Dirichlet-like metrics
Additionally, investigate theoretically the causes of decay of the numerical rank. In particular, show that both the aggregation matrices and the nonlinear activation functions can contribute to the decay. Our theoretical study restricted to linear GNNs and nonlinear and non-negative GNNs where the eigenvector of the message- -aasging matrix is also the eigenvector of the nonlinear activation function. For these models, we prove that the numerical rank of the features converges exactly one. Such results provide theoretical support to our empirical evidence that oversmoothing may occur
DDep networks (with, say, over 100 layers) that are trained but whose loss and accuracy remain far from acceptable.
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independently of the weights' magnitude and align with our perspective oversmoothing from the point of view of nonlinear activation functions.
BACKGROUND
2.1 GRAPH CONVOLUTIONNL, NETWORK
Let = (V,E) be undirected graph with V denoting its set of vertices and & S V x V its set of edges. Let IR. N be the unweighted adjacency matrix with N = | being the total number of nodes, |8| being the total number of edges of & and A the corresponding symmetric adjacency matrix normalized by the node degrees: A カ-1/2 1/2 Á AD-1/2 where 1) = D + T, D is the diagonal degree matrix of the graph 9, and is the identity matrix. The rows of the feature matrix X E Nd are the concatenation of the d-dimensional feature vectors of all nodes in the graph. At each layer 4, the node feature update of Graph Convolutional Network (GCN) (Kipt & Welling 2016) follows ((++1) = a (AX(W:)) w()) where a is a nonlinear activation function, applied component-wise, and W is trainable weight matrix.
2.2 GRAPH ATTENTION NETWORK
Graph Attention Networks (GATs) (Veličković et al.) 2017 Brody et al. (021) perform graph convolution via layer-dependent message-passing matrix A learned through attention mechanism A
(I)T (ry()X,,, + P2 (V(Txx))) where ( are learnable parameter vectors, XX., denote the feature of the i-th and jth nodes respectively, the ac- tivation Oa is typically chosen to be LeakyReLU, and softmax, corresponds to the row-wise normalization softmax, (Aiz) exp(Aij)) En exp(A1g). The corresponding feature update is ( = o x(1)).
3
OVERSMOOTHING
Oversmoothing can be broadly understood as an increase in similarity between node features as inputs propagated through increasing number of message- message-passing layers, leading to noticeable decline in GNN performance. However, the precise definition of this phenomenon varies across different sources. Some works define oversmoothing more rigorously the alignment of all feature vectors with each other. This definition is motivated by the behaviour of a linear GCN: (1+1)
A... AX (0) wo W1) Indeed, if Á is the adjacency matrix of fully connected graph, will have spectral radius equal to 1 with multiplicity 1, and A' will converge toward the eigenspace spanned by the dominant eigenvector. Precisely, A' >> uu as I + xo, where Au and A v, see e.g. (Tudisco et al.) 2015).
As consequence, if the product of the weight matrices W ((I) does not diverge in the limit I ++ oc, then the features degenerate matrix having rank aligned with the dominant cigenvector Mathematically, if we assume to be such that ||u]| = 1, this alignment can be expressed by stating that the difference between the features and their projection onto u. given by [[ uu" (0) |]. converges zero
one, where all the features are
3.1 EXISTING OVERSMOOTIING METRICS
Motivated by the discussion about the linear case, oversmoothing is thus typically quantified and analysed in terms of the convergence of some node similarity metrics towards zero. In particular, in most cases, it is measured exactly by the alignment of the features with the dominant eigenvector of matrix A. The most prominent metric that has been used to quantify oversmoothing is the Dirichlet energy, which measures the norm of the difference between the degree normalzze neighbouring node features (Cai & Wang 2020: Rusch et al. 2023a)
X
X
EDi
Σ
(1)
U
(K,))
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where u, is the i-th entry of the dominant cigenvector of the message message-passing matrix. It thus immediately follows from our discussion on the linear setting that EDIr (X(1) converges to zero as 1 + oo for a linear GCN with converging weights product W (10))..Wtl This intuition suggests that a similar behaviour may occur for "smooth-enough" nonlincarities.
In particular, in the case of a GCN, the dominant eigenvector is defined by u, vI and Cai & Wang 2020) have proved that, using LeakyReLU activation functions, it holds EDI: (X((t)) ||W(() [|2 is the largest singular value of the weight matrix W and = min, );)², where X, (0. varies among the nonzero eigenvalues of the normalized graph AAD
sy Dir X X where
Laplacian A = A =
Similarly, in the case of GATs, the matrices A, are all row stochastic, meaning that u, 1 for all i. In this case, it has been proved that whenever the product of the entry-wise absolute value of the weights bounded, that is |T | [W]]]]] < oc, then the following variant of the Dirichlet energy
decays to zero (Wu et al. 2023
Erooj (X) = || - PP
(2)
where P = uu. is the projection matrix on the space spanned by the dominant eigenvector of the matrices A0)
Note that both these metrics, Eproj and Eir. can be used only if the dominant eigenvector of A is the same for all this is, for example, the case with row stochastic matrices or when A for all 1. Moreover, both these metrics essentially measure the deviation of the feature representations from the dominant eigenspace of the aggregation matrices A So we expect them to perform very similarly in capturing oversmoothing.
particular, is not difficult show that they are equivalent metrics from a mathematical point of view, i.e. there exist constants C1, C2 > 0 such that C1 E S (X) S FX)r(X), see Lemma A.
3.2 UNIFYING PERSPECTIVE BASED ON THE EIGENVECTORS OF NONLINEAR ACTIVATIONS
We present here unifying and more general perspective of the necessary conditions to have over- smoothing in the sense that Eproj and Epi decay to zero, based on the concept of eigenvectors for nonlinear activation function. In the interest of space, longer proofs for this and the subsequent sections are moved to Appendix A
We say that a vector u E RN {0} is an eigenvector of the (nonlinear) activation function Ø: R + F" if for any E E \ (0), there exists μa E R such that o (tu) pyu.
Definition 3.1.
With this definition, we can now provide a unifying characterization of pressage-passingg operators A and activation functions a that guarantee the convergence of the Dirichlet-like energy metrics Eproj and FDDI to zero for the feature representation sequence defined by (1+11) o (A4()) xA4))xx))))))))))))) x()())). Specifically, Theorem 3.2 shows that this holds provided all matrices A share a common dominant eigenvector u, which is also an eigenvector of
Theorem 3.2. Let (1+1) = (A (0) (O(())))
I,...,,, I L, be a GNN such that u is the dominant
eigenvector of A for any and also an eigenvector of the activation a. If a is 1 1-Lipschitz namely
[[a(x) (1)1l S || - yll for any x, y, and lim I II(T PP) 22 W()) 2 = 0, then
Erooj (x(L) +
L co.
The eigenvector assumption shared by ( and recurs throughout our theoretical analysis. and aligns with existing results in the literature. For example, in the case of GCNs, the matrix n = A is symmetric, and thus | PA Az Therefore, when LeakyReLU, we obtain the result by (Cai & Wang 2020) as convergence to zero is guaranteed if ||<(0] [] < Xg. Note in fact that the choice a = LeakyReLU satisfies our eigenvector assumption since u 2 0 by the
atu with depending only the sign of t. Similarly, in the case of GATs, the matrices A are stochastic for all I, implying that with (u); = 1 for all i. If o = @Φ is a nonlinear activation function acting entry-wise through v, then a (11) (.) Therefore, Theorem 3.2 implies that if the weights are sufficiently small, the
theorem, and thus (eakyReLU(tu))
1 is the constant vector
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features align independently of the activation function used. This is consistent with the results in (Wu (et al.] (023). However, we note that the bounds on the weights required by Theorem 3.2 and those in (Wu et al. 2023) on the weights W are not identical, and it is unclear which of the two is more significant. Nonetheless, in both cases, having bounded weights along with any 1-Lipschitz pointwise activation functiin is sufficient condition for Σроoj to converge to zero as the depth grows in a GAT. In addition to offering different and unifying theoretical perspective the results in (Cai & Wang/ 2020] Wu et al.) (023), we highlight the simplicity of our eigenvector lagenvector-based proof, which provides added clarity the theoretical understanding of this phenomenon.
ENERGY-LIKE METRICS: WHAT CAN Go WRONG
Energy-like metries such as EDir and Eproj are among the most commonly used oversmoothing metrics. However, they suffer from inherent limitations that hinder their practical usability and informational content.
One important limitation of these metrics is that they indicate oversmoothing only in the limit of infinitely many layers, when their values converge exactly zero. Since they measure a form of absolute distance, small but nonzero value does not provide any meaningful information. On the other hand, convergence to zero corresponds to either perfect alignment with the dominant eigenspace or the collapse of the feature representation matrix to the all-zero matrix. While the former is symptom of oversmoothing, the latter does not necessarily imply oversmoothing. Moreover, this convergence property requires the weights to be strongly bounded. However, in most practical cases, performance degradation is observed even relatively shallow networks, far from being infinitely deep, and with weight magnitudes arbitrarily larger than what is prescribed by (Cai & Wang 2020) Wu et al., 2023) or Theorem 3.2 This aligns with our intuition and what occurs in the linear case. Indeed, for a linear GCN, even when the features x grow to infinity as - oc, one observes that X is dominated by the dominant eigenspace of A, even for finite and possibly small values of 1, depending on the spectral gap of the graph. More precisely, the following theorem holds:
Theorem 4.1. Let X(++)] (I()(() be linear GCN. Let >1; be the largest and second- largest eigenvalues (in modulus) of A, respectively. Assume the weights {W(D] are ran- domly sampled from i..dd. random variables with distribution v such that f log ([[W]])dv + I logt ([|W-1|])d < co, with log (t) = max max(())g()) 0}. 1/22 < 1, then almost surely
[[ [(I-P)x"11e =0
lim
1000
||
with linear rate of convergence Da/A1ll
In particular, the theorem above implies that x >1 (uvT + R(I) for some with R(()) O(12//A1) 1) and thus, when the spectral gap is large /2//11 < 1, (1) is predominantly of rank one, even for moderate values of I. This results in weakly expressive feature representations, independently of the magniude of the feature weights. This phenomenon can be effectively captured by measuring the rank of X (9), whereas Dirichlet-like energy measures may fail detect it, as it would, for example, be the case when A 1, having (!) +
Another important limitation of Dirichlet-like metrics is their dependence on a specific dominant eigenspace, which must either be explicitly known or computed in advance. Consequently, their applicability is strongly tied to the specific architecture of the network. In particular, the dominant cigenvector u of ( must be known and remain the same for all l. This requirement excludes their use in cases where A0) varies with 1.
THE RANK AS A MEASURE OF OVERSMOOTHING
Inspired by the behaviour observed in the linear case, we argue that measuring the rank of feature representations provides more effective way to quantify oversmoothing, in alignment with recent work on oversmoothing (Guo et al., 2023b; Roth Liebig 2023). However, since the rank of a matrix is defined as the number of nonzero singular values, is discrete function and thus not
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Figure I: Toy scenarios depicting the behaviour of over- smoothing metrics. Each plot contains 50 nodes, each with two features plotted on the x-y axis. The features are: # of the same value; # perfectly aligned with the same vector; #3 aligned the same vector except for one (red) point;
#1
#3
#
EDir
0
13.25
77.78
#4 sampled from uniform distribution. MAD (Sec. (6) and
Eproj
0.83
0.97
EDir give false negative signals #3 although features are oversmoothing by definition. EPooj can hardly differentiate between #3 and # 4, and is thus not robust in quantifying oversmoothing. EP and EDir where computed using the first feature place of
MAD
0
0.81
0.81
0.57
NumRank Erank
1
1.01
1.78
1
1.36
1.99
and
suitable measure. A viable alternative is to use continuous relaxation that closely approximates the rank itself.
Examples of possible continuous approximations of the rank include the numerical rank, the stable rank, and the effective rank (Roy & Vetterli 2007 Rudelson & Vershynin 2006 Arora et al. 2019 defined as follows
|X| |
NumRank(X) = | |
StabRank =
Erank(X) = exp (-Ep log
where |X|| =E;;; is the nuclear norm, and given the singular values > 02 > minn N.d3 of X. Pk = 0t/E, og defined as the k-th normalized singular value. These rank relaxation measures exhibit similar empirical behaviour shown in Section 6
In practice, measuring oversmoothing in terms of continuous approximation of the rank helps to address the limitations of Dirichlet-like like measures. Specifically, it offers the following advantages: (a) it is scale-invariant, meaning it remains informative even when the feature matrix converges to zero explodes infinity: (b) it does not rely fixed, predetermined eigenspace but instead captures convergence of the feature matrix toward an arbitrary swerrdiimeniionll subspace; (c) it allows for the detection of oversmoothing in shallow networks without requiring exact convergence rank one. A small value of the effective rank directly implies that the feature representations are low-rank, suggesting a potentially suboptimal network architecture.
In Figure | we present toy example illustrating that classical oversmoothing metrics fail to correctly capture oversmoothing unless the features are perfectly aligned. This observation implies that these metrics can quantify oversmoothing only when the rank of the feature matrix converges exactly to one. In contrast, continuous rank functions provide a more reliable measure of approximate feature alignment. Later, in Figure [] demonstrate that the same phenomenon occurs in GNNs trained on real datasets, where exact feature alignment is rare. In such cases, classical metrics remain roughly constant, whereas the rank decreases, coinciding with sharp drop in GNN accuracy.
5.1 THEORETICAL ANALYSIS OF RANK DECAY
In this section, we provide an analytical study proving the decrease the numerical rank for a broad class of graph neural network architectures under the assumption of linear models nonlinear models with weight matrices that are entry-wise nonnegative. Our results rigorously show that in these settings, oversmoothing can occur independently of the weight (and thus feature) magnitude and shed light on the possible causes of rank decay.
We begin with several useful observations. Let be the dominant eigenvector of A corresponding Y and satisfying ||u|| = 1. Consider the projection P = uu." Given matrix X, we can decompose X PX + (I P)X Since is a unit vector, follows that ||l|| 1, and therefore,
|| l|2 ||| X 2: |||
(3)
Moreover, since PX and (I X are orthogonal with respect to the Frobenius inner product, ( P) || Thus, we obtain the following bound:
have PX + (I P) X12 = || F.- |I(7-
NumRank |PX [I- P) || + [I - PP < 11
[[ P)
(4)
|X1
XI3
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The above inequality, together with Theorem 4.1 allows us to establish the convergence of the numerical rank for linear networks.
The Linear Case Consider linear GCN of the form X = Ax(W(( where A has simple dominant eigenvalue A satistying [^1] 2 \2/. We have already noted that || 2 || 1l2. meaning that the numerical rank converges to one if (I PI f/||X|12 decays to zero. This occurs whenever the features grow faster in the direction the dominant eigenvector than in any other direction. As established in Theorem 4.1] this is almost surely the case in linear GNNs. As direct consequence, we obtain the following result:
Theorem 5.1.
Let X(++1) Wx(W( be a linear GCN. Under the same assumptions as in the following identity holds almost surely:
Theorem 4.7
lim NumRank(XX (0) =1
71000
Extending the result above to general GNNs with nonlinear activation functions is highly nontrivial. However, a simplified setting to study is the one where the weights are nonnegative. Indeed, exactly as in the linear case, while generally the rank decreases only on average, considering nonnegative weights yields a monotone decrease.
The Nonnegative Nonlinear Case To study the case of networks with nonlinear activations, theory; we refer to (Lemmens & Nussbaum
make use of tools from the nonlinear 2012 Gautier et al. 2023) and the reference therein for further details.
We assume all the intermediate features of the network to be in the positive open cone K= := R
{ir E R x; > 0 Vi .....,N'' Here, we consider the partial ordering x Sx (x <x. y) if and only if y
(y x 0), where the inequalities have to be understood entrywise. Given two points E K. let
IG max
y/
dx(2, = loog
max
(5)
denote the Hilbert distance. Note that dæ is not a distance on K., indeed (az, By) = dH( (z, y) for any x, y E I. and 8 > O. However, it is distance up to scaling; that is, it becomes a distance whenever we restrict ourselves to a slice of the cone. Because of this property, dy is particularly useful for studying the behavior of the rank of the features, which is icale-invariant function. Indeed, the next result shows that, under mild assumptions, nonnegative weights generate layers that are nonexpansive in Hilbert distance.
Lemma 5.2. Let be nonnegative and irreducible matrix with dominant eigenvector u E K. Assume X be strictly positive, W nonnegative with ming max; Wi > 0, and a a continuous (nonlinear) function that is order preserving, subhomogeneous, and such that u is also an eigenvector of a. Then
maxdH (.A u) S 3 max (X.,,, u),
(6)
with 0 S 3 < 1. Where Y., denotes the i-th column of Y. In particular, if A is contractive in the Hilbert distance or strictly sub-homogeneous, then 3
In the above statement, order- celler-reseriig means that given any Z. y K. with x 2K y. it holds (x) 2x (yy, while (strictly) subhomogenenous means that a Ar <x) SK (o for all x E I. and any A > 1. We recall that, as discussed in (Sittoni & Tudisco 2024 Piotrowski et al., 2024). broad range of activation functions commonly used in deep learning subhomogeneous and order-preserving on K. In particular, whenever an activation function is monotone increasing on R4,, is trivially also preservig.. Additionally, as we prove in Proposition A.6 if the activation function is homogeneous, e.g. LeakyReLU, then any nonnegative vector is an eigenvector of a. By contrast, if is strictly subhomogeneous, e.g. tanh, then the only strictly positive eigenvector the entrywise constant one.
Next, we prove that for neural networks with nonnegative feature representations, the numerical rank goes to 1 as the depth grows to infinity.
Theorem 5.3. Consider GNN of the form ((+1) = = (()) with x!!) E K. for any i = -,....d d. If there exists u E KC such that lim; max; d (x(1) u) = 0, then
lim NumRank(X(()) 1.
1000
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EDi
Eproj
Accuracy
Dataset
Model
MAD
Erank
NumRank
ratio
Standard
Normalized
Standard
Normalized
GCN
-0.7871
0.6644
-0.8106
-0.8309
-0.2460
0.9724
0.5885
0.2693
Cora
GAT
-0,9189
0.6703
-0.9469
-0.6054
0.8251
0.9722
0.7612
0.2493
GCN
-0.8442
0.4350
-0.8913
-0.8667
-0.7169
0.9700
0.6795
0,4380
Citeseer
GAT
-0,9576
0.0664
-0.9585
-0.9080
0.3722
0.9915
0.8047
0,4672
GCN
-0,9068
0.7006
-0.8508
-0.1109
0.6205
0.9464
0.9268
0,5225
Pubmed
GAT
-0.8735
-0.3684
-0.8541
-0.4102
-0.3932
0.9270
0.9721
0.5564
GCN
-0.7774
0.4171
-0.7602
-0.3258
-0.8247
0.6316
0.9582
0.8457
Squirel
GAT
-0.6864
-0.5503
-0.7364
-0.7253
0.5002
0.8538
0.6840
0.7533
GCN
-0,9223
0.1504
-0.9163
-0.8201
-0.8809
0.9387
0,9014
0.6195
Chameleon
GAT
-0.8721
0.1942
-0.0099
-0.8234
0.2803
0.9446
0.8799
0,6332
Amazon
GCN
-0.9297
0.8809
-0.9079
-0.3423
0.9201
0.9301
0.8049
0,8562
Ratings
GAT
-0.9388
0.5277
-0.9089
-0.1617
0.6545
0.9248
0.8764
0.8384
GCN
0.7738
0.9194
0.5740
-0.2738
0.2822
0.9682
0.9091
0.0939
OGB-Arxiv
GAT
-0.4097
0.9439
-0.7230
0.8985
0.8492
0.7740
0.9781
0.2310
Average correlation -0.7179
0.4036
-0.7571
-0.4504
0.1601
0.9103
0.8374
Table 1: Correlation between the classification accuracy and the logarithm of metric values GNNs with LeakyReI.U and depths ranging from to 24 layers, separately trained on different homophilic (Cora, Citeseer, Pubmed), heterophilic (Squirrel, Chameleon, Amazon Ratings) and large-scale (OGB-Arxiv) datasets. For Erank and NumRank, we subtract I so that both metrics approach zero. The rightmost column reports the ratio of classification accuracy between GNNs with 2 and 24 layers. Some heterophilic datasets may be more resilient to the increasing network depth, in-line with observations from the literature, e.g. (Guo et al., 2023a). Additional results on other datasets, activation functions and additional network components are presented in Appendix E and
Theorem 5.3 requires that the Hilbert distance between the feature representation and fixed vector u goes to zero. Note that this is implied by the relative metrics Eir (X) / or (X)) going to zero, but is actually quite weaker. Note also that the bound (6) in Lemma 5.2 directly provides guidance situations where the hypotheses of Theorem 5.3 are satisfied. We discuss several such situations along with some alternative and possibly weaker assumptions for Theorem 5.3 detail in Appendix A.6 Finally, we recall that, because of Lemma 5.2 the last result applies either to GCNs with homogeneous activation function or GAT's with any kind of activation function. The convergence of the numerical rank to may not hold, as discussed in Appendix D
EXPERIMENTS
The vast majority of empirical studies in the literature that observe the decay of Dirichlet-like energy metrics are conducted over the layers of deep but untrained networks (Wang al.) 2022 Rusch et al. 2022, 2023b Wu et al. 2023 Roth 2024 Wang et al., 2025). Moreover, the measurements are done by looking at the layers of single network, rather than different networks of increasing depth. We emphasize that this is an overly simplified and unrealistic setting. In this section, we perform an extensive numerical investigation on the behaviour of different oversmoothing metrics when measured on networks of different depths / = 2,... L, trained in isolation at different depths. Our analysis shows that GNN suffer from significant performance degradation after only few-layers, which stage the convergence patterns of Dirichlet-like metrics difficult to observe, while relaxed rank metric already show a significant decrease, well well-coreelatin with the performance drop.
In particular, compare how different oversmoothing metrics behave compared to the classification accuracy, varying the GNN architectures for node classification on real-world graph data. In our experiments, consider the following metries:
The Dirichlet Energy Epir (Cai & Wang 2020 Rusch et al.) 2023a) and its variant Eproj
Wu
et al. 2023). Both discussed Section see in particular (1) and (2).
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GCN LRaLU
GCN Tanh
GCN LRe Blas
GCN Tanh
10
10
10:
10"1
10'
10a
10
luaal Layal
Co.iK)
Evg
Ane/Air
MADIOI
BranktXi
NumRark30
14
Accuracy
Figure 2: Four examples of the metric behaviours computed at the last hidden layer of separately trained GCNs of increasing depths. For Erank and Numrank, we measure Erank (X) FÈR and Num NR for some r' > 1. In these particular cases, 'ER < 1.85, ''N < 1.3. Note that the effective rank and numerical rank of the input features (0) are about 1084 and 13.6, respectively. Additional results are attached in Appendix G
Normalized versions of Dirichlet energy and its variant, Epa (x) and (X)o](X)||| Indeed, from our previous discussion, a robust oversmoothing measure should be scale invariant with respect to the features. Metrics with global normalization like the ones we consider here have
also been proposed in (Di Giovanni et al. 2023 Roth & Liebig
2023 Maskey et al. 2023).
The Mean Average Distance (MAD) (Chen et al. 2020
x. XT, (1- (1-1x,, Ix,, Ix,
MAD(X)=
[ Σ ((,))
It measures the cosine similarity between the neighbouring nodes. Unlike previous baselines, this oversmoothing metric does not take into account the dominant eigenvector of the matrices A Relaxed rank metrics: We consider the Numerical Rank and Effective Rank. Both are discussed in Section 5 We point out that from our theoretical investigation, in particular from (4), the numerical rank decays to 1 faster than the decay of the normalized Eproj energy to zero. This further supports the use of the Numerical Rank an improved measure of oversmoothing.
In Table T and Figure 2 we train GNNs of a fixed hidden dimension equal to 32 on homophilic, heterophilic and large-scale datasets in their default splits. We follow the standard setups of GCN and GAT as stated in Sections 1 and 2.2, and use homogeneous LeakyReLU (LReLU) as the activation function. For each configuration, GNNs of eight different depths ranging from 2 24 are trained. The oversmoothing metric and accuracy results are averaged over 10 separately trained GNNs. All GNNs are trained with NAdam Optimizer and constant learning rate of 0.01. The oversmoothing metrics are computed at the last hidden layer before the output layer. In Figure 2 and in Appendix G we plot the behaviour the different oversmoothing measures, the norm of the features, and the accuracy of the trained GNNs with increasing depth. These figures clearly show that the network suffers significant drop in accuracy, which is not matched by any visible change in standard oversmoothing metrics. By contrast, the rank of the feature representations decreases drastically, following closely the behaviour of the network's 's accuracy. These findings are further supported by the results shown in Table [ where measure and the classification accuracy of every GNN model. The use a logarithmic transformation based on the understanding that oversmoothing grows exponentially with the length of the network.
compute the Pearson correlation coefficient between the logarithm of every
Extensions of these results are provided in Appendices asymptotic ablation study very deep (300-layer) synthetic networks with randomly sampled, untrained weights. This study serves to validate our theoretical findings on the convergence of relaxed rank metrics and to demonstrate that such untrained settings offer little insight into the ability of existing metrics to quantify oversmoothing in realistic, trained networks.
I and In Appendix D we perform an
CONCLUSION
In this paper, we have discussed the problem of quantifying oversmoothing in message- message-passing GNNs. After simplifying the existing theoretical analysis using nonlinear activation eigenvectors and discussing the limitations of the leading oversmoothing measures, we propose the use of the rank of the features better measure of oversmoothing. We provide extensive experiments to

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validate the robustness of the effective rank against the classical measures. In addition, have analysed theoretically the decay of the rank of the features for peessag--psssig GNNs. Notably, work opens possible research directions that investigate the use of rank-based metrics in novel architectures that can alleviate the oversmoothing phenomenon. Indeed, Dirichlet-like metrics have been employed not only to measure oversmoothing but also design new architectures able to mitigate this phenomenon, e.g. by considering residual architectures where the magnitude of the residual is regulated according to the Dirichlet energy of neighboring nodes (Rusch al., 2023b Jin & Zhu 2024). In light of our results and the superiority of rank-based metrics in capturing oversmoothing is natural to wonder whether the use of rank-based metrics in place of Dirichlet- like ones could better mitigate oversmoothing in similar kind of architectures. We consider this promising future research line.
ACKNOWLEDGEMENT
KZ was supported by the EPSRC Centre for Doctoral Training in Mathematical Modelling, Analysis and Computation (MAC-MIGS) funded by the UK Engineering and Physical Sciences Research Council (grant EP/S023291/1), Heriot-Wart University and the University of Edinburgh. PD and FT members of the Gruppo Nazionale Calcolo Scientifico Istituto Nazionale di Alta Matematica (GNCS-INdAM), PD was supported by the INdAM-GNCS project Algebra Techniques for Machine Learning". FT is partially funded by the PRIN-MUR project MOLE (code 20222K5ME7) and the PRIN-PNRR project FIN4GEO (code P2022BNB97), DJH was supported by the Advanced Grant "Numerical Analysis for Stable Al" 101 198795 from the European Research Council.
- Linear
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A PROOFS OF THE MAIN RESULTS
A.1 EQUIVALENCE OF Eproj AND EDir
Lemma A.1. Assume the graph 9 to be connected and the eigenvector u to be strictly positive u4 > 0 for all i and such that || 1. Then there exist C₁ and C2 such that
C2 Epi & (X) S EDi
E Nxd
Proof. First we show that the Dirichlet energy is equivalent to the modified Dirichlet energy where all the pairs of nodes (i.j) are considered
Epi
Σ (N,))EE
X1:
j: Uj
S
X
Xj; 2 Uj
E (X).
U1
2
NEV
jEV
2
Second observe that if i E V. since the graph is connected there exists some path i 1, i2....,, lnt1 = j such that (ik. ik+1) E & for all k. Thus, using the traingular inequality we get
X 11
X
Xiga Whe
Xis-
S (X).
(8)
Wiat1
repeating the same argument for any pair of nodes (i, we observe that for some constant C
(x) S (X).
So, to prove the equivalence of EDir and Eproj is sufficient to prove the equivalence between Eproj and EDi--
If we make explicit the expression of the norms in EDit get
Xi,k
Nj.
d
X1,k
Xj.
(X)
ΣΣΣ
uu
X
U
Uj
EEV jEVA km
LEV
JEV
2
j (u=x,))ur x. 2
(10)
LEV jeV
4
X
uu- X 1|2
41V
Exroij (X).
min;
(4)
min; (1:)
To prove the opposite observe the following
2
Xix
Xi.k
XAak
EProi(X)
Ui
(u
x.,)
Wi
Σ
un
ui
uii
un
k=
k=
Xi,k
X,,k
<
[u; 5
max
(11)
Ui
Am 1
VEV
X
Xj,k²
1
S maa
EDr
14
Ui
JEV
where in the first inequality we have used that, since ||ul|2 = 1. En ux on.
Xa. is convex combination
of [ Xa.k Jn In particular, the last inequality concludes the proof.
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A.2 PROOF OF THEOREM 3.2
We start proving that
II - P)x |F - III [ x00
x
(12)
where P uu /||ull is the projection matrix the linear space spanned by u.
To this end, let и == f v E Rd be the 1-dimensional matrix subspace of the rank-1
matrices having columns aligned to Then it is easy note that given some matrix X, (I P) provides the projection of the matrix X on the subspace π, i.e.
(I P)X proj,
(13)
Indeed ((1 - P)), uo))r- = Tr(eu' ((1 - uu /||u| = 0. In particular, since the projection
realizes the minimal distance, have that
- < ||X uv I| Vu E R4. Now observe that oun' T xU- u' T (((-1) ((-))))( we have that the i-th column of ofuu² x(- is equal to (u:u) = U,u for some E; because is eigenvector of a. As a consequence we have
(14)
To for some v. Indeed, writing T
[I(T 1I - | oun A4-1) (((0))- FF
= ||o
(1-) -A- 11- W - 1)) (uu) A - ((-1)
(15)
< - ()
where we have used the 1-Lipschitz property of a. This concludes the proof of (2
To conclude the proof of the theorem observe that, in the decomposition
(I-P)A)) (I - P + (1 - A (I-)), P))
the matrix (I- (T-P))(()) is zero because A (0) = X/u for any L. Thus
(- = -I-P) ( P);
and, from 12 and the inequality ||A|| S ||A||2|
we have
[[( - 1IX < (n(=01) |I(]]((T PAA 112/1/) (x
So the thesis follows from the hypothesis about the product 11/-0110 [I(1 P) a ||]|]|
A.3 PROOF FOR THEOREM 4.1
Start by studying the norm of (7 (T-P) ()))) Then looking at the shape of the powers of the Jordan blocks matrix it is not difficult to note that T = ((())))))))) N for I larger than N. In particular if we look at the explicit expression of (1 X()
(1 - P)x (o (I - P)M (o
0
M M -x-x w(-1
(16)
7
we derive the upper bound
II P) ||5 S (N) Ael Aa (1x(0) | w(ll1)
(17)
for some positive constant C' that is independent on I. Similarly we can observe that |PX00 |P Ilu A (1))
1 N (N)
=1 + Too
MO
2/-N1 M M x (0100 -((1
> |Ail [|0] (1))
(((-1))| 11 - -11--111 |
M' 10 ( pp)
N
w(( w (1- > 1 (m
d2
||x (1x00) ... W...
A
|1 X(O w(-11|/
(18)
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Now observe that under the randomness hypothesis from (Furstenberg & Kifer 1983) and more generally from the Oseledets ergodie multiplicative theorem, we have that for almost any the limit w E IRd lim log ||w WT W(-1)) c(v) exists and is equal to the maximal Lyapunov exponent of the system. In particular for any and 0 there exists ww,e sufficiently large such that for any / w..e
Tw W1 < du) +e c(v) - - log |1 7
(19)
i.e.
(((v)-=) < || T W ....) cl(c(v)-))
1l L,n..
(20)
Now take vector first the rows of (0) and then the vector o/ X x then almost surely for any there exists le such that for any / > le
(o( < w W() ... w(l-1) < c(cle))t))
(21)
holding for any 1 2 l, and any w {1i U (Xj e
Next recall that [x (1()) w |r = E, lee x0o meaning that almost surely, for
I 12 le:
Nell(()- < |X ( w F S Ne(o(()))))
(22)
In particular for any e, there exists sufficiently large such that
(
Az X
|x(
Wa-Dile " - =
((N)
A2 A1
N
I0
X
W(-D)|
21c
(23)
and thus, since 1aa| [Arl and we can choose arbitrarily small, almost surely it has limit equal to zero. In particular we can write
[ )
12g I ... V |F 0
lim
~ lim I |A W W
(24)
|PX0
where we have used the same argument before to state that the limit is zero.
A.4 PROOF OF LEMMA 5.2
Since A is nonnegative, then from Perron Frobenius theory (Lemmens & Nussbaum 2012). we know that
dh (AAX)...)) dh ((1X) An dd ((AX),.: Au) S 3d (X,,) u)
Vi.
(25)
for some 3 I 1, where we have used (A) > 0 and the scaling invariant property of the Hilbert distance. In particular if A is contractive in Hilbert distance B < 1.
Then note that, for any i, we can write (AXW):, as follows
(AXW)
Σ (AX)))
(26)
Thus we CLAIM that given 21, x2, y K. then
dn(ri + x2. v) S max(dn (x1, (er,)) du (1((22.)))
(27)
Observe that if the claim holds, by induction can trivially be extended from to d points yielding
du (AAXW).,, S max du (W (AX):. u) s & max du ((AX).,. u) < B max du (X.s u
(28)
where we have used the scale- invariance property of the Hilbert distance and the fact that max; Wiy > O for all
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We miss to prove the claim. To this end, exploiting the expression of the Hilbert distance write
(x1); + (x2))
(y)s
(11 + 22, v) = log
sup
(v)
(x1); + (22)}
< log
supsup max m1
(2i))
(a2);:
(8)
(29)
(x1)
(x22
(v)
max 21:23
(du dn (22,
concluding the proof.
Next we prove that, continuous subhomogeneous and order-preserving function with eigenvector u in the cone, is not nonexpansive in Hilbert distance with respect to u. Formally we claim that
dæ (a((), u) I dæ (v.u) by E K.
(30)
To prove it, let E IC and assume w.l.o.g. that ||v|l1 = =>> and
||u]| = 1, then
M(y/tu)= iI....N N t(u),) 2
max Vi
||ll|
m(y/tu) = min yi ||ul|
1. (31)
t||u]l
i=I....N t(u); t||u||a
By definition given x, y K. m(y/x)æ SK y Sx M(y/x)x. Moreover we recall that since an cigenvector for any > there exists d > such that o(tu) = A.u. Thus we use the subhomogeneity of a and the fact that is eigenvector of to get the following inequalities:
m(yytu))Attu Sx (m((/tu))tu) Sx f(w) Sx (M/tt)t)) Sx (u//ze))ete
(32)
where the inequalities are strict if a is strictly subhomogeneous. In particular we have m (t)lly()t)t 2 Xem(y/tu /t) and M (f(y)/tu) < (y/tu). Finally the last inequalities and the scale invariance
property of du yield the thesis:
(f((f)): u) = du (f(v), tu) = -log Mffl))/tu) - log M(u/tu)) du (y, = du (V. u). m(y/tu))
m(f(v)/tu)
(33)
with the inequality that is strict if a is strictly subhomogeneous.
Then the thesis of the Lemma follows by applying (30) 28
A.5 PROOF OF THEOREM
We will prove that, consequence of the hypothesis lim;-so max, du (x.), u) 0.
- (0I
lim 1-+00
0,
(34)
||P
where P = uuu uuu//|ul13 Indeed 34 is equivalent to proving that the numerical rank goes to I as
goes to 0o:
I(I-P)X'01
II(I
I[[ I(I-P)x(11
1 < NumRank(x()) +++
s1+
(35)
|
To prove (4), recall from Lemma 2.5.1 in (Lemmens & Nussbaum 2012) that for any such that u w
[w - Pw|la (e -1),
(36)
where dr(#, y) = (max (M ((((://)))((//)))))m((( m(x/u)=1)) and where since the dual cone of R is R
itself, we are considering the norm induced by on the cone, i.e. || u x for any x in the cone. In practice the norm induce by 2 Ilu can be defines by the Minkowki functional of the set Ω = ConvexHull { U {- where 2 {I K u x 11
Then since ||Pwlla = ||w||a = u w. we have that M (w/Pw) 2 and m(w/Pw) S 1. Thus dy (w. Pw) dw, Pw), yielding
||l
Pwrll S ||Pw|] (oPull((e (eduuu 1).
(37)
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From the equivalence of the norms there exists some constant 0 such that equivalently write
|| - Pwl12 2 S Cl -1)
(38)
Now recall that the squared frobenius norm of matrix is the sum of the squared 2-norms of the its columns, so we can apply the last inequality to the matrix X() columnwise obtaining:
II( P) P I x (e (emax: (dy (x.), px - 11)
(39)
the proof is concluded using the hypothesis and observing that by the scale invariance property the Hilbert distance du (X p = dи (x 1,6 u). yielding:
1 1 (NumRank(X()) (1) 1 - P) 113
ai) -1) 1)² (40)
S 1
and concluding the proof.
A.6 SOME SITUATIONS WHERE limy-+00 max, du (X u) -0
In this section we explore three different situations where the Hilbert distance of the features from the dominant eigenvector of the thea aggregation matrices is guaranteed converge to 1.
1st situation A first situation where the Hilbert distance of the features from the dominant eigen- vector u converges to zero is the case of all matrices A contractive.
Lemma A.2. Let A be nonnegative and irreducible matrices with dominant eigenvector u K. Assume also (0) to be strictly positive, w(i) nonnegative with min, max; w > 0 and a E C(R R nonlinear function that is order preserving, subhomogeneous and such that is also an eigenvector of a. Any matrix A is known satisfy (AW)s Al) S di((a, v) for all T, E K. then if limp-soo I1 31 =
lim max dr (x : u) 0.
(41)
t0:0
Proof. The proof trivial consequence of Lemma 5.2 know that
Indeed iterating the result in the thesis we
max du (o(x!!))
Be max d (X u),
(42)
concluding the proof
In particular we remind that from the Perron-Frobenine theory any strictly positive matrix A is known to be contractive of parameter 3 < see (Lemmens & Nussbaum 2012).
2nd situation A second situation where the Hilbert distance of the features from the dominant cigenvector u converges to zero is the case of a strictly subhomogeneous activation function.
Lemma A.3. Let Al)) be nonnegative and irreducible matrices with dominant eigenvector u. E and
K. Assume also x()) to be strictly positive, W()) nonnegative with miny max, w:l!) functiio that is order preserving, strongly subhomogeneous and such that is also an eigenvector of Since is strongly subhomogeneous, for any I there exists X(W u) S -Bdu(((Al-)))) (1-1) (1)))
o E CRR N a nonlinear function
3t < such that max;
). u), then if
lim l 31 = 0
lim max di (x.1) u) = 0.
(43)
1000
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Proof. The proof is again trivial consequence of Lemma 5.2 we know that
Indeed iterating the result in the thesis
max dp (x(
B max dh (x u)
(44)
concluding the proof.
Note that whenever we have strongly concave activation function a R4, e.g. tanh, then it is strongly subhomogeneous and so the result above applies.
3rd situation Here we discuss a third situation, possibly weaker than the previous ones. This is essentially adaptation of the ergodic theorem proved in (Nussbaum) 1990). Consider the cone
K" TR xd and introduce the set II (Y K'' s.t. Y., 0ş Vi 1,..-,d} d). Moreover let s E [k') and define Ey- = (Y E K'' s.t. (Y) = 1}. Finally given X E KC' define
क (X) = arg min (X.
(45)
YEE,
Definition A.4 (Hypotheses H). We say that GNN x = =x ( E K' satisfies hypotheses [H] if the following conditions are verified:
with
H1 A is nonnegative and min, max; w(O for any I.
H2 The function a is subhomogeneous and differentiable in R+.
H3 u E IC is the dominant eigenvector of all of the matrices A and it is also an eigenvector of σ.
H4 There exists integer and sequence of dN x dN strictly positive matrices {B(1)
such that VX with m (x ((X(r)) (ipp))))x((() ((X(1)) X < M (x(())),,((())))
0xg (X) ((0) A24
where (X) = /((+1)p-1)
(x).
H5 VI 2 0 there exists (0 0 s.t. B T (X (1)) g( (0 (x(x((p)).
H6 lim -o E.oo ( exppp A(B()) oo where A(B) = supz,eek du (Bx, By) < oo is the projective diameter of the matrix B.
Theorem A.5. Let
Letx ( = (x(1)) = x(0x)), with ( E IC', be a GNN satisfying
hypotheses [H]. Then
lim max (d : u)) = O. t0
Proof. We claim that under hypotheses [H]
lim de (X II] 0.
(46)
10
As a consequence of this it is easy to note that lime max; (dn(X u)) 0. Indeed it is not
difficult check that
M(X , (x(1)
maxmax ja π (X(1)) Aak
x())
exp (d (xx)=(((1)) (x(x)))==
m(X(1, T (x())
ft
x x (X(1),,
(47)
x!I π (X(1) mi
max ezp (x (2))
max max max
x x (x (x
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where we have used that by definition π ((x()),; a, for all i, where necessarily a; > 0 for all d. Otherwise, since X K'', we would have dn (X(()) (4) n (X!")) oc, against the minimality.
Next we prove the claim. The proof is adapted for our scopes from the proof of the weak ergodic
theorem 2.1 proved in (Nussbaum) 1990) for homogeneous mappings. To simplify the notation we
denote by X == ( == ( ( == g(), m == m(X (()) ((((())) )), B == B( 77 == 17
and M := (() T (()()))). Then consider 2 2t) = (1-t)mm++ +tX and 2 2t) = (1-t)
Then from hypothesis [H4] we have the following inequalities:
g(X) - g(mx)) = L. 0x)(==(0)(xx - -mT) 2 B(X mm)
(48)
g g(Mm) g(x) = J. ())((( x) z B (MT x).
In partieular, since g is subhomogeneous and by definition of π m S and M 2 1 we have:
mg( B(X- mm) S g(mx) + B(X mm) S (
(49)
((M) B(M - X) S MT( - B(M - x)
Since B(X- - mT) + B(M X) = (M - mm) m))m.B. we can use Lemma 2.2 in (Nussbaum we know that
1990)
B(X mo) 2 r(M Bn or B(M* X) 2 7M m) Bx,
(50)
where y = (1/2)erp(AA(BB Without loss of generality assume that ( mm) 2 ((M m) BT
the second case can be handled analogously. Then, since B(M X), from 51 we have
mg(T) + 7M( - m)g(T) S mg(()) + (M - m - 9(X) S Mg(T).
(51)
In particular
M
M
dn (n(g((), S log(
=log
m + 77(M m
m 17 m)
log
MM
(52)
=log(M/n) /m)
m
log(M/m)
s log (0g(M//m)/)
177) = (1-17) 11-77)dH(( x).
In the last we have used the following fact: if (s) = log + (s(1+mp(s - 1) and (2(s) = log(s)
with s > 1 then since S1, and + 77(5'' - 1) > for s' 1, for any > there exists
some s' >> such that
= = =
-- (1- m))
(53)
Now note that dis (X, (X, by the minimality of and the fact that du (X, oY) du (X,Y for any > 0. Second, using [HI] and [H3] it is very easy to observe that g(T) E II. So In conclusion we have proved that
du (g()), II) S dg(gg(X).
dn (x((1))) II) di (g (x(1), II) S (1- exp( - AA(B))
dn(("))))
(54)
2
In particular iterating, using 47 and recalling that max, dH (X((1)) (u) s max, (X!1)) ,u) if l > l1 (see Lemma 5.2) we have that for any L > lp:
max (x u) S maxdy (1)) u) S du (x(11),1m) .I) s
1- 1-) exp( 10p0-A110 -A(BOT)
(55)
II 5=0
dn (x(0),,,
2
Moreover, have that
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- 1710) expl A(B6))
lim
II
=0
+00
2
j=0
1 1-1110 exp( - A(B6))
lim 1-000
log
x
(56)
2
jm0
1111) (u)erp(( - A(Bb))
lim 1-700
06,
which concludes the proof.
Next, we discuss briefly the hypothesis H4 which is the most technical one. In particular, such hypothesis recalls the property of a primitive matrix. Actually we note that, if the matrices A are primitive, under few additional mild assumptions, always satisfied. Assume that there exists an index p such that:
possible prove that the hypothesis H4 is
1. The matrices A with 1 0,.... adjacency matrix of the graph; i.e. A; = the edge (i,j) belongs to the graph, A = otherwise. And assume that A" is primitive of index smaller then p, i.. ((A)))) > 1 for all i,j
........ p satisfy A an A" where A is the unweighted
2. (Xij) 2 ag for all X with x and X* among the matrices that satisfy m m(x(0), x (x(1)))(((()) S X S M ( (()))
f o (x0) varying O.....P
3. (W(0) w(p-1)) > a3 > 0 for i,j.
Then for any matrix X satisfying 7(x") s s M( x(((())) (xx we have
(Rxo(X)), 2 afagas Vij
where g(X) = f(o-1) o...o (X).
Indeed, vectorizing the matrix X write 0x ( (X) diag ('((TT A)vec(X))) (WT
)
A) 2 a2(W"' A). Then we can apply the chain rule to the function g to get
(0xs(X)) 2 oz (w((-1))))" ((P-1) (w) T x (0).
(57)
Finally we use the properties of the matrices wi and A and the properties of the Kronecker product to obtain
(Dxo(x))) (W((--1) w(1) @ (A(--)) ..... A.A) 2 agafas1
(58)
where 1 is the matrix with every entry equal
A.7 EIGENPAIRS OF ENTRY-WISS SUBHOMOGENEOUS MAPS
We conclude with a formal investigation of the eigenpairs of activation functions that are entry-wise subhomogeneous. Let N & with v E C(R,, K) that is subhomogeneous on R.. Then one can easily show that a is itself subhomogeneous on R We have the following result,
&N a with E C(R4, R4) be order preserving. Then: 1) If a is homogeneous, any positive vector is an eigenvector of o, 2) If a is strictly subhomogeneous, the only eigenvector of in K. is the constant vector.
Proposition A.6. Let
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We start from the homogenous case. Note that since & is homogeneous have that necessarily ft) = ct for all f 2 0, this in particular means that every u E R. is an eigenvector of with corresponding eigenvalue \
Then we can consider the subhomogeneous case. Assume that we have u E IR that is an eigenvector of with eigenvalue À and u; > 0 for all i, then
( Au;
Vi I,..,,N. N.
(59)
By strict subhomogeneity this means that necessarily is constant, indeed if us > uj > 0 then
Auj = (1²) xb
ug
Ui
Mj (bu))) t4
Xuj-
(60)
yielding a contradiction. In particular any constant vector in R is easily proved to be an eigenvector of relative the eigenvalue ((())ll////lll = (l,)) (4 where u, any entry of u.
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B LIMITATIONS
Although a small effective rank or numerical rank indicates oversmoothing and can be subsequently linked the underperformance of GNNs, a large effective rank numerical rank does not necessarily correspond to a good network performance. Prior study has suggested some degree of smoothing can be beneficial (Keriven 2022). and as an extreme example, features sampled from a uniform distribution and with randomly assigned labels would almost surely have large effective rank, but they cannot classified accurately due to the lack of any underlying pattern. We note that this limitation is not specific to the two relaxed rank measures, as the same argument is directly applicable to all other oversmoothing metrics.
Consequently, shown in Table 4 when additional components are used to (partially) alleviate oversmoothing, particularly when residual connections are used, the accuracy ratio may remain large over the layers, and all oversmoothing metrics correlate poorly with the accuracy of GNNs. This in turn suggests these oversmoothing metrics become less informative as the oversmoothing problem is mitigated or alleviated.
C COMPUTATIONAL COMPLEXITY ANALYSIS
Let N, D, E the number of nodes, features and edges of graph G, the computational cost of the Dirichlet energy is O(F x D), while the cost of the Projection Energy is O(N x D). In contrast, most of the computational cost for the numerical rank and the effective rank is given by the computation of the spectral radius of X, and of all the singular values of X for the effective rank, respectively.
Standard results show that it is always possible compute the full singular value decomposition (SVD), and subsequently the spectral radius, in O(N x D x min(N, N, DJ). However, in typical cases, the latter cost can be drastically reduced using different strategies. Firstly, in the case of numerical rank, the computational cost of the 2-norm is generally much smaller than the cost of the full SVD. Indeed, using Lanezos or power methods to compute it, the cost scales as O(N D), and the methods converge typically very fast. In addition, both the effective rank and the numerical rank can be efficiently controlled by computing only the k-largest singular values of X. particular, truncated SVD containing the k-largest singular values can be computed in O(N x D k) using either deterministic algorithms or randomized SVD methods.
In general, the computational cost of metrics to quantify oversmoothing is marginal compared to the cost of training. In production, measuring the emergence oversmoothing is done by training the model and checking the performance the validation and test sets the number of layers changes. The cost of additionally computing effective or numerical ranks is marginal. Moreover, we note that computing the metrics subsection of graph can be sufficiently informative, and consequence, most oversmoothing metrics studied in this paper can be computed in less than 10 ms for graph (or subsection of with less than 2000 nodes.
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D ADDITIONAL EXPERIMENTAL RESULTS WITH SYNTHETIC WEIGHTS
In this section, we conduct an asymptotic ablation study using randomly sampled (synthetic) untrained weights. The aim of these experiments is twofold:
to demonstrate that similar untrained asymptotic experiments are inherently unrealistic, despite being extensively used in the literature (Wang al. 2022 Rusch et al.) 2022 2023b Wu et al.
2023
Roth 2024 Wang et al. 2025), as they fail to reliably capture oversmoothing in shallower GNNs where the performance degradation occurs.
to examine the convergence properties of different oversmoothing metrics with weight size control and to empirically validate Theorems
5.1 and 5.3
We construct a 10-node Barabasi-Alber graph with each node having 32 features. The weights are either an identity matrix randomly sampled at each layer from uniform distribution (0,, s), where depends the settings: small weights (s 0.05) lead to an exponentially decaying [X("))p and large weights (s 0.1) lead to exploding [|xx(0) || for uncapped activation functions. For identity weights, [|x || F is roughly constant (LRcLU) or slowly decaying (Tanh). The feature initialization ( is sampled from (/(0, 1), and is iterated over 300 layers.
In this asymptotic synthetic setting, as presented in Table the normalized EDir and Eproj exhibit decay patterns similar those of the effective rank and numerical rank, suggesting these metrics equally sensitive to asymptotic rank collapse. However, this behaviour stands in stark contrast to results on trained networks presented in Tables and and Appendix G where the normalized EDit and Eproj often fail to detect oversmoothing.
Moreover, these asymptotic results validate Theorems 5.1 and 5.3 showing that the numerical rank converges to one for GCN LReLU and GAT + any subhomogeneous activation functions. Without making any additional assumption the normalization of the adjacency matrix, the effective rank and numerical rank do not generally decay to one when subhomogeneous activation functions, e.g. Tanh, are used in GCNs.
Epir
Eproj
Architecture
MAD Erank NumRank
standard normalized standard normalized
GCN+L.Rel.U+identiyy weights
x
GCN+Tanh+identity weights
GAT+LReLU+ideniity weights
GAT+Tanb-+idenitty weights
GCN+LReLU+small weights
GCN+Tanh+small weights
GAT+LReLU+small weights
-mml weights
GCN+LReLU+Large weights
X
GCN+Tanh++arge weights
X
GAT+LReI.U++arge weights
GAT+Tanh+lagge weights
Table 2: Additional results on very deep (300 layers) synthetic networks with randomly sampled
weights. For Erank and NumRank, subtract 1 so that both metrics converge to zero. / indicates a decay of the corresponding metric to zero, X indicates otherwise. Note that GAT has similar asymptotic behaviour to GCN with adjacency normalization D
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E ADDITIONAL EXPERIMENTAL RESULTS ON ACTIVATION FUNCTIONS AND DIFFERENT DATASETS
We extend Table 1 to subhomogenous Tanh activation function and few additional datasets. The experimental setting is consistent with that of Section 6
EDir
Prioj
Accuracy ratio
Dataset
Architecture
MAD Erank NumRank
GCN+L.ReLU--0.78
0.6644
-0.8106
-0.8309
-0.24600.9724
0.5885
0.2693
GCN+Tanh 0.5243
0.9403
0.8610
0.9768
0.9734 0.9923 0.9784
0.1937
Cora
GAT+LReLU -0,9189
0.6703
-0.9469
-0.6054
0.8251
0.9722
0.7612
0.2493
GAT+Tanh 0.8300
0.8501
0.8676
0.9066
0.8603 0.9600
0.9515
0.1900
GCN+LReLU--0.444
0.4350
-0.8913
-0.8667
-0.71690.9700
0.6795
0.4380
GCN+Tanh 0.3420
0.8957
0.4631
0.9045
0.9605 0.9906
0.9457
0.3509
Citeseer
GAT+LReLU -0,9576
0.0664
-0.9585
-0.9080
0.3722 0.9915
0.8047
0.4672
GAT+Tanh 0.9234
0.8949
0.8276
0.8997
0.9176 0.9287
0.9024
0.3045
CCN+LReLU-00900
0.7006
-0.8588
-0.1109
0.6205
0.9464
0.9268
0.5225
GCN+Tanh -0.2330
0.2657
0.2029
0,9137
0.8745 0.9328
0.9940
0.3883
Pubmed
GAT+LReLU -0.8735
-0.3684
-0.8541
-0.4102
-0.39320.9270
0.9721
0.5564
GAT+Tanh 0.2977
0.8411
0.7160
0.9331
0.8546 0.9303
0.8551
0.4464
MON++RReUU-00777
0,4171
-0.7602
-0.3258
-0.82470.66
636
0.9582
0.8457
GCN+Tanh 0.6026
0.7736
0.1689
0.9377
0.7727 0.9680
0.9837
0.8152
Squirrel
GAT+LReILU -0.6864
-0.5503
-0.7364
-0.7253
0,5002 0.8538
0.6840
0.7533
GAT+Tanh -0.3606
-0.6557
-0.0363
0.8714
-0.70330.8911
0.8861
0.9103
GCN+1RReI.U-0.9222
0.1504
-0.9163
-0.8201
.0.88090.9887
0.9014
0.6195
GCN+Tanh 0.2742
0.8796
-0.2541
0.8492
0.9272 0.9869
0.9841
0.7093
Chameleon
GAT+LReLU -0,8721
0.1942
-0,9089
-0.8234
0.2803 0.9446
0.8799
0.6332
GAT+Tanh 0.4090
0.5699
0.0743
0.8230
0.6006 0.9613
0.9143
0.7052
GCN+LReLU-009297
0.8809
-0.9079
-0.3423
0.9201 0.9301
0.8049
0.8562
Amazon
GCN+Tanh -0,6960
-0.6289
-0.6910
0.8576
0.9423
0.9871
0.9327
0.8574
Ratings
GAT+LReLUU -0.9388
0.5277
-0.9089
-0.1617
0.6545
0.9248
0.8764
0.8384
GAT+Tanh 0.8354
0.8507
-0.6955
0.9245
0.9418 0.8954
0.8883
0.8382
GCN+LReeLU-00.5 -0.5635
0.7703
-0.6772
0.2575
0.6420
0.5833
0.5368
0.3891
Roman
GCN+Tanh 0.8018
0.9225
-0.6808
0.8359
0.8124 0.8570
0.8090
0.4067
Empire
GAT+LReLU -0,9174
0.5390
-0.9407
0.1868
0.7582
0.7221
0.8722
0.3705
GAT+Tanh 0.5767
0.5844
-0.4716
0.8819
0.7263 0.7332
0.8589
0.3652
GCN+LReLU 0.7738
0.9194
0.5740
-0.2738
0.2822
0.9682
0.9091
0.0957
GCN+Tanh -0.6409
0.7041
-0.6808
0.9494
0.9487 0.9900
0.9876
0.1204
OGB-Arxiv
GAT+LReLU -0.4097
0.9439
-0.7230
0.8985
0.8492
0.7700
0.9781
0.2310
GAT+Tanh 0.7834
0.7896
-0.9277
0.9393
0.8222 0.7671
0.7861
0.2376
Average correlation
-0.1956
0.5137
-0.3886
0.2669
0.4960 0.9007 0.8684
Table 3: Additional correlation coefficient results homophilic (Cora, Citeseer, Pubmed), het- erophilic (Squirrel, Chameleon, Amazon Ratings, Roman Empire) and large-scale (OGB-Arxiv) dataset.
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F ADDITIONAL EXPERIMENTAL RESULTS WITH DIFFERENT NETWORK COMPONENTS
Prior literature indicates that when adding additional components, such as bias (Rusch et al. 2023a or residual terms (Scholkemper et al. 2024), the Dirichlet energy does not decay. Therefore, we compare the correlation coefficients
Table 2 between existing oversmoothing metrics when additional components are added, such as bias, LayerNorm, BatchNorm, PairNorm (Zhao & Akoglu
2019), DropEdge (Rong al., 2019 Huang et al. 2020) and residual connections (Scholkemper
et al. 2024).
All experiments follow the setup described in Section 6 In addition, DropEdge has a probability of 0.5 in removing each edge at each layer. The residual connection is implemented as follows
x = (AX + w!!"
Table demonstrates that both the effective rank and numerical rank achieve a higher average correlation with the classification accuracy than alternative metrics. This finding confirms their superior consistency in detecting oversmoothing across a variety of architectural variants.
Furthermore, we note that when oversmoothing is effectively alleviated, c.g. when residual connection used, all metrics have poor correlation with the classification accuracy. This generic limitation discussed in Appendix B
FDi
Eproj
Erank NumRank Accuracy
Architecture
MAD
ratio
GCN+LReLU+Bias
-0.9300
0.7505
-0.9414
-0.3642
-0.2828 0.9847
0.7833
0.2110
GCN+Tamb+Bizs
0.4086
0.8950
0.7479
0.9026
0.8996 0.9926
0.9814
0.2133
GON+LLRe.U+LLLyrNNoo---
-09913
0.8445
-0.9424
0.5591
0.9753 0.9736
0.9736
0.4882
GCN+Tamb-LLaerroom
-0.7119
0.2560
-0.2069
0.9716
0.8695 0.9576
0.9684
0.1886
GCN#LRReLUU-BBacNNNonn
0.8789
0.7300
0.8872
0.7175
0.5094 0.6005
0.6577
0.8761
GCN+TTanh-BBachNNrmm
-0.6033
0.2883
-0.6587
0.7717
0.4984 0.7377
0.7038
0.8157
GCN+LRcLLU+PainNom
-0,8165
0.3731
-0,8106
0.3885
0.4850 0.5597
0.6244
0.8556
GCN+TTanh+PairNomm
-0.5963
0.0346
-0.5401
-0.3358
-0.0631 0.9838
0.9298
0.3123
GCN++LReLUU+rrooEdgge
-0.7497
0.7185
-0.7939
-0.7974
-0.46190.9720
0.5782
0.2515
GCN+Tanb-+roopEdee
0.2319
0.8388
0.5872
0.8887
0.8763 0.9934
0.9558
0.2096
GCN+LReLUURResidull
-0.0857
-0.3072
-0.0559
-0.1611
-0.1656-0.2466
-0.1296
1.0046
GCN+Tann-RReiddull
-0.6751
-0.7406
-0.3031
-0.4019
-0.6043-0.5425
-0.4897
1.0122
Average correlation
-0.3801
0.3901
-0.2525
0.2616
0.2946 0.6638 0.6280
Table 4: Additional correlation coefficient results on GCNs with different network components.
26

---

Published as a conference paper at ICLR 2026
G METRIC BEHAVIOUR EXAMPLES ON CORA, CITESEER AND PUBMED
We extend Figure 2 to additional datasets and network components. The experimental setting is consistent with that of Section 6
GCN LReLU
GCN + Tanh
1.450100, FİR
1.207918
ER 1.690520,
T'R 1.007135
ER
20"
97
10"
07
Medicc 1000
066
06
10T4
10a4
05
10
10
0.2
30
Layer
Layer
GCN + LReLU + Bias
GCN + Tanh + Bias
1.454981, FRR= 1.020319
ER 1.858602, VNR 1.021634
ER
10a
9.7
10"
07
Me0icc V8N
0.6
06
10-1
10-1
03
10a9
10a9
02
GCN + LReLU LayerNorm
GCN Tanh + LayerNorm
FR 1.017424, FÄR 1.000101
FRR 1.528688, VNR 1.003682
10a
0.7
20"
67
Wettle 3000
0.6
06
10
10-1
10:9
10-1
02
Layer
GAT + LReLU
GAT Tanh
YER
2956233, FİR = 1.079250
ER 1.002191,
FÀR 1.000006
10"
10
0.7
20"
67
0.6
06
20
10
10:9
10:1
0.2
02
Layer
Layer
ExxiX]
E
Emi
En
MADXX1
trank60)
NumRarkixi
[x
Accuracy
Table 5: The table showcases the behaviour of different metrics and the classification accuracies for GNNs separately trained on Cora Dataset. This table is an extension figure
27

---

Published as a conference paper at ICLR 2026
GCN LReLU
GCN Tanh
1. 1,750663,
IiR 1.142120
/R 1.436012,
'NR 1.003035
FRR
10
0.
08
07
Me0rc Vake
107
101
10
1-
0.4
10"
16-4
03
10-5
03
10
10-
201
Teaa Laye
Total Layer
GCN LReLU Bias
GCN Tanh Bias
1.526297,
TĂR = 1.045451
TRR 1.752850, VĂR 1.007417
ER
10F
08
N00 0000
10a
101
10a
0.6
10"
03
63
10a4
10
10
207
Traal Laysr
Total Layer
GCN LReLU LayerNorm
GCN Tanh LayerNorm
ERR 1.006405, rin = 1.000009
1.473417, FNR 1.002555
EER
10a
107
08
Ma0ic 0007
10a
101
10a9
0.4
10
0.3
033
10m1
10"
207
10
Tral Laysr
Total Layer
GAT LReLU
GAT Tanh
1.541711, TiR = 1.148837
TR 1.003797,
U 1.000005
ERR
16
0.8
10"
08
107
101
10
044
10"9
0.3
03
10
16-
201
1C1
Thaal Layar
Total Layer
Eaili
F
MADIXI
Eranki3)
NumRankiXi
Table 6: The table showcases the behaviour of different metrics and the classification accuracies for GNNs separately trained on Citeseer Dataset.
28

---

Published as a conference paper at ICLR 2026
GCN LReLU
GCN Tanh
1.420913,
TiR 1.085279
/R 1.838982,
'N 1.014851
FRR
10
0.8
107
08
07
107
101
10
1-
0.4
10"
16-4
03
10-3
03
10
10-
201
Teal Laye
Total Layer
GCN LReLU Bias
GCN Tanh Bias
1.450100, IĂR = 1.207918
TR 1,443576,
FĂR 1.018361
ER
08
N00 C000
10a
101
10a
0.4
10"
63
16a4
10
10
207
Trall Laysr
Total Layer
GCN LReLU LayenNorm
GCN Tanh LayerNorm
ERR 1.002088, rin = 1.000074
1.668289, FNR 1.009275
EER
20a
107
08
Ma0ic 0002
10a
101
10
0.4
10
0.3
033
10m1
10"
207
10
Trall Laysr
Total Layer
GAT LReLU
GAT Tanh
1.577163 TNR 1.119213
ER 1.016096,
FNR 1.000019
ERR
16
0.8
10a
08
Me0i 90N0
107
101
10m
044
10"9
0.3
0.3
10
10-
201
XC1
Thaal Layar
Total Layer
Eaili
F
MADIXI
Erank(3)
NumRankiXi
Table 7: The table showcases the behaviour of different metrics and the classification accuracies for GNNs separately trained on Pubmed Dataset.
29


Reviews:
[{"review_id": "Nxv079GslR", "reviewer": "Reviewer_EXwb", "summary": "This paper argues that dirichlet energy based measures may fail to capture oversmoothing in real application scenarios, so the author introduces rank (numerical or effective rank) as an effective alternative measurement of oversmoothing. The main contributions of the paper are: 1. The authors provide novel rank relaxation functions to measure oversmoothing. 2. The authors compare current oversmoothing metrics with their relaxed rank metrics and demonstrate rank metrics correlate better with performance, 3. The authors theoretically justify the aggregation matrices and nonlinear activation functions will both contribute to the decay of rank, whereas energy-based measures fail to detect oversmoothing. Empirical results shows that the performance drop from GNN is well-correlated with the decrease of rank metric based on comparisons with different energy and rank metrics.", "questions": "What are certain techniques that can mitigate oversmoothing from the view of rank since the authors show ranks metrics are more effective than energy-based metrics. Also, could the rank metric be leveraged as a regulation metric to prevent oversmoothing, or to achieve better performance?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. This paper provides a strong theoretical proof of the decay of the rank as a measurement of oversmoothing.  \n2. Extensive experiments and ablation studies show the rank-based metrics correlate better with performance degradation than energy-based measurement with extension to modern architectures with dropout, residuals, or normalization layers.  \n3. Unlike Dirichlet energy, which only shows meaningful trends for very deep networks, the proposed rank metrics can detect degradation even in moderately deep networks (10 or fewer layers).", "weaknesses": "1. Research of related work part is not sufficient as many previous works have already shown energy-based metric is not a proper or sufficient measurement of oversmoothing. For example, [1] has shown that oversmoothing can be mitigated without explicitly dirichlet energy based control.\n2. The paper shows rank metrics are effective measurement of oversmoothing but does not provide practical techniques to mitigate it using the rank insight.\n\n\n[1] Y. Jin and X. Zhu, \"Graph Rhythm Network: Beyond Energy Modeling for Deep Graph Neural Networks,\" 2024 IEEE International Conference on Data Mining (ICDM), Abu Dhabi, United Arab Emirates, 2024, pp. 723-728, doi: 10.1109/ICDM59182.2024.00083.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JeBeeDFIWo", "reviewer": "Reviewer_BZvC", "summary": "This paper critiques traditional oversmoothing metrics in graph neural networks (GNNs), such as Dirichlet energy, for their limitations in detecting performance degradation in realistic, trained models. The authors argue these metrics only work asymptotically in deep, untrained networks and propose rank-based alternatives (numerical and effective rank) that measure feature matrix collapse to low rank.", "questions": "None", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper's novel critique of widely used metrics is timely and well-supported, exposing flaws like scale-dependence and eigenspace reliance with examples and simplified theorems. Theoretical contributions, including proofs of rank convergence independent of feature magnitude, extend to GCNs/GATs and offer a unifying eigenvector perspective, and some nonlinear analysis. Empirically, the extensive evaluation across homophilic/heterophilic datasets, depths, activations, and ablations robustly demonstrates rank metrics' superiority, with high correlations and methodological rigor (e.g., log-transformed Pearson analysis). This could practically guide deeper GNN designs.", "weaknesses": "The theoretical analysis is restricted to linear/nonnegative models or shared eigenvectors, limiting generalizability to signed graphs or complex activations. Experiments focus solely on node classification, neglecting graph/edge-level tasks or larger/dynamic graphs, and don't prove causation between rank decay and performance. While rank metrics excel in trained settings, the paper somewhat overlooks scenarios where energy metrics succeed (e.g., untrained asymptotics) and lacks discussion of computational costs or hybrid approaches.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hGmuZg8NcA", "reviewer": "Reviewer_eo32", "summary": "The paper argues that existing energy-based oversmoothing metrics for GNNs (Dirichlet energy and a projection variant) are unreliable in trained networks, and proposes rank-based measures (numerical rank and effective rank) as better metrics. The authors present (i) a unifying framework based on eigenvectors of nonlinear activations, (ii) a linear analysis predicting dominance of the leading eigenspace, and (iii) a nonlinear analysis using the Hilbert projective metric that yields rank collapse under nonnegativity and subhomogeneity conditions. The authors conduct experiments on GCNs and GATs across multiple datasets and small depths demonstrate that rank-based measures correlate well with model accuracy, whereas energy-based metrics often remain insensitive to oversmoothing in some realistic training settings.", "questions": "Could the authors explain why effective/numerical rank aligns with accuracy more faithfully than Dirichlet‑style energies in trained networks? In the linear setting, at first glance, Thm. 4.1 and Thm. 5.1 suggest that both energy‑like deviations and rank collapse shrink comparably. How do you reconcile this with the empirical gap you observe? Could you provide intuition or formal statement showing a sensitivity advantage for rank metrics?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper is overall clearly written. It clearly points out the failure modes of energy metrics. For example, it shows these metrics are informative mainly in the limit, can be confounded by scaling, and rely on a fixed dominant eigenspace, which often violated by trained GNNs. It also provides a unifying theoretical perspective for previous results on the oversmoothing phenomena measured by Dirichlet-like energies. The nonlinear case with Hilbert-metric argument are technically interesting.", "weaknesses": "Despite strong empirical results, the paper lacks a theoretical justification that rank‑based metrics dominate Dirichlet‑style energy measures as oversmoothing indicators in trained GNNs. The analysis shows when energies vanish and that ranks can collapse, but offers no formal comparison, dominance result, or sensitivity guarantee for rank.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "392eKTL1Yx", "reviewer": "Reviewer_472b", "summary": "This paper investigates widely used measures of over-smoothing in graph neural networks such as Dirichlet energy and investigates alternative metrics based on the rank of the feature matrix and argues that rank based measures are better for measuring oversmoothing. The paper reviews existing results on oversmoothign and extends them to rank based measures. Finally, the paper experiments on trained GNNs and shows that the rank based measures correlates better with performance on many datasets.", "questions": "1. What does Section 3.2 offer that prior work does not?\n2. For Theorem 5.3 could you maybe expand on “Note that this is implied by the relative metrics [like normalized Dirichlet going to 0], but is actually quite weaker ”. Does the theorem lead to a concrete example where the rank is 1 but the normalized Dirichlet energy or E_proj is not zero? This might help the reader understand in what ways this measure offers a more nuanced perspective.\n2. Table1: What is the sample size used to compute each correlation? Can you report confidence intervals (or p-values) for the correlation coefficients?\n3. Table 1: In many cases standard measures have a strong negative correlation with performance, sometimes even surpassing some of the rank based rank-based measures in absolute correlation. How do you explain this?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. **Significance of problem.** Developing improved tools for understanding the internal mechanisms of GNNs is an important and timely direction.\n\n2. **Insightful discussion of limitations.** The paper provides a thorough critique of standard oversmoothing metrics, clearly articulating where existing measures fall short.\n\n3. **Potential of rank-based metrics.** The proposed rank-based metrics offer a promising way to capture more nuanced behavior than conventional oversmoothing measures, potentially enabling a more fine-grained understanding of representation collapse in GNNs.", "weaknesses": "1. **Lack of relevance** Over-smoothing as a problem has been widely studied, and remedies have been found, the simplest of which is adding a residual connection. This paper proposes a new method to measure a phenomenon that is no longer a problem in practice which weakens its relevance today. This is further shown in Table 4 where little correlation exists between performance and the proposed measures once residual connections are added.\n2. **Lack of novelty** A large portion of the paper discusses already established results rather than focusing on what new insight rank based measures can offer.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)