Geometric Imbalance in Semi-Supervised Node Classification
Authors: Liang Yan
Shengzhong Zhang
Bisheng Li
Menglin Yang
Chen Yang
Min Zhou
Weiyang Ding
Yutong Xie
Zengfeng Huang
Venue: NeurIPS 2025 poster ()
Keywords: Geometric Imbalance
GNNs
Semi-Supervised Node Classification
Primary area: social_and_economic_aspects_of_machine_learning
TL;DR: Propose a novel approach that redefines pseudo-labeling and confidence estimation in the latent space.

Abstract: Class imbalance in graph data presents a significant challenge for effective node classification, particularly in semi-supervised scenarios. In this work, we formally introduce the concept of geometric imbalance, which captures how message passing on class-imbalanced graphs leads to geometric ambiguity among minority-class nodes in the riemannian manifold embedding space. We provide a rigorous theoretical analysis of geometric imbalance on the riemannian manifold and propose a unified framework that explicitly mitigates it through pseudo-label alignment, node reordering, and ambiguity filtering. Extensive experiments on diverse benchmarks show that our approach consistently outperforms existing methods, especially under severe class imbalance. Our findings offer new theoretical insights and practical tools for robust semi-supervised node classification.

Geometric Imbalance in Semi-Supervised Node Classification
Liang Yan', Shengzhong Zhang', Bisheng Li', Menglin Yang', Chen Yang',
Min Zhou", Weiyang Ding', Yutong Xie Zengfeng Huang!
Fudan University Shanghai Innovation Institute *MBZUAI
'Hong Kong University of Science and Technology (Guangzhou) 'Logs AI
Project Page: https://divinyan com/UNREAL Code:
com 1ong///aa///nnn/n11/////////1/
Abstract
Class imbalance in graph data presents significant challenge for effective node classification, particularly in semi-supervised scenarios. In this work, we formally introduce the concept of geometric imbalance, which captures how message pass- ing on class-imbalanced graphs leads to geometric ambiguity among minority-claas nodes in the riemannian manifold embedding space. We provide rigorous theoret- ical analysis of geometric imbalance on the riemannian manifold and propose a unified framework that explicitly mitigates through pscudo-label alignment, node reordering, and ambiguity filtering. Extensive experiments on diverse benchmarks show that our approach consistently outperforms existing methods, especially under severe class imbalance. Our findings offer new theoretical insights and practical tools for robust semi- cumersspeeeieed node classification.
Introduction
Class imbalance is prevalent challenge in real-world graph datasets, where the disproportionate distribution of class labels often hinders the effectiveness and fairness of Graph Neural Networks (GNNs) [73, 37]. While GNNs have achieved remarkable success in various graph- learning tasks [25, 54], their message passing mechanisms can amplify the negative impact of class imbalance, especially for minority classes. This challenge is further exacerbated cemi-supervised settings, where only small fraction of nodes are labeled, making the reliable propagation of information even more difficult.
Self-training (ST) [63, 48], which iteratively incorporates high-confidence pseudo-labeled nodes into the training set, offers a simple yet effective approach for leveraging unlabeled data to address class imbalance. Compared to oversampling [73, 37] or loss function engineering [6, 47], ST avoids the complexities of synthesizing new nodes or edges and thus is naturally well-suited for graph- structured data. However, despite its practical success, existing ST-based work for imbalanced node classification [75, 71, 27] lacks a principled thcoretical and empirical understanding self-training in imbalanced node classification scenarios, especially regarding its inherent limitations under severe imbalance.
In this work, for the first time formally introduce and analyze the notion of geometric imbalance in the context semi cemi-supervised imbalanced node classification. We rigorously characterize geometric imbalance on riemannian manifold based on the Mises-Fiste (vMF) distribution, and establish its direct relationship with prediction uncertainty and class imbalance ratio. Our theoretical results show that message passing, especially class-imbalanced graphs, induces geometric ambiguity
"Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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in the embedding space, making minority-clase nodes particularly vulnerable to unreliable pseudo- labeling. This previously overlooked phenomenon motivates the need for targeted solutions beyond existing self-training strategies. It is worth noting that our proposed concept of geometric imbalance fundamentally distinct from previously studied topology imbalance [6. 37J. Specifically, geometric imbalance is defined in the riemannian manifold node embedding space, whereas topology imbalance is defined on the original graph structure. Moreover, topology imbalance focuses on the influence of labeled minority nodes in the gressage-passing process on the raw graph, while our work emphasizes the positional ambiguity of unlabeled nodes in the riemannian manifold space (A detailed discussion of this distinction can be found in Appendix G.).
Building on this foundation, we propose a unified and modular framework that explicitly addresses geometric imbalance through three complementary components: (1) DualPath PseudoLabeler that aligns clustering and classification perspectives for more reliable pseudo-labels; (2) a Node- Reordering strategy that fuses geometric proximity with classifier confidence; and (3) a lightweight mechanism for discarding geometrically ambiguous nodes. Extensive experiments on nine benchmark datasets- including both synthetic and naturally imbalanced settings demonstrate that our frame- work consistently outperforms state-of-the-un baselines, especially under severe imbalance. Detailed ablation and sensitivity studies further validate the effectiveness of each module. Our contributions advance the theoretical understanding of geometric imbalance in GNN-based self-training and offer a robust solution for real-world semi-supervised imbalanced node classification.
2 Preliminaries
Notations and Task Definitions. In this work, we focus on the problem of semi-supervised imbal- anced node classification (SINC) on an undirected and unweighted graph. denoted as G = (V,E, L). Here, V represents the set of nodes, & denotes the set of edges, and L is the set of node labels. The la- beled nodes form a subset Dilbel c V. while the unlabeled nodes are denoted as Dunlabel > Dlabel The feature matrix X E _add contains node features. Here, n = | is the number of nodes, and d is the feature dimensionalityy To represent the graph structure, we use the adjacency matrix A E (0. 1 "xn" For each node v, its I-hop neighbors are denoted by (v).. Nodes in the labeled set are divided into C' classes, denoted as C2..C..... Cc) with the number of nodes in each class (ax:( (i.) where E 21, C). The objective is to design robust model fa that effectively utilizes the imbalanced labeled set to perform accurate and fair node classification. Table 8 in Appendix A summarizes all notations used throughout the paper.
represented as (n1, P2.......... ncc.) To quantify class imbalance, we define the imbalance ratio as
Message Passing Neural Networks. Message Passing Neural Networks (MPNNs) form the foundation of many graph-based learning tasks. An MPNN typically comprises three compo- nents: a message function my an aggregation function 6;, and node feature update function 4/z. At each layer, the features of node v are updated. Let !!!) represent the feature of node v at the 1-th layer. Then, the feature update for the (1 + 1)-th layer is given by n (1+1) ((e(()))) where (w) denotes the set of neighbors of
thr (n,), Ot (fmy/ (119), ,e.))))))))))) | | node v, and cr represents the edge weight between and é. To complete the node classification task, a classification layer, typically implemented as fully connected or softmax layer, is appended after the final GNN layer to map the node embeddings class probabilities.
Graph Self-Training Paradigm. In graph semi-supervised learning [25, 54J, graphs encode re- lationships between data points as edges, enabling information sharing across the graph structure. This facilitates the propagation of labels from labeled to unlabeled nodes. Self- Self-training meth- ods [26, 63, 48] align well with this framework by iteratively incorporating pseudo-labela data into the training process, thereby expanding the labeled set and improving the model's performance on un- labeled data. The self-training pipeline can be summarized as follows. Let Dlabel = {(w2 (o"))) ()))} Ieeae
denote the labeled dataset and Dunlahel = {u') ]== | the unlabeled dataset. Initially, a GNN fo is trained on Dlabel At each iteration t, the model predicts labels jó for nodes in Dunlahel, A subset of pseudo-labeled nodes Meeeadd {(w), } selected based on prediction confidence (c.g.. selecting predictions with probabilities above a predefined threshold), is added to Dlahel The model is then retrained iteratively with the augmented labeled set, gradually incorporating more pseudo-labeles data in each step.
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3 Geometric Imbalance
In this part, we provide a novel theoretical analysis for SINC and introduce the geometric imbalance problem induced by class imbalance through message passing.
Theoretical Background. We conduct the analysis on the unit hyperspher (a simple riemannian man- ifold) because GNN embeddings by message passing are typically normalized before classification to ensure stability and improve training dynamics [25, 29]. This normalization maps embeddings onto a riemannian manifold, which is consistent with the assumptions of von Mises-Fisher (vMF) distribution [14]. The riemannian manifold further allows us to rigorously define and analyze geo- metric imbalance in terms of angular distances and class separation [28, 55]. Following the notation introduced in Section 2, the class prior distribution is given by Jabel (i) =
ni Given a node v
sampled from the node set V, the feature vector h. is extracted by a message passing neural network. Specifically, we adopt simple message passing neural network, which computes node embeddings iteratively. For node v, its representation after one layer of GNN message passing can be written as h =W (EneNti) Gnch )),, where W E Raxdd is the trainable weight matrix at the I-th layer, a()) denotes a nonlinear activation function (e-g.. ReLUU,, and ave is the edge weight, representing the contribution of node to node 0. Through message passing, the representation ! is a weighted combination of information from its neighbors.
The embeddings 1h are then projected onto the riemannian manifold gd-1 using 5. A
and subsequently fed into a von Mises-Fisher (vMF) classifier 2 The latent representations corresponding to the C' classes are modeled as a mixture of C' von Mises-Fisher (vMF) distributions defined on the riemannian manifold $d-1 Each class is parameterized by compactness K; E R and a unit orientation vector jı; E Ixx The vMF classifier is defined as M), where K. = fKi...., and M M (ja,.... ... are the sets of learnable compactness and orientation vectors for the C' classes, respectively. The probability density function (PDF) of the i-th class, denoted p. R
|
as p(h (h [K;, (b)), is given by p(h!") |N,, j.,)) Ca(Ks)e
- is where Ca(k) (2x) (e)
the normalization constant for the d-dimensional von Mises-Fished distribution, ensuring that the density integrates one. Here, d is the dimensionality of the embedding space, and I, (x) denotes the
modified Bessel function of the first kind of order r [23]. Using Bayes' theorem [21]. the posterior probability p(y'
ilhh)) (0) is:
plabel (i) ((( Kig pa)) Σド= PD (j) p(ho [Rj. pbj)
p ply" =(hh9)
If v E DDunlahell the posterior probability pt, also represents the classifier's confidence in assigning the pseudo-label to the unlabeled node v. For node u² in the unlabeled set Dunlabel the information entropy of the predictions for u' is defined as H(²)) = = -E pe log pi,, and the average entropy | Hww)).
over all unlabeled nodes is A = Dunske
To further analyze the uncertainty introduced by class imbalance through message passing in the riemannian manifold embedding space. introduce the notion of geometric imbalance. This concept captures the ambiguity of a node's embedding position with respect to class centers on the hypersphere, especially for minority class nodes. Intuitively, a node that is nearly equidistant from multiple class centers is geometrically ambiguous and more likely to be assigned a wrong pseudo-label. We formalize this phenomenon by defining e-geometric imbalanced sample and provide a theoretical connection between geometric imbalance and prediction uncertainty in terms of entropy. The following definition and theorem establish the quantitative foundation of our analysis.
Definition 1 (e-Geometric Imbalance the Riemannian Manifold). To characterize directional ambiguity in hyperspherical embeddings obtained by message passing, we define the geometric imbalance score for unlabeled nodes as follows. Let w² E Dunlabel be unlabeled node with
This choice is motivated by the fact that the von Mises-Fishee (vMF) distribution is natural analogue of Gaussian distribution on the riemannian manifold. It has been widely used to model directional data and has shown effectiveness in normalized embedding spaces [14, 1J. Compared to Euclidean Gaussian modeling, vMF offers better geometric interpretability when embeddings are fz- n--ormalized.
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(9) Cors-GON (Case 1)
[6] Com-BCN (Case 2]
[d] Cona. MLP (Case 3)
[6] Com-MMLP (Cass 4)
Figure 1: Illustration and quantitative analysis of Geometric Imbalance (GI). (a) Conceptual illustra- tion of GI under different pretrained and fine-tuned settings. (b)-(e): t-SNE visualizations of node embeddings in the Cora dataset under four representative cases, showing intra-class compactness and inter-class separation patterns. (f)-(g): Quantitative relationships between GI and (f) average entropy, and (g) class imbalance ratio.
normalized embedding 6 E S produced by the 1-th layer of GNN, Let {ūe E 3d-11C I--1 denote the class center embeddings on the hypersphere for C classes, and let yd E (l,...,00 0). be the ground-trute truth label of w². We define the geometric imbalance score of u² as:
G(w)):= = Ecsccs llãc, 122,1?"
We say that us is an c-geometrically imbalanced sample if G((W) > e, where e >> 0 is user-defined threshold.
We define V=
yo Crnino) the set of all nodes whose ground-truth labels belong minority classes, where Cminor C {1, (1,..., C} denotes the set of minority class indices. The following theorem establishes direct relationship between geometric imbalance and the prediction entropy of pscudo-labels. It shows that higher geometric imbalance is associated with greater uncertainty, especially among minority-class nodes.
Theorem 1 (Geometric Imbalance Information Entropy). Let Dunlabel C V be the of unlabeled nodes, minnor == {u E V Yo E } denote the set of all nodes whose ground-truth labels belong the minority class set Caiinor (l..... ......C), C). Define the intra-class compactness and inter-class separation as: Dinma
Σ
7. - ir jryes
Dimer
Σ llec -Acal1 C17C2
W E F
Then the average information entropy A of pseudo-label predictions over Dumlabel satisfies A. x Damm i.e., the expected prediction uncertainty increases with greater intra-class compactness and decreases with greater inter-class separation. The proof is in Appendix C.
Definition and Theorem offer quantitative lens through which understand geometric im- balance. Specifically, we define the intra-class compactness of minority class nodes as D
Eu V.VamnnDDriaa 4J Ae. and the inter-class separation as Dinter Σc,#c, llãe, Ac. 2
From Theorem 1, it follows that reducing Dimm or increasing Dinter leads to decrease in average entropy H. thus improving pseudo-label reliability. This underscores the importance of mitigating

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geometric imbalance in embedding space. Furthermore, we investigate how the degree of geometric imbalance is affected by the class distribution in the training set. The next theorem shows that the geometric imbalance metric grows with the class imbalance ratio.
Theorem 2 (Imbalance Ratio vs. Geometric Imbalance). Let denote the class imbalance ratio, defined as the ratio between the number of samples in the majority class and the number in the minority class. Let the geometric imbalance of unlabeled minority nodes be measured by:
Gminor
Σ
G(u'),
Vminer
Dunlabel
WI Valual DDiiiaaee
where G(w)) is the geometric imbalance score as defined in Definition 1. Then, under fixed feature extraction and class centers, the expected geometric imbalance Gminor increases monotonically with the class imbalance ralio pr Guinor X p. The proof is in Appendix D.
This result characterizes the empirical observation that datasets with greater class imbalance exhibit higher average geometric ambiguity among minority-clast unlabeled nodes, highlighting the structural difficulty in correctly pseudo-labeling minority nodes when training data is skewed. Importantly, this theoretical formulation is particularly relevant to GNNs, where message passing mechanisms tend to amplify the effects of class imbalance, resulting in more pronounced geometric confusion for minority- class nodes compared to non-graph-basee models such as MLPs. As we will later demonstrate, MLPs-lacking structural aggregation- are far less affected by class imbalance in this manner. Together. Theorem I and Theorem 2 highlight the dual challenges posed by geometric ambiguity and class imbalance in semi-supervised GNN settings. These insights motivate the development of principled approach to detect and mitigate geometric imbalance, which we introduce in the next section.
Empirical Illustration. The above theoretical analysis is visually and quantitatively substantiated in Figure 1. Figure I(a) presents a conceptual overview of geometric imbalance under different pretrain- ing and fine-tuning scenarios for both GNNs and MLPs. Figure l(b)-c) use t-SNE visualizations to directly contrast the evolution of node embeddings: for GNNs (Case and 2), fine-tuning on imbalanced data leads to substantial increase in intra-class dispersion and a reduction in inter-class separation for minority classes, resulting in pronounced geometric imbalance. In contrast, MLPs (Case 3 and 4) preserve compact and well-separated clusters, showing minimal change regardless of class imbalance. This empirical contrast highlights that geometric imbalance a distinctive and aggravated issue in GNNs, primarily due to their message passing mechanisms, while it remains marginal in MLPs. Furthermore, Figure I(f) and (g) quantitatively validate our theoretical results, demonstrating a strong positive correlation between geometric imbalance and prediction entropy (as predicted by Theorem 1), as well as a monotonic relationship between geometric imbalance and the class imbalance ratio (as characterized by Theorem 2). Detailed experimental setups and result analyses for Figure 1(b)-(e) and Figure ((f)-(g) are provided in Appendix E and F, respectively. Taken together, these empirical observations in Figure/ 1 not only substantiate our theoretical claims but also underscore the need for GNN-specific mitigation strategies, which we propose in the following section.
4 Method
A straightforward strategy mitigate geometric imbalance is to compute the geometric imbalance score for each unlabeled node and discard those with high scores during self self-training. However, this approach suffers from two major limitations: (1) it computationally expensive, as it requires evaluating pairwise angular distances between each unlabeled node and all class centroids; and (2) it relies ground-truth labels to determine which class node should be close to, or alternatively, depends on potentially noisy pseudo-lahels-mmkiin the approach unreliable and indirect in practice. To address these challenges more efficiently and effectively, we propose unified and modular framework, UNREAL, named to reflect its nature as a pseudo-labeling algorithm. Our UNREAL framework mitigates geometric imbalance through three flexible and complementary components: (1) DualPath PseudoLabeler that enhances pseudo-label quality via alignment between clustering and classification; (2) a pode-Reordering mechanism that jointly considers geometric proximity and classifier confidence; and (3) Discarding Geometrically Imbalanced Nodes (DGIS) module that filters out samples with ambiguous geometric positioning. Notably, DGIS can be viewed as a lightweight, approximate alternative to direct geometric imbalance scoring, enabling scalable filtering without requiring true labels or dense computations. These components can be used independently or in combination, allowing the framework to adapt to various scenarios and computational budgets.
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Figure 2: The pipeline of our UNREAL framework.
The overall pipeline is illustrated in Figure 2, and we elaborate on each component in the following subsections.
4.1 Mitigating Geometric Imbalance with Pseudo-label Alignment
Traditional node pseudo-labeling methods rely on embeddings generated by a graph encoder, which finaly fed into a classifier to get pseudo-labels. However, the geometric imbalance problem address originates in the embedding space itself, creating a mismatch between mitigating geometric imbalance and improving pseudo-label generation. To address this, we propose novel approach, the DualPath Labele,,, as illustrated in Figure 2. This method generates pseudo-labels via two pathways: (1) unsupervised clustering (using k-means segment the embedding space) and (2) supervised classification (using a dual-layer MLP for node prediction).
DualPath PseudoLabeler Overview. Let the embeddings of labeled nodes are represented as Helabel E R|pidl|xd and the embeddings of unlabeled nodes as Humlabel E TR| |R|Dimie xd Each row of these matrices represents node embedding, denoted h2 hus which corresponds to point in the d-dimensional Euclidean space. (1) Unsupervised Clustering. We apply k-means clustering" to the embeddings of unlabeled nodes Humlabel partitioning them into K' clusters, where K C and denotes the number of classes. This clustering process produces cluster centers and cluster assignments: feluser (7 (?nunla + /A1,HKc, K2, JK2..... KK MrC,. }, where K.; is the i-th cluster and μic, is its center. For labeled nodes, we compute the embedding center for each class: Me., mean(fh, v² c.)), where mean(.) is the mean function. We assign pseudo- label to the cluster K. by minimizing the distance between cluster centers and class centers as n' arg min, distance(μc, p..)). Nodes with the same predicted label grouped into sets ü, such that Dunlabel = - U. (2) Supervised Classification. Simultaneously, the GNN model
Although embeddings in our model are approximately distributed on a hypersphere, we employ K-Means clustering for its computational efficiency and robustness (See Appendix J.6.3 for extended discussion.).
"Thi overclustering strategy allows for finer-grained partitioning of the embedding space, which is particu- larly beneficial imbalanced or geometrically entangled regions. By assigning more clusters than classes, we aim to capture local structures within each class and reduce the impact of overlapping or ambiguous boundaries. The strategy (K >> C') has been commonly adopted in deep clustering and seeudolabeling to improve separation in ambiguous regions [20, 53, 18, 10, 36].

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generates predictions 5 for each node in Dunlahel Nodes with the same predicted label i are grouped
into sets 24, such that Dunlaael = - U4
Dual Pseudo-label Alignment Mechanism (DPAM). To address the geometric ambiguity of clustering-basee pseudo-labels and the majority bias of classifier predictions, DPAM aligns both sources by retaining only nodes with consistent labels from clustering and classification, i.e., 4ffinal = U4 24, This intersection acts as a filter, improving pseudo-labe reliability and mitigating geometric imbalance, while remaining broadly applicable to different clustering and classification models.
4.2 Node-Reordering
While classifier confidence is often used for pseudo-label selection, it does not explicitly address geometric imbalance expecially in early training stages where node embeddings may overlap or drift between class regions. On the other hand, geometric proximity to class centroids provides valuable structural information, but lacks semantic certainty. To reconcile these two perspectives, we propose aode-Reordgring (NR), a dynamic ranking strategy that adaptively fuses geometric and confidence-based rankings for more robust pseudo-label selection.
Ranking Definitions. We define two types of rankings over the candidate pseudo-labeled nodes U4 nu:
Definition 2 (Confidence Rankings (CR)). For each node the classifier outputs confidence score confidence(u) = max (softmaax (sxootnaax(logits.))) where logits, is the output vector of the final classification layer. Nodes then ranked in descending order of their confidence scores, yielding per-class rankings {T1, ... ....
Definition 3 (Geometric Rankings (GR)). For each node u, let pe, denote the centroid of class i. The geometric distance is defined as óμ = distance (hu: He.), where hy is the embedding of node u. Nodes are ranked in ascending order of their distances, forming the geometric rankings (2,,S2...,,S))
Fusing Rankings via RBO. To combine the two rankings, we adopt Rank-Biased Overlap (RBO) [62], metric that measures the agreement between two ranked lists, assigning higher weights top- ranked elements. For each class m, we compute the similarity score rm = RRBO(Sm. Tm) between its geometric and confidence rankings. Weighted
Based on the computed RBO score, we construct fused ranking New by adaptively weighting the two sources, N/W max{Tm, 1-rm +min 1-rm J J.Tm This formulation ensures that when geometric and confidence rankings diverge (i.e., low rm the dominant weight is assigned to the geometric perspective, which is more reliable in early stages (Detailed ablation results are shown in Appendix J.3 and J.4.). As training progresses and r' increases, the classifier's s confidence gradually plays a greater role in node selection. Theoretical Justification. The following theorem characterizes the contribution of geometric rankings the fused list: Theorem 3. The proposed ranking New guarantees that geometric rankings dominate the node selection process when the disagreement between confidence and geometry is high, thereby reducing the effect of geometric imbalance in early iterations. The proof is provided in Appendix H.
Empirical Dynamics. We empirically observe that as training proceeds, the similarity between CR and GR improves. Figure 3 illustrates the evolution of RBO values over iterations, confirming that Sn and Tm become increasingly aligned. This observation supports our design: geometric criteria dominate early on to stabilize selection, while confidence scores gain influence as the classifier matures. Additional analyses are provided in Appendix J.4.
4.3 Mitigating Learning Bias by Discarding Geometric Imbalanced Samples While the previous components (DPAM and NR) effectively alleviate geometric imbalance by enhancing pseudo-label quality and selection, a more direct strategy is to explicitly identify ge-
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ometrically imbalanced nodes based on the for-
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from training. Although intuitive, this approach incurs high computational cost if applied naively to all unlabeled nodes, as it requires evaluating distances to multiple class centroids. Moreover, depends on reliable pseudo-labele or ground- truth labels to determine geometric deviation, which may not be available or accurate during early
0.20
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Figure 3: Fluctuation of RBO values between rank-
ings as iterations progress.
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training stages. To overcome these limitations, we defer this operation until after DPAM and Node- Reordering have been applied, thereby reducing the candidate pool and improving label quality. In practice, we adopt simplified and computationally efficient version of the geometric imbalance score-serving as a lightweight surrogate for full hyperspherical analysis.
Discarding Geometric Imbalanced Samples (DGIS). We define a Geometric Imbalance (GI) index to approximate the degree of ambiguity in a node's 's embedding. Let 6, be the distance between node u's embedding and its closest class centroid, and let 3. denote the distance to its second-close centroid. Since 8, S Bu. a smaller gap between them indicates that the node is nearly equidistant from multiple centroids, and thus more likely to be geometrically ambiguous. The GI index is defined as GI S. _. Nodes with lower GI indices are more prone to geometric imbalance. We therefore introduce a threshold on the GI index and discard nodes whose values fall below this threshold. This filtering mechanism allows us to eliminate geometric imbalanced samples with minimal computational cost, further reducing learning bias and improving model robustness.
5 Experiment
In this section, we conduct comprehensive experiments to address the following research questions: (RQ1) How does model perform under varying levels of class imbalance? (RQ2) How does it perform on datasets with naturally occurring label imbalance? (RQ3) Is the model scalable and effective on scale datasets with severe imbalance? (RQ4) How sensitive is the model to its key components and hyperparameters?
5.1 Experimental Setups
(I) Datasets. We conduct evaluations under various benchmarking settings on 8
[68],
Citeseer [68], Pubmed [68], Amazon Computers [43], Computers- RRantermr---dnnn [67], CS-Random [67], Flickr [69], and Ogbn-arxiv [17]-capturing a wide range of class imbalance scenarios. To answer RQ1, construct varying levels of class imbalance (p = 10, 20, 50, 100) settings on four datasets (Cora, Citeseer, Pubmed, and
following the methodology from [73, 37, 47]. Specifically, we designate half of the classes as minorities and convert portion of labeled nodes into unlabeled ones to achieve the desired imbalance ratio (p). For the citation networks (Cora, Citeseer, and Pubmed), we use the standard splits from [68] to create imbalance settings with = 10 and = 20. For more extreme imbalances (p = 50 and 100), which require more labeled nodes per class, we adopt random splits. For Amazon- ComnzonnComputes, we generate splits with varying degrees of class imbalance (p = 10, 20, 50, 100) based on the procedure in [67]. To answer RQ2 and RQ3, we extend our exper- iments to naturally imbalanced and large-scale datasets, Computers- Random, CS-Random, Flickr, Ogbn-arxiv, reflecting real-world con- ditions. Here, random sampling is employed to construct training sets that reflect the original label distributions for Computers-Ranne Randted and CS- AS-Random, following the protocol in [67]. For Flickr and Ogbn-arxiv, we adopt their publicly available splits, as the settings are inherently highly imbalanced. Appendix K details our ex- perimental framework, including label distribu-
Table 1; Ablation analysis for UNREAL.
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tions, evaluation protocols, and algorithm implementations. (II) Baselines. We evaluate our frame- work against several classic techniques and state-of-the-art methods for addressing imbalanced node classification, including GraphSMOTE (GS) [73), GraphENS (GF) [37], ReNode (RN) [6]. TAM [47], GraphSR (GSR) [75], BIM [71]. We also compare our method with cross-entropy loss with Re- Weighting (RW) [19], PC Softmax (PS) [16], and Balanced Softmax (BS) [40] GS and GE are representative over-sampling methods, while RN and TAM modify the loss function. GSR and BIM are state-of- sfute-of-the-eee the- self-training based models for imbalance node classification. We also test the performance of TAM when combined with different base models, including GE, RN, and BS, as described in [47]. See Appendix K.5 for implementation details of the baselines. (III) Evaluation. We evaluate the performance of our method on several mainstream GNN architectures, including GCN [25], GAT [54], and GraphSAGE [12]. We report the averaged balanced accuracy (bAcc., %) and Macro-F1 score (%), along with the standard errors over repetitions on the GNN architectures. The reported metrics include: balanced accuracy (bAcc.) and Macro-Fl (F1).
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Table 2: Experimental results of our method and other basclines on three -aassiimallanced node classification benchmark datasets with
Table 3: Experimental results of our method and other baselines on three class-imbalanced node classification benchmark datasets with
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Table 4: Experimental results of our method and other baselines on naturally imbalanced setting Computers- Random (p R 17.7).
Table 5: Experimental results of our method and other baselines on naturally imbalanced setting CS- Rannond (p x 41.0).
GCN
Metiic
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98.70.413
83.55
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125
20.50 11
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71.88pmm
67.58
32.69093
48.77-11
65
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24.85
35.6
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57 Manes
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68.35.xx
73.00.10s
68:71:10
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5.2 Experimental Results and Analysis
(1) For RQ1: Results Under Varying Levels of Class Imbalance (Table 2, Table 3, Ta- ble 20, Table 21, Table 22, Table 23). We evaluate our method under different class imbal- ance ratios (p- 10. 20, 50, 100) across multiple benchmark datasets and GNN architectures. As shown in Table 20-23, our method consistently outperforms all baselines in both balanced accuracy and Fl score, regardless of the imbalance severity. The performance advantage re- mains stable even as the imbalance ratio increases, demonstrating the robustness of our method under skewed label distributions. In contrast, some baselines suffer from drastic drops in per- formance under severe imbalance or even become inapplicable due to data sparsity in minority classes. These results highlight the effectiveness and generalizability of our method across di- verse imbalance settings. Owing to space limitations, we include partial results from Table 20 and Table 23 (aligned with Table 2 and Table 3) in the main text as representative examples. (2) For RQ2: Results On Naturally Imbalanced Datasets (Table 4, Table 5). Additionally, we val- idate our model on three intrinsically imbalanced datasets: Computers-Randon (p R 17.7) and CS- Random (p 41.0), where the unlabeled data also exhibits imbalance (refer to Table 34). The compo- sition of the training. validation, and testing sets is detailed in Appendix K. We present the experimental results in Table 4 and Table 5. More importantly, on these two datasets, our method consistently outper- forms other approaches. (3) For RQ3: Results On Large-Scale Naturally Imbalanced Datasets (Table 6, Table 7). We further evaluate our method on two large-scale datasets with naturally occurring class SiaallnceeFFicke and Ogbn-arxiv. As shown in Table 6 and Table 7, our method achieves competitive or superior performance across all GNN backbones. Despite the high imbalance ratio (especially in Ogbn-arxiv), our model maintains stable performance and outperforms existing methods in most cases. Notably, several baselines encounter out-of-memory issues, while our method remains computationally feasible. These results highlight the scalability and practicality of our approach in real-world imbalanced settings. (4) For RQ4: Ablation Studies and Sensitivity Analysis (Table 1, Figure 4). Here we investigates the individual contributions of the key components within our method. We examine three ranking strategies CR. GR, and GR-as well as the impact of the DGIS module, which targets geometrically imbalanced nodes. As shown in Table 1, the combination of NR and DGIS consistently achieves the best F1 scores in three out of four settings, highlighting the effectiveness of our design. We also conduct a sensitivity analysis on two key hyperparameters: the number of clusters k' in K-Means and the
715
15.0
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Threanol of Ozi
(a) Sensitivity perfor- mance of K² of K- Means.
(b) Sensitivity perfor-
mance of the threshold ? of DGIS.
Figure 4: Sensitivity analysis.

---

Table 6: Experimental results of our method and other baselines on large-scale naturally imbalanced setting Flickr (p R 10.8).
Table 7: Experimental results of our method and other baselines on large-scale scale naturally imbalanced setting Ogbn-arxiv (p R 775.4).
Model
GCN
GAT
SAGE
Mivel
CCN
GAT
SAGR
Metrc
bAkc:
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bass
Metii
Mos.
Marila
24.52
25.87
25.32
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16.11 16
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27.63
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23.53
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threshold y in DGIS, as illustrated in Figure 4. Model performance tends to stabilize when k' is sufficiently large, but degrades sharply when k' is too small, likely due to increased pseudo-label noise from coarse clustering. Similarly, performance remains stable at low values of y suggesting that overly conservative filtering risks discarding useful nodes. In contrast, excessively high y introduces substantial noise into the training process.
Additional Analyses in the Appendix. Appendix J.2 presents experiments verifying the effectiveness of DPAM, while Appendix J.4 details further analyses on Node-Reordering and DGIS. The variation in RBO similarity across iterations is discussed in Appendix J.3. Moreover, we provide extensive discussion and analyses on geometric imbalance, self-training. and our framework, please refer to the appendix for details.
6 Related Work
Class imbalance in graphs poses unique challenges due to the interplay between topology and feature distribution [44, 61, 73, 31, 39, 6, 37, 47, 67]. Existing methods can be broadly categorized into three lines of work: (1) Node generation methods, which synthesize nodes to balance class distributions. For example, GraphSMOTE [73] interpolates minority class embeddings and predicts new edges; ImGAGN [39] jointly generates node features and topology by modeling global minority distributions; and GraphENS [37] enhances diversity via ego-network mixing. While effective, these methods often incur high computational cost and struggle to ensure structural consistency across generated nodes. (2) Topology-aware adjustment methods exploit structural priors to refine model training. ReNode [6] rewcights nodes based on their topological distances to class decision boundaries, while TAM [47] calibrates logits using local topology and class statistics. However, such approaches rely heavily on labeled nodes to infer structural bias and lack a general framework for quantifying or mitigating topological imbalance. (3) Self teeiiitrainnn methods attempt to exploit unlabeled nodes in imbalanced settings by generating labels to supplement minority classes. GraphSR [75] combines similarity-bait based filtering with reinforcement learning to select reliable useudo-labeled nodes; BIM [71] frames the task as an influence maximization problem to balance class influence across receptive fields; and IceBerg [27] proposes a Double Balancing mechanism that simultaneously adjusts pseudo-label distributions and loss calibration, while disentangling GNN propagation for better supervision in few-shot regimes. Despite their promise, these methods often rely on heuristic selection strategies and lack theoretical guarantees or standardized metrics for pseudo-label quality, which may lead to instability under severe class imbalance.
7 Conclusion and Future Work
In this work, present self-training framework that addresses geometric imbalance in graph- structured data under imbalanced settings. By redefining pseudo-labeling and filtering unreliable nodes, method improves performance on real-world benchmarks and outperforms existing self-training techniques. Future work includes extending the model to imbalance problems in link prediction tasks.
Acknowledgements
This work was supported by National Natural Science Foundation of China No. U2241212, No. 62276066. W. Ding was supported by the National Natural Science Foundation of China (No. 12471481, U24A2001), the Science and Technology Commission of Shanghai Municipality (No.
10

---

232R 15ZR1403000) and the Open Foundation of Key Laboratory Advanced Manufacturing for Optical Systems, CAS (No. KLMSKF202403). Liang Yan would like to thank Xun Qian from Shanghai AI Lab for his very valuable discussions and insightful comments.
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14

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Appendix
Table of Contents
Notations
B Comprehensive Related Works
Proof Theorem
20
Proof Theorem
Visualization Analysis of Geometric Imbalance
23
Experimental Setup
23
Results Analysis
23
Analysis of Geometric Imbalance under Varying Class Imbalance Experimental Sctup
25
E1
25
Quantitative Analysis
25
Distinction Between Geometric Imbalance and Topology Imbalance
27
G.1 Geometric Imbalance
27
G.2 Topology Imbalance
27
G.3 Summary of Differences
27
Proof Theorem 3
28
Comprehensive Experimental Results
29
L1 Comprehensive Results Under Varying Levels of Class Imbalance
29
1,2
Comprehensive Results on Natural Imbalanced Datasets
29
1.3
Comprehensive Results On Large-Scale Natural Imbalanced Datasets
29
1.4
Experimental Results of Our Method with Less Training Rounds
30
1.5
Adapting Our Framework to Imbalanced Image Classification
30
1.6
Analysis of Computational Complexity
31
1.7
Comparison of UNREAL and Recent Soft Pseudo-Labeling Methods
31
18
Experimental Results on Imbalanced Heterophilous Graphs
32
I.9
Comparison to General Self-Training Methods
33
J
Comprehensive Abaltion Study
41
J.1
Analysis for Decoupling Representation and Classifier for Imbalance Node Classi- fication.
41
J.2
Detailed Analysis for DPAM
42
J.3
Detailed Results and Analysis about Fluctuation of RBO Values in Node-reordering
43
J.4
Detailed Analysis for Mode-Reond and DGIS
44
J.5
Detailed Ablation Analysis
44
J.6
In-Depth Comparison between Self Self-nraining and Our Method
44
K
Experimental Setup
50
K.1
Imbalanced Datasets Construction
50
K.2
Details of GNNs
50
K.3
Evaluation Protocol
51
K.4
Implementation Details
51
K.5
Baselines
51
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K.6 Configuration
51
L Pseudocode of Our Algorithm
53
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A
Notations
Table Notation table used throughout the paper, categorized by roles.
Indices
Number of nodes, n = [V].
Dimension of node features.
Number classes.
Indices of laheled and unlaheled nodes.
Index of self training iterations.
Index of GNN/MPNN layers.
R,
Nodes in the graph.
K'
Number of clusters k-means clustering (K'
Parameters
(V,E,)
Undirected and unweighted graph: nodes V, edges and labels C.
E
Node feature matrix.
AE (0, 1p
Adjacency matrix of the graph.
N(v)
Set hopop neighhors of node Labeled node set: Disbal Unlaheled node set: V (Ttaae
Ditke
D.pess
Pscudo-labeled nodes added iteration
Nodes labeled/onlabeled datasets. Ground truth label of node vf. Predicted label node of by model. Number of labeled nodes in class 1.
y
8¹
Imbalance ratio, (ax, mingtry?
6v.b
Edge weight berween nodes and 0.
do
Degree of node including sell lolp.
6'
Trainable parameter matrix the 7-th GCN layer. Feature node at layer fg normalized embedding of node on the unit hypersphere. Compactness parameter of class vMF distribution. Unit mean direction vector class the hypersphere. Normalization constant in vMF distribution. Posterior probability of node being class i. Entropy of prediction over all classes unlabeled node Average prediction entropy over all unlabeled nodes.
659
P.
Cafes)
ph
H(r))
i
Vance
nodes with labels minority class.
Cune
minority class indices.
Dam
Intra-class compactness: of squared distances to class centers. separation: sam squared distances between centers.
C(w²)
Geometric imbalance score of node w/.
Grinz
Average geometric imbalance for minority unlabeled nodes. Distance
&
centroid.
Bu
Distance from node to its second-closest class centroid.
GL.
Geometric Imbulance index: Gl. 8a-6.
Functions and Sets
fo
GNN model parameterized by
me(-)
Message function layer
8c(-)
Aggregation function at layer I.
sx(-)
Node update function layer 1.
Embeeliing labeled unlabeled
Pscudo-labeled sets from classifier and clustering for class Intersection of classifer and cluster predictions for class Cluster from unsupervised clustering. Center of cluster K.,.
MAc,
Embedding center labeled nodes in class
Sm.
Geometric and confidence rankings for class m.
Rank- Biased Overlap (RBO) score between S., and Ti.
Fused node ranking for class RBO-weighted fusion.
17

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B Comprehensive Related Works
Imbalanced Learning in General Machine Learning, Real-world datasets often exhibit inherent class imbalances, making it challenging to train fair models that avoid bias towards majority classes. To address this issue, variety of approaches have been developed. Ensemble learning methods [1I. 30, 74, 59, 33, 3] combine the outputs of multiple weak classifiers to improve overall model performance imbalanced data. Data resampling techniques [5, 13, 45, 42, 22, 56] address im- balance by synthesizing or duplicating minority class samples to modify the label distribution in the training set. A third category of solutions involves modifying the loss function, where class- specific weights or adjusted class margins are used to emphasize minority classes during training [76, 51, 4, 50, 65, 40, 57]. Also, post-hoc correction methods [22, 52, 35, 16] aim to mitigate imbal- ances during the inference stage, after model training is completed. Finally, there is also some work focusing on developing effective generative models [2, 34] under class imbalance situations [38, 66]. While these techniques have demonstrated effectiveness in i.i.d. data scenarios, extending them to graph-structured data introduces unique challenges due the interdependence between nodes and the graph topology.
Pseudo-labeling Methods in Graph Learning, Recent research has predominantly focused on leveraging pseudo-labeling techniques to train graph neural networks (GNNs) when confronted with limited labeled data. Notably, co-training [26] has emerged a prominent approach that utilizes Parwalks [63] to generate confident pseudo-labels, thereby facilitating the training of GNNs. Conversely, self-training [26] expands the label set by acquiring pseudo-labels from previously trained
GNNs. Moreover, the M3S [48] method employs a clustering technique to enhance the accuracy of pseudo-labeling, effectively filtering out labels that do not align with the clustering assignments.
Semi-supervised Imbalance Node Classification. Recent advances have tackled the challenge of imbalanced node classification in graphs. with particular emphasis on leveraging topological structures to improve performance [44, 61, 73, 31, 39, 6, 37, 47, 67]. These methods can be broadly grouped into three categories: (1) Node generation methods, which synthesize new nodes to balance class distributions; (2) Topology-aware adjustment methods, which exploit graph structure to adjust model weights or decision boundaries; and (3) Self-training methods, which incorporate pscudo- labeled nodes to enhance generalization. Node generation methods include GraphSMOTE [73], which interpolates minority class nodes in the embedding space and constructs edges via link prediction; ImGAGN [39], which generates both features and topology by modeling global minority distributions; and GraphENS [37]. which augments diversity by mixing ego-networks. Despite their effectiveness, these approaches often suffer from high computational cost and poor scalability on large graphs. Furthermore, generating semantically meaningful and structurally consistent topology remains a non-trivial, dataset-dependent task. Topology-aware adjusiment methods reweight or reshape training signals using structural information. For instance, ReNode [6] assigns weights based on nodes' topological distances to decision boundaries, while TAM [47] calibrates prediction logits by integrating local topology and class distribution statisties. However, these methods typically rely on labeled data to estimate structural bias and lack a unified framework for quantifying or mitigating topological imbalance. Self-training methods aim to exploit unlabeled data in imbalanced settings by generating Aseudo-labels, GraphSR [75] adopts two-step strategy that combines similarity-based filtering with reinforcement learning to select reliable pseudo-labeled nodes for minority classes. BIM [71] formulates the problem through the lens of influence maximization and selects nodes to balance class influence within receptive fields. IceBerg [27] introduces a simple yet effective Double Balancing mechanism that simultaneously calibrates pseudo-label distribution and the loss function, while also disentangling GNN propagation to improve supervision in few-shot scenarios. While promising, these methods still lack thcoretical guarantees and standardized metrics for evaluating pseudo-label quality. Their performance often hinges on heuristics- as confidence thresholds or influence approxinations-whiie can be unreliable under severe class imbalance.
Topology Bias in Graph Learning. Topology-an -wwar learning in graphs has recently emerged as a crucial research direction, especially in the context of class-imbalanced node classification. Traditional class rebalancing techniques often overlook structural biases rooted in the non-uniform distribution of labeled nodes across the graph topology. A growing body of work has started to explicitly address this issue, proposing both theoretical formulations and practical remedies for topological bias. Renode [6] first formalizes the notion of eopology-imbllane identifying structural
18

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asymmetries among labeled nodes as a new source of bias in semi- sumi--spee node classification. Their method, ReNode, detects node influence conflicts and adaptively re-weights node influence to mitigate the imbalance. Following this, several works proposed augmentation-haal baased solutions. GraphENS [37] generates ego-networks for minor-class nodes to simulate balanced topological contexts, while TAM [47] introduces topology-awan margin loss that adapts to local connectivity patterns to improve representation learning under imbalance. RGE [46] approaches the problem from the perspective of noisy topology. It introduces Repulsive edge Group Elimination strategy that selectively removes groups of edges with consistently harmful influence, enhancing robustness against pruucturr-inducer bias. Beyond these, [49] proposes PASTEL, structure learning framework by optimizing node positions to improve intra-class information
that mitigates aapology-imbllane flow, particularly addressing under-reaching and over-squashin effects. Recent work by [32] challenges the necessity of class rebalancing altogether. They propose BAT, lightweight topological augmentation technique that mitigates class imbalance by enhancing local structure, without altering label distributions. [72] explores topology-level imbalance at a coarser granularity. They argue that dominant sub-topology groups bias learning and propose two-stage method to extract and regulate these groups to ensure more equitable representation. From a metric design perspective, [60] introduces Topological Concentration (TC) to quantify local neighborhood informativeness for link prediction. Their analysis shows how low-TC nodes consistently underperform due to topological constraints, and propose re-weighting schemes to correct this. Meanwhile, [7] unifies topology and class imbalance under the lens of AUC optimization. Their TOPOAUC framework jointly optimizes class-wise margins and topological influence via a TAIL (Topology-Awer -wware Importance Learning) module, offering practical and effective solution. Finally, [9] proposes neously constructs the graph structure and trains the semi-supervised model, thereby overcoming biases introduced by pre-constructed graphs and addressing topological imbalance in large-scale settings. [58] proposes Topological Sample Selection (TSS) scheme that promotes informative node selection by leveraging graph position, especially around class boundaries-crruch under noisy labels. [64] designs CE-GSL, a community-entre based GSL framework that reconstructs the graph by connecting uncertain nodes and enhancing supervision using both class-level and node-level entropy measures. Together, these works establish diverse methodologies from augmentation and margin- maagen- regularization influence modeling and graph construction for addressing it across different learning paradigms.
as a fundamental limitation in GNN-based learning and offer
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C Proof of Theorem 1
Proof of Theorem I (Message Passing Perspective). We aim to prove that the average entropy 1i of pseudo-label predictions for unlabeled nodes is positively correlated with intra-class compactness Dintra, and inversely correlated with inter-class separation iitters i.c.,
D Dintra Dinter
A
Step Message Bessing-Inducee Representation Shift
Consider a GNN where each node representation is updated via:
( asyhi + cxatt ))
h (1+1)
Σ
JEN((
Assume initial representations ( Idd are normalized to lie the unit hypersphere, i.e., ||A!00| We ignore nonlinearity a and weight matrix W(()) for clarity,
h
Σ
aighi (0)
JEN((
Let node i belong class The updated representation after one message passing step can be decomposed as:
h (1 B:) h (c) + B, h
where:
(c) is the contribution from same-class neighbors.
is the contribution from different-class neighbors.
3 E [0, 1] quantifies the influence of different-class neighbors.
This update shifts h." away from the class center Mee especially when 3, is large when minority class nodes are sparsely connected.
Step 2: Definition of Geometric Terms
(,000 Kya 1|2
Dintra
v³ Viisse D'siiiie
Dinter
Σ Fra
|ites Mo 110
Minority node representations deviate from their true class centers due to influence from majority neighbors, increasing Dintr and potentially decreasing inter-
Step Entropy under von Mises-Fisher Posterior
The posterior is:
exp K cos(huus (is))
p.d ΣC,
- exp (x- cos(Thus ūix)
Using cos(r, v) ~ 1 - 211 2 for unit vectors:
exp (- paall))
(- 1h (1x113)
p.
exp
Z
k=1
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The entropy is then:
C
Huu) log Z +
l uusl? - -Ail?
-111
22
Step 4: Bounding the Average Entropy
Upper bound: Suppose for most classes * yu the distance |l - Dill2 2s Dimer/ C, and for the
correct class y"² it is D intra Then:
7
exp
mi)
i=1
Dinter
exp
Dintra)
+ (C
exp
Using log(1 + z) & x and Taylor expanding:
H Hw")) log Z
m,e mm
27
log C
D Dinter
Dintra// / D Pinter o(1),
20
Averaging gives:
A < log
k
(Pieer
( Dinter
Dintra
20
Lower bound: Assume the posterior is highly confident:
pau yos Re R8 and par izy E/(CC with
E < due to hn being close to 7 Nyu..
Using entropy expansion:
H(w)) - ((- -) Elog
K Dintra
log - 2 Dimerr
where the last inequality holds since e -x(Dmur-Dam Dram) decays with class separation.
So conclude:
Dinta
A log C'
Dinter
So, we obtain the sandwich inequality:
6 log C'
Dinter
Dintra
A 2 log C
K
Dintra
20
Dineer
2
D inter
which implies:
Dinra Dinter
H
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D Proof of Theorem 2
Proof of Theorem 2. Let p Paaix be the imbalance ratio between majority and minority class nodes.
Viaine
Fix the feature extractor and class centers {
Recall the geometric imbalance score:
Prus Hggu
G(w²) Eestor lIies #e,ll?"
Gminor
Σ E((").
Vinior
Dunlabel
We aim to show that Gminor & p. To do this, we analyze the effect of imbalance on the representation shift of minority nodes.
From the message-passing formulation (see Theorem 1), we have:
1 - 3,) h (0) + B5 -
h
where 8 is the proportion neighbors from other classes. Under class imbalance, minority nodes more likely to have majority neighbors, hence:
E[ ya Cminor] f with
Since 1,22) convex combination class-consistent and inconsistent information, increasing 3 pushes h, away from its own class center jıc. Specifically, the deviation magnitude satisfies:
|| Rell 2 6 |
(0) - -
Therefore, when p increases, so does 3i, and consequently:
G(w') x li - Byns 112 xx 8 x p².
Averaging over all w³ Vinnnor Dunlabel and observing that Eortes ll ioo2l is constant under fixed class centers, obtain:
Gminor & [[ x p².
Thus, Gminor increases monotonically with p. and conclude:
Gminor p.
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E
Visualization and Analysis of Geometric Imbalance
E.1 Experimental Setup
To demonstrate the phenomenon of geometric imbalance in semi-supervised imbalanced node classification (SINC), we design an illustrative experiment using the Cora dataset. We construct controlled scenario that isolates the effects of class imbalance geometric properties of embeddings by manipulating both model architecture and training settings.
Initial Pretraining Phase:
We begin with a balanced labeled training where each class contains exactly 50 labeled nodes. On this balanced data, we pretrain two types of encoders:
- a Graph Neural Network (GNN) encoder,
a Multilayer Perceptron (MLP) encoder.
These pretrained models serve as feature extractors, capturing representations a fair, geometry preserving way.
Fine-tuning Imbalanced Data:
We then fine-tune the pretrained models on an imbalanced variant of Cora, where class-wise labeled node counts
to [100, 30, 100, 100, 100, 30),, inducing imbalance ratio p R 3.3. We consider two fine-tuning settings:
1. Frozen (Pretrained): the encoder parameters are fixed and not updated during fine- tuning.
Unfrozen (Fine-tuned): the encoder is fully trainable.
Visualization Protocol:
To inspect geometric structure, extract latent node embeddings (from the final hidden layer), project them using t-SNE, and visualize labelcd/unlabeled nodes across major- ity/minority classes.
E.2
Results and Analysis
Figure and Figure 6 illustrate the change in geometric structure before and after fine-tuning, across both GNN and MLP backbones.
Case 1 (GNN-Precee :
The initial GNN encoder produces geometrically well-separated clusters with clear class boundaries.
Minority class nodes, both labeled and unlabeled, exhibit compact intra-class distributions and are reasonably separated from majority classes.
Case (GNN-Finetune):
After fine-tuning on the imbalanced set, the embeddings show signs of geometric degrada- tion:
Dinter (distance between class centroids) shrinks significantly.
Dintra (dispersion within minority clusters) increases.
This leads to reduced separability and greater geometrie confusion for minority nodes typical of geometric imbalance.
Case 3 :
The MLP, lacking structural bias from graph topology, preserves geometric structure during pretraining.
Clusters are reasonably separated and compact, although slightly less so than GNN due to lack of message passing.
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Case 4 (MLP-Finetune):
Interestingly, the MLP does not suffer from significant degradation after fine-tuning.
-inter remains stable.
mimra does not increase notably.
This suggests that geometric imbalance is amplisied by message passing in GNNs, especially when label imbalance exists.
Conclusion of Analysis:
The results empirically confirm our theoretical claim (Theorem and 2)
that:
Class imbalance leads greater prediction entropy due increased Dunra and reduced Dinerr
And that GNNs due to their structural bias and message propagation- are more vulnerable this effect than MLPs.
This observation motivates the need to explicitly account for geometric imbalance during self-training and pseudo-label propagation, as do the proposed UNREAL framework.
80
Minottyy (Labele))
Minoily (Uriabeled)
Majority (Laceled)
20
0
20
-20
0
Minetity Clees (Labalec)
Minciity Class (Urllaaied)
Majonty Class (Labe ed)
-40
60
40
20
-75
-50
-25
25
50
75
(a) GIN-GNNNNetnnn Pretrained
(b) GNN-Fintune
Figure 5: Illustration of geometric imbalance across different GNN encoder cases.
Minctity Clasa (Labelcc)
Minctlly (Urlabaled)
40
Majortl Class (Labeied)
0
-20
-20
-40
Minotiy Class (Labeled)
Minetty Class (Unlsbeted)
Majorit Chess (Labered)
-60
-40
-20
0
20
60
40
-20
20
40
(a) MLP-Pretrained
(b) MLP-Fintume
Figure 6: Illustration of geometric imbalance across different GNN encoder cases.
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Analysis of Geometric Imbalance under Varying Class Imbalance
F.1 Experimental Setup
To comprchensively analyze the effect of class imbalance on geometric properties of node embeddings, we conduct controlled experiments on the Cora dataset using three mainstream GNN architectures: GCN, GAT, and GraphSAGE.
For each architecture, simulate 50 different imbalance scenarios by systematically varying the class distributions- rranggg from balanced to highly imbalanced cases. For every setting, we train the model and extract node embeddings, based on which we compute two key metrics:
Geometric Imbalance (GI): quantifies the structural disparity of embeddings between minority and majority classes, considering both intra-class compactness and inter-class separation.
Average Entropy: evaluates the uncertainty of the classifier's predictions across all unla- beled nodes.
Additionally, we record the Imbalance Rate, defined as the logarithmic ratio between the sample sizes of majority and minority classes
F.2
Quantitative Analysis
Figure 7 illustrates the relationship between Imbalance Rate and Geometric Imbalance across the three architectures. We observe monotonic increasing trend, confirming that higher label imbalance leads greater geometric distortion the learned representations. This effect is consistent across GCN, GAT, and GraphSAGE, although the absolute GI values vary slightly depending on the architecture.
Figure explores how Geometric Imbalance correlates with Average Entropy. Again, we find a strong positive correlation, indicating that as geometric imbalance worsens, model confidence unlabeled nodes degrades, leading to more uncertain predictions. Notably:
GCN and GAT exhibit smooth, concave-upward trends, suggesting gradual degradation in certainty.
GraphSAGE shows a peak entropy followed by plateau, possibly due to over-smoothing reduced model sensitivity extreme imbalance levels.
These results empirically support the theoretical intuition that geometric imbalance serves as an intermediate variable linking class imbalance to model uncertainty, and underscore the necessity of designing imbalance-aware algorithms to preserve embedding quality and classification reliability.
27.5
25.3
12.5
23
10
22
23
23
27
22
22
(a) Cora-GCN
(b) Cora-GAT
(c) Cora-SAGE
Figure 7: Relationship between Imbalance Rate and Geometric Imbalance across different GNN
architectures.
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339
Fited Curve
Riteed Curve
132
Fited Curve
21
4.0
Cesmane sca8)
(a) Cora-GCN
(b) Cora-GAT
(c) Cora-SAGE
Figure 8: Relationship between Geometric Imbalance and Average Entropy across different GNN architectures.
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G Distinction Between Geometric Imbalance and Topology Imbalance
Geometric imbalance and topology imbalance [6] are two fundamentally distinct forms of imbalance that challenge semi-supervised node classification graphs. Understanding their differences is crucial for designing effective algorithms for imbalanced graph learning.
G.1 Geometric Imbalance
Geometric imbalance refers to the ambiguity in node representations within the embedding space induced by class imbalance and GNN message passing mechanisms. Specifically, characterizes how minority-class nodes, due to limited and biased label propagation, are more likely to have embeddings that are close the boundaries between classes or even equidistant from multiple class centers on the hypersphere. This phenomenon is rigorously analyzed by modeling embeddings using the von Mises-Fisher (vMF) distribution and quantifying ambiguity via geometric imbalance scores, which measure the relative position of node's embedding to the centers of all classes. Higher geometric imbalance is directly linked to greater prediction uncertainty and error pseudo-labeling. particularly for unlabeled minority-class nodes. The severity of geometric imbalance is shown to increase monotonically with the class imbalance ratio, and it is especially pronounced in GNNs due to their structural aggregation process, while being marginal in non-graph models like MLPs [41].
G.2 Topology Imbalance
In contrast, topology imbalance is defined on the original graph structure and focuses on the asym- metric topological distribution of labeled nodes across classes. Unlike quantity imbalance, which concerns the number of labeled nodes per class, topology imbalance arises when labeled nodes in the same different classes occupy distinct structural roles-such such as being closer or farther from the class boundary or center. This leads to issues like influence conflict (labeled nodes near boundaries causing ambiguous label propagation) and influence insufficiency (labeled nodes far from target regions failing to effectively propagate information). Topology imbalance is quantified via metrics like Totoro, which captures the degree of conflicting influence among labeled nodes across the graph. Its perniciousness lies shifting the effective decision boundary away from the true class boundary, thereby degrading GNN performance even when the class labeling ratio is balanced [6, 37, 47].
G.3 Summary of Differences
Domain:
Geometric imbalance concerns embedding space positions of (especially un- labeled) nodes, while topology imbalance concerns the graph structure and the relative positions of labeled nodes.
Target: Geometric imbalance primarily affects unlabeled minority nodes by making their embeddings ambiguous, leading to unreliable pseudo-labels. Topology imbalance focuses the distribution and influence of labeled nodes, particularly those near class boundaries.
Measurement: Geometric imbalance is measured via geometric distances (e.g., on the hypersphere) between node embeddings and class centers. Topology imbalance is measured by influence metrics like Totoro, which consider the network-wide propagation of label information.
Mechanism:
Geometric imbalance is amplified by message passing and normalization in GNNs, while topology imbalance arises from the structural arrangement of labeled nodes and their impact on message propagation.
Orthogonality: The two forms of imbalance are orthogonal: one can exist without the other. For example, a graph may have balanced class sizes (no quantity imbalance) but still suffer from severe topology imbalance if labeled nodes are poorly distributed structurally.
In word, while both geometric and topology imbalance deteriorate node classification performance in semi-supervised learning, they originate from different sources and demand distinct mitigation strategies. Geometric imbalance calls for techniques addressing ambiguity in embedding space, especially for unlabeled nodes, whereas topology imbalance requires careful selection or re-weighting of labeled nodes according to their topological positions in the graph.
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H Proof of Theorem
Proof of Theorem 3. Recall the construction of the proposed ranking:
N max(rm: Tm Sm + min rm T
(I)
where RBO Tm)) measures the similarity between the geometric ranking S. and the confidence ranking Tm
We consider the two extremal regimes of disagreement:
Case 1: Tm 0 (maximum disagreement). In this case,
max 1 Tmn = 1 - Tm- min(rm: 1 rm = mm
Hence,
lim N New
lim ((1 - Tm) Sm + mm Tm)
Pm -0
I'm -0
Sm.
Case 2: Tm + (maximum agreement). this case,
max 1 Tim Tim
min 1 rmm] 1 Tmm-
Thus,
lim N New lim (rm Sm + (1 - -mm) Tm)
Sm.
In both limiting cases, the ranking New converges the geometric ranking Sm. Therefore, S.. dominates the selection process particularly in early stages when Tm is small, i.e., when disagreement between geometry and confidence is high. This confirms that the design reduces the effect of geometric imbalance by emphasizing Sn until Tm becomes sufficiently aligned.
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I Comprehensive Experimental Results
I.1 Comprehensive Results Under Varying Levels of Class Imbalance
Due to space limitations, we present the complete experimental results here. In Table 20, Table 21, Table 22, and Table 23, we report results under different imbalance scenarios (p- 10, 20, 50, 100). Several consistent trends can be observed across varying imbalance levels.
In Table 20, when p 10, our method outperforms all baselines by clear margin most datasets. Notably, on Cora and CiteSeer, our method achieves substantial gains in both balanced accuracy and Fl-score. On PubMed and Amuzon-Compulee our model still ranks among the top performers. We attribute this to two factors: (1) PubMed has only three classes, resulting in a less severe imbalance scenario. Most methods setting reasonably well. (2) The Amazon -riizonn-mmuuerss dataset has been shown to be less sensitive to label sparsity, which narrows the performance differences among competing methods.
although the improvements are relatively smaller,
still handle this
As the imbalance ratio increases (Tables to 23), three key observations emerge:
(1) The performance gap between our method and the baselines consistently increases. For example, p 100, our method outperforms the second-best method by up to +10.60% in balanced accuracy and +7.85 % in Fl-score on Cora. This demonstrates that while baseline methods degrade significantly under severe imbalance, our method maintains stable and superior performance.
(2) Oversampling-baat based strategies such as GraphENS [37] or GraphSMOTE [73], although effective in mild imbalance settings, show limited scalability under higher p. We hypothesize that generating synthetic nodes and edges may introduce noisy or redundant samples that interfere with representation learning, especially when minority groups extremely sparse.
(3) BalancedSoftmax [40] performs competitively across all p levels by correcting the bias in the classifier head. However, it lacks representation-level adaptability, limiting its potential compared to our method, which additionally improves node representations through geometric ranking and uncertainty-awan selection.
Overall, these results validate the robustness of our method in handling various degrees of class imbalance. outperforming both classical re-weighting techniques and recent augmentation-bun based approaches.
1.2 Comprehensive Results on Natural Imbalanced Datasets
The experimental results on the naturally imbalanced datasets CS-Random and Computers-Ranie Randdom (Table 24, Table 25) further demonstrate the consistent superiority of our method UNREAL. On Random, our method achieves the highest balanced accuracy and Fi-score across all three GNN architectures, with particularly notable gains GAT. While methods such as BalancedSoftmax and Renode perform competitively in some cases, they suffer from performance instability across different backbones and fail generalize under higher imbalance.
On the more challenging Computers-Randee dataset, Our model delivers even more substantial improvements, especially with GCN and GAT, achieving gains up to +2.47 % in balanced accuracy and +6.71 in Fl-score. Although the improvements on SAGE are relatively modest-part) due to the dataset's 's structure enabling easier classification-
method still ranks among the top performers.
These findings confirm that our model is not only effective in synthetic long-tail settings but also robust and generalizable in real-world class-imbalanced scenarios, regardless of architecture choice.
I.3 Comprehensive Results On Large-Scale Natural Imbalanced Datasets
We further evaluate method on large-scale real-world datasets with naturally occurring class imbalance: Flickr and Ogbn-Arxiv (Table 26, Table 27). These datasets present distinct challenges: Flickr features moderate imbalance with noisy labels, while Ogbn-Arxiv has extreme imbalance ratio exceeding 700.
On Flickr, our method achieves consistent improvements over baselines across all three GNN architectures. Specifically, our method outperforms prior methods by up to +4.79 in FI-score and +3.45% % in balanced accuracy. While several augmentation-based methods (c.g., GraphENS, ReNode)
29

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fail due to memory or scalability issues (OOM), our method remains robust and computationally feasible. Notably, on GCN and SAGE, our performance gains are both statistically significant and practically meaningful.
For Oghn-Arxiv, the extreme long-tail label distribution poses severe challenges. Most oversampling or memory-intensive methods cannot run on this scale. In contrast, our method achieves the best overall performance, surpassing all baselines by small but consistent margin. Despite the relative difficulty of obtaining large improvements under such extreme imbalance, our method shows gains up +0.07 in FI-score and +0.34% balanced aceuracy, demonstrating its scalability and reliability.
These results confirm the applicability our method to large-scale, naturally imbalanced benchmarks where many existing methods fail to scale generalize.
To further validate the scalability and effectiveness of our proposed method, we conduct experiments on the large-scale natural imbalanced dataset Ogbn-Products. Following the same experimental settings as BIM [71]. we evaluate our method using three widely adopted graph neural network backbones: GraphSAGE, GAT, and GCN. We report the performance in terms of FI score, Accuracy (ACC), and AUC-ROC. All experiments are repeated five times with different random seeds, and the mean and standard deviation are reported. Tables 9 summarize the results on the ogbn-products dataset. Across all three architectures, our method consistently outperforms the baselines, demonstrating superior robustness and generalization under serge-scale natural class imbalance. Compared to BIM, which is specifically designed for imbalanced graph learning, our approach achieves an average improvement of around 2-3% in both F1 score and AUC-ROC across all backbones. These results verify the scalability of our method and its strong empirical effectiveness on large-scale graphs.
Table 9: Experimental results of our method and other baselines on the large-scale natural imbalanced We report averaged Accuracy (ACC, %), Fi-score (%), and AUC-ROC (%) with standard deviations over 5 repetitions three representative GNN architectures.
dataset
Dataset (eghn-prodact))
GCN
GAT
GraphSAGE
AUC-ROC
Vanilla
57.4
60.12-035
93.96=0.15
19,27=
62.10
94.2520.14
58.181121
61.03:0.82
94.1220112
(RN) ReNode
67.80-044
94.75+0.13
68.55-0.28
69.2740.36
9501+0..0
67,89+0.35
68 19+0.41
94.86-0.14
HIM
6805=0021
68:47-023
95.41=0.13
69.88-0016
70.05:00211
95.6340.11
69.3640.18
09:55+00.19
95.51=0.12
Ours
70.7120:13
70.33-0.10
96,4520.12
72,04:0.09
71.951013
96.710010
718510.10
716621011
96,59_0.14
1.4 Experimental Results of Our Method with Less Training Rounds
We also validated method the Cora (p 10) dataset with fewer iteration rounds (node selections). The experimental results are presented in Table 10, demonstrating that our method outperforms state-of-the-art methods even with reduced number of rounds.
Table 10: Experimental Results of Our Method under Less Rounds.
Model (FI)
Cora-GAT Cora-GCN
Cora-SAGE
GraphENS (w TAM)
72.14
70.00
70.40
Our method
76.44
75.99
73,63
Our method (2 rounds)
70.62
69.32
69.83
Our method (4 rounds)
71.83
72.93
70.21
Our method (6 rounds)
73.41
73.55
71.09
Our method (8 rounds)
75.63
74.54
72.90
I.5 Adapting Our Framework to Imbalanced Image Classification
Although our theoretical analysis and methodology primarily address the issue of geometric imbalance node classification, we have also validated the versatility of our method. For instance, by applying our method within an image classification framework, we conducted experiments on CIFAR 100LT using the our method framework in conjunction with ResNet-32. The results is presented in Table 11, and the results demonstrate method's commendable performance.
30

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Table 11: Recognition results (%) of different training data. All configurations are evaluated on the testing set of normal CIFARIO0.
Model
Accuracy
Precison
Recall
ResNet-32 (CIFARIOOLT)
37.74
42.12
37.54
Our method
39.31
44.32
40.43
Table 12: Refined running time analysis of vanilla GNN models and our method. (a) Training over 1000 epochs, and (b) selecting 100 nodes into the training set. Results are averaged over five runs.
Model/Time (1000 epochs)
Cora
CiteSeer
PubMed
Model/Time (Selecting 100 Nodes)
Cora
CiteSeer
PubMed
Vanilla (GCN)
6.221(6)
7,212(s)
10.771(s))
Vanilla Self-Training (GCN)
2.244())
2.234(6)
2.675(s)
(GCN)
6.208(s)
7.191(s)
10.858(s)
(GCN)
2.367(8)
2.M05(s)
2.790(s)
Vanilla (GAT)
8.201(s)
9.574(s)
14.443(s)
Vanilla Self Self-Trannig (GAT)
2.624(6)
2.520(s)
2.431(s)
Ours (GAT)
8.218(8)
9.517(s)
15.088(s)
Ours (GAT)
2.697(6)
2.670(s)
2.411(6)
Vanilla (SAGE)
6.522(s)
15.475(s) 11.170(s)
Vanilla Self-Training (SAGE)
2.424(6)
2.780(s)
2.032(s)
Ours (SAGE)
6.452(8)
11.256(s)
14.658(s)
Ours (SAGE)
2.461(6)
2.827(s)
2.115(s)
1.6
Analysis of Computational Complexity
Time Complexity Analysis. Our framework, as described in Algorithm 1 (see Appendix L), consists of three main modules per round: Dual-Path Pseudo-label Alignment (DPAM), Node-Reondering (NR), and Discarding Geometrically Imbalanced Nodes (DGIS). (1) The dominant computational cost arises from the DPAM step, which clusters the embeddings of unlabeled nodes using k-means (complexity (n, d'f.ter,, with n., unlabeled nodes, k' clusters, d-dimensional embeddings, and Titer k-means iterations). Computing class centroids and intersecting with glssiffer-predictte labels are both linear-time operations. (2) NR involves ranking nodes by geometric distance and classifier confidence (each (n,, log n,))), with ranking fusion being negligible. (3) DGIS computes distances from candidate nodes to their closest and second-closed centroids (O(,,.Cl per round, where C' is the number of classes). (4) Overall, the per-round complexity is primarily determined by k-means, but since k' < nh and k-means can be efficiently implemented (e.g., on GPU), so, the total complexity remains comparable to standard self-training pipelines. Memory usage is linear in the number of nodes, as we only store embeddings and label predictions. Importantly, our method is significantly more efficient than prior over-sampling or augmentation approaches such as GraphENS and GraphSMOTE, which may have O(n²) worse complexity on large graphs. In summary, the only notable overhead beyond vanilla GNN training comes from clustering and ranking, both of which are scalable and can be accelerated by modern hardware.
Empirical Scalability and Runtime. To address practical efficiency, we conducted comprehensive scalability analyses on both medium- and large-scale datasets, comparing our method ("Ours") to vamilla GNN backbones and representative baselines. (1) On Cora, CiteSeer, and PubMed, the training time per epoch for our method is almost identical to the backbone GNNs (see Table 12), confirming the negligible overhead of our extra modules (clustering and ranking). (2) On large-scale graphs (e.g.. Flickr, agbn-arxiv), our method remains tractable and runs smoothly on single RTX 3090 GPU (Tables 26 and Table 27). By contrast, prior data augmentation and oversampling baselines (e.g.. GraphSMOTE, GraphENS) either become prohibitively slow or run out-of out-of-memory, synthesizing large numbers of new nodes/edges is expensive.
I.7 Comparison of UNREAL and Recent Soft Pseudo-Labeling Methods
Experimental Setup. Here, we compare UNREAL with soft pseudo-labeling strategies such as SCR [70] and ConsisGAD [8]. We have conducted comprehensive experiments comparing our method with these approaches under class-imbalanced settings. For all datasets, we followed the initial hyperparameter search ranges provided in Appendix A.3 of the SCR paper and Appendix D.2 of the ConsisGAD paper. We further expanded these ranges as needed, and employed Bayesian optimization (using Wandb) to efficiently search for the optimal hyperparameters for both methods
31

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Table 13: Results with p 10 (GCN) on three class-imbulanced node classification benchmark
datasets for comparison of UNREAL and Recent Soft Eseudo-Labellig Methods.
Method
Cora bAcc.
Cora F1
CiteSeer bAce.
CiteSeer F1
PubMed bAcc.
PubMed F1
GSR
0.44
71.37:10..3
59.28:10.22
55,96:0.95
73.61 1.25
71.88
BIM
12.1910.42
72.67.10:44
58.54.10.61
56,91
74.62.1.15
72.93.1..21
Ours
78.33+
76.44+1.00 1.06
65.63+1.33
64,94-1.83 1.38
75.35 1.41
73.65+1.44
SCR
0.57
65.43++0.91
1..9
39.12:+1.59
67.88-0.67
62.70:+1.88
ConsisGAD
66.50.10.59
65.91
44.7911.099
39.33
68.05.10.62
62.75.11.90
Table 14: Results with 100 (SAGE) three class-imbalanced node classification benchmark datasets for comparison of UNREAL and Recent Soft Pseudo-Labeling Methods.
Method
Cora bAcc.
Cora F1
CiteSeer bAce.
CiteSeer FI
PubMed bAce.
PubMed F1
GSR
66.45+22.10
64,42=1183
53.52==1.47
13.01=1.75
74.09+2.11
72.97=11.0
BIM
63.75+2.13
64.68+1.95
53.83=1.62
53.29+1
74.38+22.08
73.24+1.83
Ours
73.4742.31
68.30+2.11
59.77=2998
58.92+3.07
77.11+0.559
74.03+0.81
SCR
59.31++2.91
55.34-1..22
36.51
27.19+2..4
63.41:-0.76
1..97+1664
ConsisGAD
59.63++2.89
55.18:4.43
36.39=1.28
27.28+2.87
63.27:0.74
16.33+1660
on the various imbalanced datasets. This ensured that both SCR and ConsisGAD were evaluated under their best possible configurations in experiments.
Results and Analysis. Empirical results (see Table 13-17) show that both SCR and ConsisGAD experience significant performance drop under severe class imbalance, particularly in terms of Macro-F1. In contrast, our method consistently achieves much stronger performance in these challenging scenarios. We believe this is due to the following reasons: (1) Soft pseudo-labeling methods, including SCR and ConsisGAD, propagate pseudo-label information uniformly across all nodes with global consistency constraints. In imbalanced graphs, the majority classes dominate the label distribution and, consequently, the consistency regularization process, causing minority class signals to be overwhelmed ignored. This leads to suboptimal decision boundaries that do not adequately account for the geometric marginalization of minority nodes. (2) In contrast, our method specifically motivated by the phenomenon of geometric imbalance non -on-Euclidean manifolds, and is designed to selectively generate reliable pseudo-labels for ambiguous nodes. By doing so, we explicitly enhance the representation learning for minority classes and avoid overfitting majority class signals.
1.8 Experimental Results on Imbalanced Heterophilous Graphs.
To further evaluate the generalization capability of method, conduct experiments het- erophilous graph benchmarks, including Chameleon, Squirrel, and Wisconsin. Unlike homophilic graphs, where neighboring nodes tend to share similar labels, heterophilous graphs pose additional challenges due to the weak correlation between node features and their adjacency structure.
As shown in Table 18, our model consistently outperforms existing baselines on all three datasets. This demonstrates that our approach not only handles class imbalance effeetively but also generalizes well to complex graph topologies where conventional message- message-passing passing schemes tend to fail. The improvement observed across both balanced accuracy and Fl score indicates that the proposed model achieves a better trade-off between precision and recall under highly non-homophilic conditions.
Furthermore, the performance gain highlights the robustness of our method's representation learning mechanism. By dynamically adjusting pseudo-label confidence and leveraging relational consistency, our approach mitigates the over-smoothing and noise propagation that commonly affect GNNs on heterophilous graphs. These results verify that our framework remains effective even when node similarity is low, suggesting its potential for broader application scenarios such as citation, social, and web networks characterized by heterophily.
32

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Table 15: Experimental results on Computers- Random (p 17.7) for comparison of UNREAL and Recent Soft Pseudo-Labeling Methods.
Model
GCN bAcc.
GCNFI
GATbAcc.
GAT F1
SAGE bAcc.
SAGE F1
GSR 83.82+0.74
77.78+0.42
76.79-0.68
73.61+0.63
77.63+0.32
72.56+0.51
BIM
84.03++0.73
77.96+0.45
77.01-0.70
73.82+0.60
77.76+0.65
72.09+0.377
Ours
85.32++0.22
80.4340.56
82.52-0.35
78.90:10.88
15.81=1.86
71.86=+1.66
SCR
80.41
75.18+0.57
71.90=082
70.71
66,41=1.45
16.05
ConsisGAD
80.56++0.46
75.15+0.556
71.97-0.79
70.62+0.54
66.54-11.49
66.13+11.4
Table 16: Experimental results CS-Random (p 41.0) for comparison of UNREAL and Recent
Soft Pseudo Pseudo-Labeling Methods.
Model
GCN bAcc.
GCNFI
GAT bAce.
GAT F1
SAGE bAec.
SAGE F1
GSR
86.73=0.22
85.9111..1
85.34=0.13
86.56:10.29
85.44:0.27
86.46:10.23
BIM
86.89=0.23
85.9940.21
85.63=1.87
86.65-0.35
85.65=0.28
86.73+40.22
Ours
88.94=0.09
89.87+0.06
87.65-0.12
87.65=0.11
88.03==0211
88.65++0007
SCR
87.41
88.7331.111
83.61=0.41
84.68:10.3
85,8011.23
87.3710.118
ConsisGAD
87.4440.16
88.6840.12
83.58=0.38
84.75:0.34
85.73=0.25
87.30:00.15
I,9
Comparison to General Self- Self-Training Methods
We also compared our method with other self-training frameworks the CS-Random dataset, as shown in Table 19. By explicitly addressing hard samples in the representation space, our method achieves superior performance compared existing frameworks.
33

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Table 17: Experimental results on Ogban-arXiv (p ~ 775.4) for comparison UNREAL and Recent Soft Pseudo-Labeling Methods.
Model
GCN bAcc.
GCN
bAcc.
GAT F1
SAGE bAcc.
SAGE FI
GSR
50.31
49.7010.17
51.31=0.41
49.33:10.26
50.86=0.30
49.53:10.20
BIM
50.33:+0.22
49.7110.20
49.36-0.28
50.87=0.18
49.56=0.23
50.2110.26
Ours
51.21=0.36
50.3240.32
51.84-0.27
51.78=0.24
01.34=0.32
51.36+0.27
SCR
50.23:10.41
49.6911.13
51.10=0211
49.67-1026
50.83:1.200
49.80.10.23
ConsisGAD
50.2540.39
49.7210.14
51.13=0.37
49.63:1027
50.80
49.77
Table 18: Results heterophilous graphs. We report balanced accuracy (bAcc.) and F1 scores three benchmark datasets: Chamelcon, Squirrel, and Wisconsin.
Method
Chameleon bAcc.
Chameleon F1
Squirrel båcc.
Squirrel F1
Wisconsin bAcc.
Wisconsin Fl
GiR((GraphSR)
38.45-40.75
07.67+0..2
27.9440.44
27.01+00.2
31.54+2.30
28.914-244 28.79=2.01
BIM
38.26:10.68
37.48-0.66
27.8340.46
27.16::0.37
31,59-2.25
Ours
43.59 0.60
42.24 0.63
30.04-1.488
29.12:10.46
3.51
41.52 4.00
34

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Table 19: Comparison with General Self-Training Frameworks.
Model (FI)
CS-GCN
CS-GAT
CS-SAGE
Self-Traine Self-Training [26]
87.54
85.32
86.54
Co-Training [26]
87.56
85.32
87.21
M3S [48]
88.12
86.54
87.43
Our method
89.87
87.65
88.65
Table 20: Experimental results of our method and other baselines four class-inbalanced node classification benchmark datasets with 10. We report averaged balanced accuracy (bAcc., %) and Fl-score (%) with the standard errors over repetitions on three representative GNN architectures.
Donaset
CieeSeer
PabMed
Amazon Computers
(o
Vanilla
62.82 1.43
61.07 = 1.59
38.72
1.88
28.74
3.21
65.64 1.72
56.97 + 8.17
8001 = 0.71
71.56
Beighightt
65.36 1.15
64.7 = 1:39
44.6 1.78
3861 ::
69.06 1484
64.08 2:97
80.93
73.99
2.20
70.27
GrapISMOTE
66.39 0.56
65.49
0.9
1.12 4487
39.20
1.62
67.91
0.04
62.68 + 1:92
79.48 0.47
72.63
0.76
BalatcedSSofmax
69.58 0.58
68.65 = 0.55
55.52 0.97
33.74 1.42
0:89 73.73
71.53 + 1.06
81:46 0.74
74.31
0.51
BalutcedSofmaa TAM)
69.94
56.13
0.71
72.25
72.94
L.43
Remide
67.03
67.16
1.67
23.47
2-22
37.52
71.40
61.27
61.89
0.77
73.13
1.60
Renoda TAM)
68.26 1.84
68.11
1.97
46.20
1.17
39
2,76
2.03 12.13
72.63
68.28 3:30
80.36
72:51
0.68
GraphliNS
1.33
1.07
74:26
1.05
GraphENN or TAM)
71.69 0.36
74.14 0.53
58.01
0.88
$6.32 1: 1.03
74.14
1:42
72.42 1.39
81.02
0.99
70.78
1.72
GraphSR
70.85 0.44
71,37 0.63
59.26
00.72
55.96 10.95
73.61
71,88 1.33
$3.09 0.29
72.03
0.98
Ours
78.33 a: 1.04
76.44 1.06
65.03 1.38
64.94 :: 1.38
75.35
1.41
73.85 :: 1.43
85.08
0.38
75:27 2 0.23
+6.14
+3.77
+6.35
+8.13
+0.73
+0.72
+1.99
+0.96
Vinilla
62.33 1.56
61.82
38.84
1.13
31.25
64.60
55.24
2:50
70.04 1.60
70.00
2.50
Re- Weight
66.87 0.97
66,62
1.13
45.47
2.35
40.60
68.10 10.10.12...
3.54 63.76
80.38
0.66
69.99
0,76
Sotumax
1.30
50.78
2.08
72.88
71,09
71.33
GrapISMOTE
66.71 + 0.32
65.m = 1.21
0.93 45.68
38.06
0.97
67.43 1:23
2.54 61.97
1.97 70.38
69.76
2.51
RalancodSohmmax
67.89 0.36
67.96 041
$4.78 + 1.25
51.83
72.30 1:20
69.30
1.79
82.02 1.19
72014
BalateedSohmax TAM)
69.16
73.50
71.36
Renade
67.33
68.05
1.16
44.48
2.06
37.01
2.87
69.03 +.100++..100
66.27
2:00
76.01
1.08
66.72
1.42
Renode TAM)
67.50 0.67
68.06 L 0.96
45.12
1.41
39.29
1.79
70.66 2.13
66.94
3.54
74.30
1.13
66.13
L.75
GraphiNN
70:45
1.32
51.45
2.08
73.15
71,90
71.23
GraphENN orr TAM)
70.15 0.18
70,00 1 0.40
56.15
1.13
$4.31
1.68
1.07 73:45
72.10
0.36
81.07
1.00
71.27
1.98
GraphSR
70.86 0:22
70,61
0.38
56.85
1.09
55.02
1.55
78.38
72.85
0:33
61,72
71.91
L.87
72:58
Ours
78.91 0.59
75.99
1147
64.10
1.49
63.44 :: 1.47
74.68 1.43
72.75 :: 0,39
KS.62
0.44
75.34
0.99
+7.38
+4.65
+6.56
+7.68
+0.50
+0.13
+3.14
+2.40
Vinilla
61.82 0.97
60.97 1.07
43.18
0.50
36.66 1.25
68.68
64.16 2.58
72.36 + 2.30
64-32 1 2.21
Re Weight
63.94 1.07
63.82
1.30
46.17
1.32
40.13
L.68
69.39
68.71
2311
76.08
65.76
1.40
Sofumax
50.66
1.66 35.18
71.49
70.23
GraphSMTTE
61.65 0.34
0.98 60.97
2.87 42.75
1:75
66.63 + 0.66
2.54 61.97
0.98 71.85
68.92
0.73
RalanosdSohmax
67.43++006 0:61
67.66
0.40
51.74 -.3---3.33 2.32
49.01
3.16
71.36
69.65 1.51
73.67 +
65:33 2.44
SAGE
BalateedSohmmx TAM)
2.19
3.25
72.28
71,02
2.28
Renode
66.84 1.78
67.08
1.75
48.65
1.37
44.25
71:37
1:33
67.78 1:38
0.74 77.37
68:42
Renode TAM)
67.28 1.11
67.15
1.11
43.39
1.76
2.31 43.56
1.07 71.25
68.69 0.98
74.87
66:37
2.52
GraphleS
55.51
70377
N2.57
71.95
0.51
GraphENN or TAM)
70:45+
70.40
0.75
1.13 $4.60
53.56
1.86
1.355 73.61
72.50 + 1.58
82.17
0.93
72:46
1.00
GraphSR
09.24 0.42
68,82
53.98
0.74
51.92
1,10
71.43
0.75
70,46 1.15
80,97 0.44
12:34 0.55
5488
1.08
55.21
75.75
Ours
75.99 0L98
73.63 1.23
66.45
0.39
65.83 = 0.30
74.78
1.30
72,80 +: IL.4
83.21 1.50
7031 1,70
+5.40
+3.08
+11.22
+12.12
+1.03
+0.14
+0.24
+1.65
35

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Table 21: Experimental results of our method and other baselines on four iaasliimallnceed node classification benchmark datasets with p 20. We report averaged balanced accuracy (bAcc., %) and Fl F1-score (%) with the standard errors over 5 repetitions on three representative GNN architectures.
Danasei
Cora
CheeSeer
PubMed
Amazon Compusers
Imbalance Ratio (p 20)
bAcc.
F1
bApc.
F1
bAcc.
F1
bAcc.
FI
Vanilla
53.20
0.88
47.81
35.3
21.81
0.12
61.13
0.35
46.85
65.42
3.00
Re- Re-Weight
57.51 1.05
54.63
1.08
36.99
1.79
27.33
2.32
66.52 2.42
58.22
3.65
72.45
2.06
65.85
1.46
PC Softmax
61.74 1.50
1.97 60.55
42,53
1.53
36.54 1.13
1.99 68.26
66.54
1.87
73.84
2.64
66.30
2.97
64.06
62.88
47.29
44.08
69.71
68.31
76.93
BalamcdSofmmx TAM)
63.46
1:70
3.02
Renade
59.40 + 1.00
56,88
1.52
38.25
1.60
27.61 2.25
67.45 + 3.34
60.40=5.74
74.15 1.72
67.27
0.92
Renade (w TAM)
59.88
1.16
58.05
1.66
41.11
-.44
31.58
2.62
68.53 3.53
64.82 4.32
73.46
1.77
67. 07.50++4
1.18
GriphENS
67,50
66.82
46.39
42.58
71.37
69.37
1.69
75,41
69.30
GraphENS TAM)
66.34
66.67
2.98
45.06
71,92
69.35
75.78
68.58
GraphSR
67.98 = 1.42
67.53 = 1,36
47.03
3.40
43.06
4.06
72,05
1.72
70.01 + 1.64
75.97
1.70
69.96 1.54
BIM
67.94
1.32
67.51
1.76
46.98
3.26
42.91
3.95
12.05 + 1.68
69.95
1.52
76.04
1.61
69.91
1.44
A
46.62
+7.01
+8.81
+1.01
+0.76
+8.60
+4.85
Vanilla
$1.51
46.59 0.61
34.74
60.22
46.00
0.70
60.08 2.76
Re- Woight
58.68 3.44
55.98
3.97
36.78
: 0.94
26.63 1.61
63,47
1.73
54,63
71
2,42
62.86
1.94
PC Sofimax
59.62 1.41
1.95 58.77
2.01 43.38
37,76
2.12
70.81
1.41
70.25
1.30
71.16 1.15
62.26 0.87
62.05 1.62
61.14
1.71
47.89
44.84 1.35
69.91
67.43
1.73
72.91 1.93
62.79
BalancedSofmax TAM
63.30
62.81
1.29
2,90
Renade
59.52:+22.38
57.16
2.47
37.21 - 2.01
27.09 3.17
64.56 1.65
55.87 2.83
69.
2.35
59.00
1.67
Renode (o TAM)
61.32 1.128
59.19
2.64
39.85 1: 1.20
30.63
2.63
66.28 3.24
$8.99
3.04
65.81 2.57
$6.73
1.62
GraphENS
64.52:42.005
62.52
43.74
37.47
69.00
65.54
71.78
2.30
61.83
GraphENS TAM)
1.62
1,79
44.81
70.33 2:33
23.55 3.04
1.32
GraphSR
64.76 11.76--2200
62.75
1.79
43.96
3.70
37.73 4.10
69.21
2.61
65.76
11776-3.44
3.48
72.03 + 2.25
62.04 1.72
BIM
64.72 2.03
62.81
1.88
43.91
3.79
37,72 4.18
69.21
2.65
65.77
3.52
72.01 :: 2.33
62.06 1.76
79.10
76.21
56.11
+13.22
+12.41
+10.36
Vanilla
54.61
50.95
37.36
62.04
54.18
62.70
$5.39
Weight
57.37 0.61
55.30
0.72
37.69
27.92
2.01
65.01
2.69
58.34
2.19
68.31
2.06
60.45 2.40
PC Softmax
$9.25 a 0.74
58.55
0.81
42.77
1,82
40.08
1.82
70.55
1.19
67.60
1.59
70.57 2.86
62.73 2.69
BslancedSofmax
61.93 1.26
60.89
1.36
43.64
1.33
38.31
1.13
69.89
1.40
68.12
0.78
68:45
2.92
62.12
3.10
RalancaSSofmax TAM)
0.94
1.10
40.17
$9.22
SAGE
Renode
58.46 0.97
55.39
0.94
10.65
1.36
3T.76
66.30 3.65
56.72
1.16
66. 06.66
T.56
61.6
3.00
Renode (w TAM)
$4.77 2:220
$1.98
=
42,50
0.93
35.11 1: 1.84
67.31 a:
60.63
3.49
2.32 66.42
$8.62
1.95
GriphENS
63.54:100:1
62.30
44.89
40.48
71.3F
69.37
1.66
75.47
67.49
GraphENS TAMI
61.66
1,96
75.75
GraphSR
63.75:1...9
62.42 00.42:000 0.89
45 03.06:00
2.48
40.71
2.91
71.59
1.76
69.61
1.67
75.71
2.18
67.74
1.66
BIM
63.98 + 0.93
62.68 0.88
45.29
5.50
40.95
2.90
71.84
1.78
69.86
1.66
25:05
3.2.
67.97
1.64
57.51
73.67
71.15
A
+8.94
+5.69
+12.43
+15.54
+1.83
+1.29
+2.93
+0.14
36

---

Table 22: Experimental results of our method and other baselines on four iaaslimmallnceed node classification benchmark datasets with p 50. We report averaged balanced accuracy (bAcc., %) and F1-score (%) with the standard errors over 5 repetitions on three representative GNN architectures.
Donaset
Cora
CheeSeer
PubMed
Amazon Compusers
Imbalance Ratio (p- 50)
bAcc.
F1
bApc.
F1
bAcc.
F1
bAcc.
FI
Vanilla
$1.81 1.662
43,98
37,59
0.17
23.54
0.13
61.65 0.34
47.95
17.36
69.68
Re-Weight
58.54 2,39
54.13
38.19 1.28
27.43
2.34
65.70 1.59
56.35
1.55-4.26 4.26
79.10
2.44
71.40
2,86
PC Softmax
64.87 2.33
3.14 62.01
2.14
2.70 38.83
69.21 + 0.59
69.40 0.87
81.00 1.63
74.34
2.13
BalencesSShhmmx
65.94
64.00
47.62-11.11
70.40
69.04
0.83
BalancdSoftmax
1.58
1:27
3.42
3,80
1:45
81.74
Renade
62.22 + 1.76
61,18
2.24
41,23
1.66
33,06
2.69
68.67 + 1.21
63.05 1.47
81.71 = 0.99
72.55 1.61
Renode (w TAM)
63.93
1.96
61.64 2.71
48.17 17
41.07
2.94
69.63 2.55
64.30 14.30-1.1
80.55 1.75
72.33
1.63
GriphENS
63.47
62.21
48.17
44.07
69.83
64.30
81.63
72.37
GraphENS TAM)
65.05
62.11
45.03
42.65
81.60
72.00
GraphSR
64.12:00..4
62,89 = 1.58
48.84
1.52
41.76
2.26
70.31 =: 2.48
64.98
=.49
82.28
1.33::2..33 2.30
73.21
2.28
BIM
65.72 1.07
62.80
1.90
45.68
1.29
43.33
1.88
70.42 0.74
71.46 0.66
82.34
2.17
72.36 1.71
A
+7.05
+5.34
+6.54
+6.92
+2.57
+1.91
Vanilla
45.53
Re- Weight
59.78 1.92
56.69
2.21
38.70 2.23
29.38
3.06
66.27 1: 0.68
$7.34
1.41
73:46 2: 3.07
67.00 1:00::100 2.60
PC Sofimax
59.44 1. 2.62
58.06
2.69
43.13 1.56
37.04 2.07
70.86 1. 0.44
70.96
0.54
0.54
71.21
2.90
69.17 2.89
SslmmcedSofmmx
64.71 64.71 1:2.28
62.55
2:61
$1,89
49.36
70.94 1:
20.33
0.99
77.49
70:44 2.33
Balancedoolmax TAM
1.03
1.48
Renade
63.81 1.72
60.63
2.76
41. 2.30
33.94 4:4
70.35 1.26
67.43
1335-0000 0.01
72.30
2.75
65.23
3.35
Renode (o TAM)
64.40 1.83
63,48
2.83
43.54 1.54
35.80
2.43
71.23 2.04
66.61
4,31
76.07
2,70
68:43 2.68
GraphENS
61.41
45.23
41.12
69.66
66.83
78.
2.74
70.44
GraphENS TAM)
65.33
65.34
48.
72.58
GraphhRR
65.17-42.44
62.11
10:11-330 3.08
45.89
2,89
41.79 4.10
70.31
0.98
67.49
11:40-0.31 0.91
79.05 2.66
71,12 2.46
BIM
65.98 2.60
66.03 2:4.7
48.63
1.42
48.87 1.38
72.19
1.22
73.28
1.03
80.65
3.27
73.03 2.42
$7.70
56.81
79.41
77.38
86.06
+7.37
+4.04
+3.76
Vanilla
53.02.1
45.58
38.81
25.28
61.41
$0,46
56.53
48.53
Weight
58.03 0.81
54.32
0.99
38.49
1.34
30.41 1.82
62.41
0.90
$1.37 2.62
70.36 2.21
61.52 2.73
PC Softmax
62.33 1.62
1.98 $9.97
41.79
1.19
36.90
0.84
1.09 69.58
67.13 0.95
73.53 a: 2.02
66.12 3.19
BslancedSofmax
64.57 0.77
62.22
0.82
41.84
1.72
40.09
1.04
70.43 1: 0.38
68.99
0.99
73:27
2.30
68.30 1.97
RalaneaSSolmax TAM)
65.97
0,88
1.65
1:83
66.45
Renode
61.35"11 1.86
58.85
2.53
40.37
1 2.33
32.57
3.62
61.34
3.05
59,77
70.76
3.65
6233
7,50
SAGI
Renode (w TAM)
62,79 1. 0.47
61.05
0.82
43.04
1.30
1.92 37.44
36.97
71.79 :: 1.33
67.80 2.45
74.55
2.95
66.06 2.16
GriphENS
63.95 0.96
62.65
41.99
66.07
61.63
76.21 2.84
68
GraphENS TAMI
GraphSR
64.58 1 0.91
63,32
2.05
42,67
1.49
38.13
2.35
66.78
1.08
62.31
1.75
76
2.78
68. (9.74 2.49
BIM
65.60 0.91
64.32
2.06
43.20
1.50
39.13
+.3.
67.84
1.07
63.37
1.76
77.92 2.79
69.78 2.49
72.99
58.70
75.27
72.16
82.03
+10.06
+7.46
+5.81
+7.61
+2.03
+1.57
+1.28
+0.67
37

---

Table 23: Experimental results of our method and other baselines on four insasaimblencee node classification benchmark datasets with p 100. We report averaged balanced accuracy (bAcc., %) and Fl F1-score (%) with the standard errors over 5 repetitions on three representative GNN architectures.
Danasei
Cora
CheeSeer
PubMed
Amazon Compusers
Imbalance Ratio - 100)
bAcc.
F1
bApc.
F1
bAcc.
F1
bAcc.
F1
Vanilla
$1.62
43,91
0.5
38.83
1.71::00.5
61.28
47.55 0.16
76.09
69.33
3.44
Re-Weight
59.11 1.06
54.04
1.36
42,67
1.60--2.06
33.17
3.40
67.14
55.24
5.36
81.53+2200
2.20
71.45
2.05
PC Softmax
63.75 1.02
61.19
1.48
18.34:00.71
33.65
1.42
70.85 + 0.44
0.63 70.36
82.22 1.99
72.38
2.52
63.03
61,28
48.49
46.59
68.83
83.33
74.34
BalancdSootmax
0.50
0.84
1.10
73.70
3.17
05:39
Renade
60.76 2.533
58.09
3.00
43,41 = 2.07
33,09
2.76
67.63 + 2.77
61.70
4.84
82.13 1.73
71.79
1.85
Renade (w TAM)
64.19 1:46
60.00
1.56
44.78
35:00
2.61
70.53
0.75
64.35
1.79
82.32
2.19
73.00
1.75
GriphENS
63.00
62.35
2.06 45.99
37.23
68.65
1.00
62.17
2.17 83.37
73.96
T.98
GraphENS TAM)
60,40 +:442
57.77
42.72:22.54
39.40
70.73
72.50
81.20
71.66
GraphSR
64.64
1,25
64.04
1.62
47.66
1.98
38.96
3.28
70.29
0.95
63.85
1.52
83.02 0: 2.12
73,00 #: 1.90
BIM
64.38
1.26
63.69
1.61
47.31
2.00
38.61 + 3.78
70.03 0.96
63.51
1.54
82.77
2.10
73.24
1.91
A
+3.38
+2.02
+5.06
+5.29
+5.00
+2.31
+0.60
+0.67
Vanilla
$1.58
0.21
62.07
64.87
Re- Weight
58.28 1.88
54,47
2.35
38.13
1.55
29.60 3.02
67.41
2.69
58.06
5.07
77,10 10:00::31 3.26
68.35 2.71
PC Sofimax
63.74 1.00
59.76
2.19
45.07 1.13
39.21 1 2.29
69.68 1.29
69.44 1.29
79.72 1.52
70.78 1.45
SslmmcedSofmmx
63.19 1.35
61.03
46.03 2.11
43.38
71,45 1.23
69
79.15 2.08
69
2.13
BalancedSofmax TAM
62.91
1.29
Renade
60.04
11.004+-223
58.04 09.04-.. 2.66
42,40 2.97
34.09 0.04
68.54 2.11
65.63-- 3.15
75. 33344-22 1.65
69.99
1.60
Renode (o TAM)
63.45
61.51 1.95
41.55 1.39
36.13 2.87
71.53 2.35
68.11 4,28
78.60
1.90
70.35
2.80
GraphENS
63.93-42220
61.77
39.26
68,
64.14
81.63
71.20
GraphENS TAM)
62.52:2:0.95
70.64
83.33
GraphhRR
64.89 1..99-+-200 2.62
62.74 0. 3.30
45.30
1.86
40.18
2.48
69.47 1.75
65.08 3.21
82,52
2.02
72.13 1 2.69
BIM
65.84 2.55
63.72
3.22
46.26
1.82
41.10
2.42
70.45 1.70
66.00 3.15
83.39
1.97
73.04
2.63
71.50
60.35
+7.78
+7.60
+4.50
+2.53
+2.79
Vanilla
52.65 0.65.10..34
43.79 0.47
36.63
0.09
62.29
47.00
$5.94
47.21 2.73
Weight
59.42 1.2...8888 2.88
55.26 -
36.24
1.30
27.07
2.88
63.33 0.75
55.11
1.62
70.76
3.35
62.09
3.30
PC Softmax
64.01 1.15
1.68 60.74
44.74
1.41
37.61 1.69
72.62 d: 1.42
1.70 70.95
75.96
2.44
69.12
2.90
BalancedSofmax
63.43 2:12.1
62.30
2.27
49.33
1.12
44.58 1.64
70.68 0.92
69.15 0.84
74.56
0.86
66.28
1.92
RalaneaSSofmax TAM)
64.56
72.22
71.02
Renode
62.92"110099
60,06
1.19
39.61
1 3.66
30.13" 10.1551153 1.
61.1 1.12
61.09
3,30
73.75
1..5511.33 3.26
64A7
3.39
Renode (w TAM)
62.06 2:2008
3.32
42.08
1.88
33.19 J:
69.95
1.01
65.99
2.28
74,81
3.29
67.48
3.32
GriphENS
$3.09:1:0097
61.20
42.03
36.71
69.71
63.47
3.87
81.33
72.83
GraphENS TAMI
GraphSR
66.45 2.10
64,42
1.83
53.52
1.47
$3.01
1.75
74,09 0:000::::00
2.12
72.97
1.90
81:45
0.87
72.65
1.54
BIM
62.75 2.13
64.68
1.05
55.83
1.662
$3.20
1.00
74.58
2.08
75.34
1.85
82.01
0.48
72.52
1.01
68.30
73.11
A
+5.72
+3.62
+6.04
+5.63
+2.73
+0.79
+0.91
+0.79
38

---

Table 24: Experimental results of our method and other baselines on Commuterr-Randoon We report averaged balanced accuracy (bAcc.,% %) and Fi-score (%) with the standard errors over 5 repetitions on three representative GNN architectures.
Dataset (Computers-Ranne
GCN
Imbalance Ratio (p 25.50)
bAcc.
F1
bAcc.
F1
bλcc.
F1
Vanilla
78.43 = 0.41
77.14
+ 0.39
71.35
#1.18
69.60
: 1.11
65.30
= 1.07
64.77
1.19
Re Weight
80.49 0.44
75.07
1 0.58
71.95 1 0.50
70.67
0.51
66.50
1.47
66.10
1.46
PC Soltmax
81.34 0.55
75.17
0.57
70.56
= 1.46
67.26
1.48
69.73
0 0.53
67.03
U.6
BalancedSoftmax
81.39 0.25
74.54
0.64
72.09 = 0.31
68.58 + 0.69
73.80
1.06
69.74
0.60
GraphSMOTE
71.98
1.01
Renode
81.64 1 0.34
76.87
1 0.32
72.80
0.94
71.40
1 0.97
70.94
1,50
70.04
1.16
GraphENS
0.53
BalmmedSoftmax-AAM
81.64 0.48
75.59
0.83
74.00 1 0.77
70.72 1 0.50
73.77
1.26
71.03
0.89
GraphENS+TAM
82.83 2: 0.68
76.76 + 0.39
0.72 75.81
72.62 + 0.57
78.98 = 0.60
73,59 = 0.55
GraphSR
77.78
76.79
73.61
77.63
0.51
BIM
84.03 U.73
77.96
0.4L5
77.01
0.71
73.82
0.50
77.76
0.65
72.09
0.37
1.86
A
+1.29
+2.47
+5.51
+5.08
-3.17
-1.73
Table 25: Experimental results of our method and other baselines CS-Random. We report averaged balanced accuracy (bAcc.,%) and F1 FI-score (%) with the standard errors over repetitions on three representative GNN architectures.
Dotaset (CS-Random)
GCN
GAT
SAGE
Imbalance Ratio 41.00)
bAcc.
F1
bAcc.
F1
bAcc.
F1
Vanilla
84.85 = 0.16
87.12
0.14
82.47 #0.36
84.21
+ 0.31
83.76
0.27
86.22 : 0.19
Re: Weight
87.42 1 0.17
88:70 0.10
83.55
0.39
84.73
1 0.32
85.76
0.24
87.32
1 0.16
PC Softmax
88,36 +: 0.12
88.94
0.04
85.22
85.54
0.33
87.18
17.18=0.14 0.14
88.00 + 0.19
GrapESMOTE
85.76 1 1.73
87.31 1.32
1.32 84.65
85.63
1.01
1.98 85.76
87.34
0.98
BalancedSoftmax
87.72 - 0.07
88.67 + 0.07
84.38
0.20
84.53
86.78
0.10
88.05 + 0.09
BalancedSoftmax (w TAM)
88.22 + 0.11
89.22
85.48
0.24
85,77
0.50
87.83
88.77 0.07
Rénōde
83.91
0.06
83.38
0.T9
8697
7000
86.13
0.10
87.89 1 0.09
Renode TAM)
87.55 + 0.06
89,03 + 0.05
86.61 0.30
0.24 87.42
85.21
0.33
87.01
0.31
GraphENS
85977 27729
86.68
85
0.19
Mo.31
+ 0.32
33.30 0.26
86.4T
: 0.34
GraphENS TAM)
0.13
87.25
GraphSR
86.73 1. 0.22
85,91 0.21
85.34 1 0.13
86.56 1. 0.29
85.44
0.27
86.46 10.23
BIM
Ours
88.94 0.09
89.87
0.06
87.65
0.12
87.65 0.11
88.03
0.21
88.65 +: 0.07
0.58
0.65
1.04
0.23
0.20
-0.12
39

---

Table 26: Experimental results of our method and other baselines on Flickr. We report averaged balanced accuracy (bAcc.,%) 5) and Fi-score (%) with the standard errors over repetitions on three representative GNN architectures.
Model
GCN
GAT
SAGE
Imbalance Ratio (p 10.80)
bAcc.
F1
bAcc.
F1
bAcc.
Vanilla
24.62
25.32
24.16
Re-weight
28.31 = 1.64
24.06 1.16
30.66 = 0.76
27.12 0.34
27.39 = 1.84
22.62 + 1.04
Softmax
GraphSMOTE
0OM
0OM
0OM
00M
00M
0OM
BalancedSoftmax
27.61
23,70 0.77
26.01
2881
23.50
28.24
2.10
24.98 +
BalancedSofimax (ir TAM)
27.06 + 1.03
23.97
0.60 OOM
28.24 + 0.99
25,52 + 0.89
29.79 1 0.37
27.56
0.25
Ranode
DOM
OOM
COM
Renode (w TAM)
0OM
OOM
COM
ODM
OOM
OOM
GraphENS
OOM
TOM
GraphENNS (w TAM)
0OM
OOM
0OM
OOM
OOM
OOM
GraphSR
27.63 0.59
23.73
088
26.03
2.75
23,53
3,15
28.26
2.18
25.01 1 1.62
BIM
27.87 + 0.65
23.75 0.73
26.15 2.70
23.74 + 3.10
28.34
2.00
25.03 + 1.66
Ours
30.76 =: 0.27
30.60
0.29
29.45 # 0.72
28:21 = 0.76
30.68
0.63
31.01 = 1.34
Table 27: Experimental results of our method and other baselines on Ogbn-Arxiv. We report averaged balanced accuracy (bAce..%% .)) and Fi-score (%) with the standard errors over repetitions on three representative GNN architectures.
Model
GCN
GAT
SAGE
Imbalance Ratio (p 775.40)
bAce:
Fl
bAce.
bΛcc.
Vanilla
50.21 + 0.65
49.60
0.14
0.87
49.23
0.33
50.76 0.21
49.43
0.29
Re-weight
50.24 + 0.40
49.71
0.12
51.12 0.80
49.65 = 0.25
50.81 + 0.19
49.78
0.22
PC Softmax
50.20 0.58
49.64
0.77
49.16
0.19 50.82
49.65
0.24
GS
O0M
0OM
OOM
0OM
00M
0OM
RalancedSoftmux
0.13
49.36
BalancedSoftmax (w TAM)
50.34 1 0.48
49.72 0.10
$1.36 0.72
49.98
1 0.26
$0.94 L 0.17
49.95 0/22
RaNode
00M
DOO"
OOM
GOM
COM
REmode (w TAM)
0OM
OOM
OOM
OOM
OOM
OOM
GrighTRS
DOOM
"OOM
DOO
ODM
COM
GraphENS TAM)
0OM
OOM
OOM
0OM
OOM
OOM
GraphSR
50,31 1 0.24
49.70
0.17
51.31
0.41
49.33 = 0.26
50.86
0.30
49.53 0.20
BIM
50.33 = 0.42
49.71
0.19
51.35 0.60
49.36
0.28
50.87 0.18
49.56 0.23
Ours
51.21 2
50.65
$1.84
$1.28
51.34 0.22
$1.36
+0.92
+0.40
40

---

J Comprehensive Abaltion Study
J.1 Analysis for Decoupling Representation and Classifier for Imbalance Node Classification.
We conduct more extensive experiments on the Cora and Amazon-Compure datasets using three different GNN architectures to analyze the effect of decoupling representation and classifier for imbalanced node classification. We hypothesize that even the GNN encoder is trained on skewed data, the embeddings it learns are of high quality.
Experimental Setup. As explained in Section 4.1, we can obtain two pseudo-labels for all unlabeled nodes, one from unsupervised algorithms and the other from supervised classifiers. Experiments on more datasets are conducted compare the accuracy of the two pseudo-labels for all unlabeled nodes. We chose the two benchmark datasets, Cora and Amazon-Compuln more specific, half of the classes are
to build scenarios with
varying degrees of imbalance (p= 1, 5, 10, 20, 50, 100). designated as minority classes and randomly selected labeled nodes are converted into unlabeled nodes until the training set's imbalance ratio reaches p. The GNN architecture is fixed the 2-layer GNN (i.e. GCN [25], GAT [54), GraphSAGE [12]) having 128 hidden dimensions and train models for 2000 epochs. We set the K-Means algorithm's s cluster size to 200. Each experiment is repeated five times, and the average experiment results under different imbalance ratios are shown in Figure 9.
0.700
0,700
0.60
supervised Supervised
UnSupervised Supervised
Unsupervised
0.675
0.675
0.58
Supervised
Accuracy (%)
0.650
Accuracy (%)
0.625
Accuracy
0.625
0.52
0.575
0.575
0.590
0.550
0.50
0.525
0.525
0.48
0.500
20
40
60
100
0.500
20
60
100
0.46
Imbalance Ratio
Imbalance Ratio
20
80
100
Imbalance Ratio
(a) Cora-GCN
(b) Cora-GAT
(c) Cora-SAGE
0.90
0.90
supervised Supervised
supervised Supervised
Unsupervised Supervised
0.85
Accuracy (%))
Accuracy (%))
Accuracy (%)
0.80
0.80
0.75
0.70
0.70
0.6
0.65
0.65
0.5
0.60
40
80
100
0.60
40
60
100
20
80
100
Imbalance Ratio
Imbalance Ratio
Imbalance Ratio
(d) Computers-GCN
(e) Computers-GAT
(f) Computers-SAGE
Figure 9: The experimental results on Cora and Compuess under different imbalance scenarios (p= 1, 5, 10, 20, 50, 100). We compare the accuracy of the two pseudo-labels (predictions) from unsupervised algorithms and supervised classifiers respectively for all unlabeled nodes.
Analysis.
As depicted in Figure the predictions generated by unsupervised algorithms maintain high accuracy rate even in imbalanced scenarios. The final results unveil several intriguing insights: (1) In imbalanced scenarios, both supervised and unsupervised algorithms exhibit degraded performance, particularly in extreme cases (p = 50, 100). (2) The predictions derived from the embedding space outperform the biased classifier, indicating that the classifier is the weaker component when trained on imbalanced training set. (3) Extensive experimental results demonstrate the significance
41

---

of predictions from unsupervised algorithms and classifiers, suggesting that relying on single component does not lead to optimal performance.
J.2 Detailed Analysis for DPAM
DPAM utilizes an unsupervised algorithm to derive pseudo-labels for each unlabeled node in the embedding space. Only unlabeled nodes with aligned pseudo-labels and classifier predictions are included in the candidate pool. This approach effectively mitigates the bias issue of the classifier, preventing the inclusion of low-quality nodes in the training set based skewed confidence rankings. To gain a deeper understanding of DPAM's underlying mechanism, we conduct a set of novel experiments outlined below.
Experimental Setup. We use DPAM to filter the unlabeled nodes of the whole graph, and test the accuracy of pseudo-labels (prediction of the classifier) of the aligned node set Min and the discarded node set Hout respectively. DPAM based on different GNN structures are trained on two node classification benchmark datasets, Cora, and Amazon-Computer., We process the two datasets with traditional imbalanced distribution following [73, 37, 47]. The imbalance ratio p between the numbers of the most frequent class and the least frequent class is set as 1, 5, 10, 20, 50, and 100. We fix architecture as the 2-layer GNN (i.e. GCN [25]. GAT [54]. GraphSAGE [12]) having 128 hidden dimensions and train models for 2000 epochs. We select the model by the validation accuracy. We observe the accuracy of pseudo labels for unlabeled nodes which are filtered out and absorbed into by DPAM respectively. We repeat each experiment five times and present the average experiment results in Table 28 and Table 29.
Analysis. DPAM partitions the unlabeled nodes of the entire graph into two subsets, namely, Uin and Vout- The accuracy of pseudo-labels for these two subsets is examined evaluate the effectiveness of DPAM. It is evident that the accuracy of pseudo-labels differs significantly between Uin and Houe in various imbalanced scenarios. Generally, the pseudo-label accuracy for Mmn is high, while it is comparatively lower for Uout, thereby validating the efficacy of DPAM. Moreover, as the imbalance ratio (p) increases, the accuracy of both subsets decreases, which reflects the model bias resulting from the imbalanced label distribution.
Table 28: Experimental results of DPAM effectiveness on Cora with p 1, 5, 10, 20, 50, 100. We observe the accuracy (%) of the pseudo-label (prediction of the classifier) of the aligned node set Unn and the discarded node set Hout respectively. We report averaged results with the standard errors over 5 repetitions on three representative GNN architectures. All, Labeled, Unlabeled represent the size of whole nodes, labeled nodes, and unlabeled nodes on the graph. Align, Out, Align- Out-Ture represent the size of Un, Mout, nodes with accurate pseudo-labels of Un, Vous respectively.
Datoset
All
Labled
Unlabled
Align
Align: True
Ascuracy(5)
Out
Out-True True
Avcuracy(3)
p=1
2718
140
2568
2072.00
1391 00 +
67.11
496.00
10.29
253.80
16.66
47.17 3.74
5
2718
92
2616
2122,80
18.93
1392.00 34.21
65.58 +: 1.57
403.20
18.73
186.80 + 13.08
37.86 + 1.75
2708
86
2622
2134,60
23.42
1326.40 24.23
62.14 1.67
487.40
23.43
181.60 à 18.24
37.32 à: 3,13
2708
83
2625
2149.60
17.67
1310.20
86.72
60.97 3.50
475.40
17.67
169.80 + 21 21.47
35.64 3.44
50
2708
203
2505
1860.80
:: 31.15
1059.40 58.77
56.90 +: 2.62
644.20
31.14
225.80 + 10.70
35.05 + 3.79
100
2708
403
2305
1820.40
12.42
1001 10.60 4: 21.60
55.02
3.99
23.99 484.60
151,40 4: 20.74
31.78 4: 2.37
1
2708
140
2568
2072.00 =: 37.18
1412.40 d: 37.31
68.16 1.41
496.00
20.89
239.40 d: 11.37
48.29 :: 2.15
2708
2616
2141.40
26.36
1433.00
59.82
66.90
474.60
26.36
195.20
41.02 3.27
GNT
2708
86
2622
2132.60
29.94
1377 40 49.61
64.58
1.60
489.40
29.95
185
:: 12.28
37.97 :: 1.13
20
2708
83
2625
2150.60
=: 37.35
1344.60 à: 54.17
62.16 1.64
462.40 = 33.28
178.00 d: 5.05
38.60 :: 2.12
2708
140
2568
1892.40
37.18
80
57.52
1.52
612.60
37.17
271.20
44.35 1.86
100
2708
403
2305
1934.60 :: 19:65
1038.20 21 00
53.66 0.83
370.40
37.17
147.53
3.20
39.83 :: 1.36
2708
2568
1944,00
25.77
973.40
51.27
624.00
25.77
237.00 +: 13:28
5
2708
92
2616
2004.40 =: 35.
1038.20 22.53
$1.50 3.73
611.60 = 35.50
203.80 7.15
33.40 :: 1.85
2708
86
2622
2041.60
32.48
1039.00 41.32
50.89 1.88
580,
32,48
189.20 2.35
32.56 1: 4.25
2708
83
2625
2040.20
1002.20 66.97
48.95 2.66
578.80
186.60 +: 18.00
32.18
2708
203
2505
1789.40 =: 30.56
870.20 +: 24.33
48.63 1.03
715.60 = 30.56
242.40 +: 16.77
33,87 + 1.18
100
2708
403
2305
1859.00
192.42
914.41 1: 23.65
49.36 2.59
446.00 21.24
138.87 1: 6.32
31.15 1: 2.43
42

---

Table 29: Experimental results of DPAM effectiveness on Compute- with 5, 10, 20, 50, 100. We observe the accuracy (%) of the pseudo-label (prediction of the classi- fier) of the aligned node set Unn and the discarded node set Mout respectively. We report averaged results with the standard errors over 5 repetitions on three representative GNN architectures. All, Labeled, Unlabeled represent the size of whole nodes, labeled nodes, and unlabeled nodes on graph. Align, Out, Align-True Out-Ture represent the size of Min. Mouts nodes with accurate pseudo- labels of Mans Mont respectively.
Dotaset
All
Lahled
Unlabled
Align
Aligo-True
Ascuracy(55)
Out
Oun-True
Accuracy
13752
200
13552
11977.60
108.09
9603 50 93.34
80J88 +: 3,07
1554.40 40
676.60
141.11
43.58
2.83
13752
13632
11593.60
73.16
9172.50
87.30
2308.
173.54
30.74
13752
13642
1807.60
495,00
100.37
20
13732
105
13647
11866.60
17.34
8698.20
188.13
73,40 1.39
1780.40 67.36
521.00
60.76
29.20 2.41
60
15752
255
13497
11843.20 = 168.20
8994 40 175.24
75.94 + 0.75
1653.00 + 138.11
474.20 = 50.72
28.68
2.16
100
15752
505
13247
9159.00
192.42
7352.00
61.23
51.41
4.59
4088
93.99
1129:60
75.74
28.67
4.77
13008.00
9984
1544.80
101.94
580.40
190.49
43.33
0
13752
120
13632
11570.80 =: 136.11
8715.00 86.33
75.33 1: 0.54
2061.20 4: 136.13
477.00
91.07
25.39 1.33
10
13752
110
13642
8947.60
13:40
6680.40 2: 177.54
75.85 1: 6.07
4094.40 a: 134.74
591.80
13.74
15.94 :: 2.97
20
13752
105
13647
10045.80
68.00
7300 1000.80:0 64.89
71.42 1.80
3401.20 20 69
370.60
43.87
18.52 0.09
15752
255
31.56
7772.00
2.85
3303.40
457.20
19.28 1.43
13752
505
13247
11377.00
63.32
15752
200
13552
10815.20
86.50
7131.403 72.83
65.94 0.28
2736.80 + 86.50
965.40 56.42
35.26 1.31
13752
120
13632
10627.80
78.33
6728.00
53.24
63.25 0.36
3001.20 1: 78.00
978.20 59.93
32.55 1.49
15752
13642
10475.00
118.11
6015.00
5167.00 18.41
52.71
33.59
13752
13647
10653.20
87.35
0098840-1
24.57
50
13752
235
13497
11044.80 =: 129.14
6760.80 1: 50.26
61.22 1: 3.42
2442.20 d: 48
879.00 =: 91.45
35.71 1.78
100
13752
505
13047
9175.20 :: 32.53
6475 607566011 1: 80.88
72,07 +: 1.96
4071 800008001 52.63
1218.60
14.70
34.43 1.08
J.3 Detailed Results and Analysis about Fluctuation of RBO Values in Node-reordering
In Section 4.2, argue that the classifier's confidence becomes increasingly valuable as the iteration progresses, gradually balancing the training set, whereas the geometric rankings are determined in the embedding space and remain unaffected by the classifier. Consequently, we can trust that the similarities between the Confidence Rankings and the Geometric Rankings will gradually increase as the confidence gains credibility throughout the iterative process. unsupervised algorithm performs inferiorly compared to supervised methods, particularly when dealing with balanced training set. Therefore, by leveraging the combined features of both rankings, we can significantly enhance the performance of our algorithm. To validate the aforementioned hypothesis, experiments are conducted.
worth noting that the
0.34
0.35
Class (Majori (lass7(4toor)
Class1(Major)
0.32
0.40
Class7(Minor)
0.30
0.28
RBO Value
0.35
RBO Value
0.26
0.30
0.25
0.24
0.25
0.22
0.20
(Major)
0.20
0.15
0.18
(M
0.15
1
2
3
4
1
2
6
1
5
6
7
8
Round
Round
Round
(a) Cora-GCN
(b) Cora-GAT
(c) Cora-SAGE
Figure 10: Fluctuation of RBO values (p= 10) of two rankings iterations progress.
Experimental Setup. We conduct more experiments on Cora (p 10) to observe the similarities between the Geometric Rankings and Confidence Rankings. The architecture is fixed as the 2-layer GNN (i.c. GCN [25], GAT [54], GraphSAGE [12]) having 128 hidden dimensions and train models
43

---

for 2000 epochs. The UNREAL model's hyperparameter settings can be found in Appendix K.4. We choose a majority and minority class at random to compare the similarities of their respective two rankings (our setting is the first class and the last class of Cora), and we limit the number of iterations to eight. Each experiment is repeated five times, and the average experiment results are reported in Figure 10.
Analysis. As depicted in Figure 10, it is evident that the similarities between the Confidence Rankings and the Geometric Rankings exhibit a gradual increase during the initial stages of iteration. This observation substantiates our hypothesis. It noteworthy that, as the training set becomes gradually balanced, the similarity between the two rankings of the minority class surpasses that of the majority class. This finding further emphasizes the compensatory advantage of UNREAL for the minority class.
J.4 Detailed Analysis for Node-Reordering and DGIS
Experimental Setup. We conduct experiments test the accuracy of pseudo labels for unlabeled nodes on insslimbblanced graphs. All model combinations based on different GNN structures are trained on two node classification benchmark datasets, Cora, and Amaon Cotanonnr.mm.srs......m We process the two datasets with a traditional imbalanced distribution following [73, 37, 47]. The imbalance ratio between the numbers of the most frequent class and the least frequent class is set as 1, 5, 10, 20, 50, and 100. We fix architecture as the 2-layer GNN (i.e. GCN [25]. GAT [54], GraphSAGE, [12]) having 128 hidden dimensions and train models for 2000 epochs. We select the model by the validation accuracy. We observe the accuracy of pseudo labels for unlabeled nodes which are newly added to the minority class of the training set. We repeat each experiment five times and present the average experiment results in Table 30 and Table 31.
Analysis. As shown in Table 30 and Table 31, we validate the efficacy of each component of our framework by assessing the accuracy of the selected pseudo-labels for nodes using various model combinations, namely DPAM+Confidens ranking (with or without DGIS), and DPAM + Nooddeeogg (with or without DGIS). Notably, across different imbalanced scenarios, both components of our framework (Node-reorderin and DGIS) demonstrate significant importance, resulting in superior performance compared to other model combinations.
ranking (with or without DGIN), DPAM-Geometric
J.5 Detailed Ablation Analysis
Considering the space limitations of the main paper, we present the detailed ablation analysis herein. In this section, we conduct ablation studies to analyze the individual contributions of each component in our proposed method. The results from Appendix J.1 have already confirmed the necessity of incorporating unsupervised learning in the embedding space. Therefore, in this section, DPAM is applied in all comparative methods. We evaluate the performance of three different ranking techniques: confidence ranking, geometric ranking, and Node-reordering (which combines the former two rankings using information retrieval techniques). Additionally, we examine the impact of DGIS, which aims to mitigate the presence of geometrically imbalanced nodes. As illustrated in Table 32, each component of our method demonstrates performance improvements. Notably, in three out of the four settings presented in the table, Node-Reorte Reordeeig + DGIS achieves the highest Fl scores. Furthermore, across all cases, geometric ranking consistently outperforms confidence ranking, supporting our hypothesis that confidence scores may be biased and less reliable.
J.6
In-Depth Comparison between Self-Training and Our Method
J.6.1
The Comparison of Accuracy of Pseudo Labels for Self- Self-Training and Our Methods
We provide complete evaluation results more benchmark datasets, where more basic models are included in addition to the reported results in the main paper.
Experimental Setup. Since true labels for all benchmark nodes are provided, we first conduct experiments to test the accuracy of pseudo labels for unlabeled nodes on class-imbalanced graphs inventively. We select top 100 unlabeled nodes newly added to the training set through ST & Ours and evaluate the performance of ST & Ours by testing the accuracy (%) with the standard errors of these nodes' pscudo labels. We test unlabeled nodes that are selected into the minority classes and
44

---

Table 30: Analyzed experimental results of Node-Reordering and DGIS on Cora with P 1,5, 10, 20, 50, 100. We select 100 unlabeled nodes newly added to the minority class of training set through different method combinations, and evaluate the validity of Mode-Reordering DGIS by testing the accuracy (%) with the standard errors of the pseudo labels for these nodes. We report averaged results over repetitions on three representative GNN architectures.
Dataset
Cora
Imbulance Ratio (p)
p=
p=
p= 20
100
DPAM-Confidence Ranking
61.40 + 2.73
62,40 +: 2:59
60.20 + 1.02
58.40 + 1.05
57.60 + 1.86
58.40 2.15
DPAM-Cioomctric Ranking
64.00
61.20
2:89
61.20
1.31
55.60 :: 2:31
47,80
DPAM DPAMe-Node-Rerdrrnng
89.65 3.23
86.98 4: 0:21
88.32 = 0.83
85.32 4: 2.98
90.87 2.31
71.60 2.91
DPAM+Comiskenee Ranking DGIS
71.00
75,40 +: 2.15
68,20 2: 1,25
69 4:.40:110 1.28
67.80
2.75
66.60 0.16
DPAMsGeometric Ranking DGIS
69.60
+: 0.45
64.20
+: 1.57
69.00
DPAM-Nade-Reorderring + DGIS (Ours)
92.80 1,30
96.40 :: 4.27
92.20
0.85
W9.40
1.37
93.00 :: 0.82
77.80 2.50
DPAM-Confience Ranking
61.60 d: 4,26
64.00 +: 2.07
62,60 = 3.47
$7.80
1.65
58.20 +: 1.07
60.60 :: 0.79
DPAM-Goomstric Ranking
64.00 2,78
67.80 +: 3,76
65,00
4,30
$2.00 1.02
65.20 d: 2.58
40,80 2.63
OPAM--ood-Reenddring
91.79
90.45 +: 5.78
84.32
3:45
0.23
V0,32 +: 0.43
75.34 1.54
DPAM+Confidence Ranking + DGIS
69.80 2.77
72.80 d: 3.94
72.40
1.13
67. 60.60
1.59
71.60 d: 9.12
64.00 d: 1.74
DPAM-Cieometric Ranking DGIS
73.60
74:00 5.47
20
2.17
68,00 +: 1.17
MAMM--ode-Reorderti DGIS (Ours)
93.80 1.92
91.20 d: 4.60
90.40
1.69
90.00 d: 9.92
94.60 :: 4.92
78.20 d: 2.47
DPAM-Conildence Ranking
54.80 4,96
53.00 +: 2.46
51.80
1.97
43 2.57
46.20 +: 0.53
41.60 1.14
DPAM+Geometric Ranking
53.60 d: 2.78
45.40 + 1.75
40.60 :: 0.26
52. $2.60 2.47
47.40 da 4.27
44.80 d:: 2.84
SAGE
DPAM-Nind-Reondering
90,60 0.21
86.90 0.56
34 2.43
75.34 4.20
76.43
DPAM-Confdence Ranking + DGIS
66.20 =:: 5.78
39.00 +: 3.04
63.80 = 1.52
54.60
1.64
60.60 4: 1.37
57.40 4: 2.26
DPAM-Goometric Ranking- DGBS
6..60:t3..1
61,80 +: 5.21
$4,00 7,31
53 63.60 1,38
63,00 +: 1.23
45,20 1.96
OPAMr-odd-Reorderingg (Ours)
97.50 1.78
1.32
89.20
1.39
8.04
Table 31: Analyzed experimental results of Node-Reordering and DGIS on Amazon-Compute with 1, 10, 20, 50, 100. We select 100 unlabeled nodes newly added to the minority class of training set through different method combinations, and evaluate the validity of Node Groe-Reengg & DGIS by testing the accuracy (%) with the standard errors of the pseudo labels for these nodes. We report averaged results over 5 repetitions on three representative GNN architectures.
Dataset
Compuleen
Imbulance Ratio (p)
p=
p=
p=
p
100
CoAMiConfidecce Ranking
75.40 :: 2,50
70.20 +: 3.03
74.88 + 3.11
68. 20 à: 4,20
63,60 +: 2.30
61.40 1.51
DPAM-Cioometric Ranking
76.00
74.80
4.71
76.80
80
3.27
64.80
3.70
65.60
DPAM DPAM--Nod-Reeddring
82.80 2.38
79.60 +: 3.64
78.20 = 0.26
74.00 à 3.28
65.20 :: 1.87
66.00
2.82
DPAMHComisknnee Ranking DGIS
76.40 :1 2.07
67.20
4.32
75.80 :: 2.38
66. 20 3,70
62.80 :: 0.12
59,20
1.30
DPAM:Geometric Ranking DGIS
78.20
+:
66.00
+: 3.83
DPAMoNodde-Reordeeinng IGGS (Ours)
84.40 3.60
82.20 :: 2.16
80,40
3,46
80.60 it 1.51
69.60 :: 3.04
66,40 3,20
DPAM-Confidence Ranking
84.60 2.40
79.20 4: 1.78
73.00 d: 2.12
74.80 2.16
65,00 +: 1.73
68.60 +: 1.40
DPAM-Geomstric Ranking
86.00 3,80
79.80 +: 2.94
74,80
3.42
75.0
2.91
70,80 :: 2.16
69.40
1.10
W7.40 2.30
80.60 +: 3:04
80.40 - 2.19
79.00
3.67
75 +: 1.22
73.40 :: 2.52
DPAM--onfidence Ranking + DGIS
84.20 d: 1.64
79.40 4: 2.07
76.40
6.50
76.00 à: 2.34
66.00 4: 0.12
72.00 d: 1.84
DPAM-Cieometric Ranking DGIS
N3.80 1.09
80.20
76.20
2.58
71.60 0.89
TAAM--Nede-Rordeeiin DGIS (Ours)
89.00 2.54
86.60 d: 2.50
85.60
4.44
83.40 3.31
78.00 J: 3.39
79.80 A: 3.03
DPAM+Conillence Ranking
N5. 20 3.38
80.20 +: 6.26
84.8 0.83
77.
0.89
61.00 :: 0.70
65,40 2.65
DPAM-Geometric Ranking
86.00 0.70
81.20 d: 2.16
83.40 d: 1.14
78.00 d: 1.22
61.40 d: 0.54
65.00 d:: 1.72
SAGE
DPAM-Nudd-RRerdderig
N6.00 1.58
83.20 3:27
84.60
79.20
1.92
61,80
0.44
67.80
DPAM-Confdence Ranking + DGIS
86.40 c:: 2.07
81.60 4. 3:20
83.40 =: 1.14
79.20
0.44
61.20 4: 0.44
70.40 4: 3.59
DPAM-Giemettrcc Ranking- DGIS
87.00 -t: 2.12
80.80 +: 2/48
84,20 1.30
78. 20.30-2
1.48
61.20
0.47
68.20 1.72
DPAM+Node- Reooddeeen (Ours)
88.20 2.16
87.60 d:
78.00 1.55
d: 2.86
45

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Table 32: Ablation analysis on different components.
Modules
Confidence ranking
Geometric ranking
Node-roordering
DGIS
73.93
72.74 + 0.63
75.85 = 0.82
Cora+GCN 10)
75.34 0.63
75.00 0.97
76.44 1.06
46.09 4.08
47.76 1.06
50.32 = 3.75
CAE (p 20)
53.32 3.75
55,43: 2.14
57.51 4.92
76.34 0.39
75.42 0.39
77.32 + 0.21
PubMed+GAT (p- 50)
76.89 1.43
76. 2.63
77.38 0.39
70.86 + 1.73
68.86
1.2
72.32 2.43
Computers-GAT 100)
73.65 0.67
74.03 2.53
75.83
0.74
unlabeled nodes that are selected into the majority classes separately. We evaluate the performance of each method on Cora, CiteSeer, PubMed, Amazon Amazon-Compunn Computers under different imbalance scenarios. We process the datasets with traditional imbalanced distribution following [73. 37. 47]. The imbalance ratio between the numbers of the most frequent class and the least frequent class is set as (balanced), 5, 10, 20, 50, 100. We fix architecture as the 2-layer GNN (i.e. GCN [25], GAT [54], GraphSAGE [12]) having 128 hidden dimensions and train models for 2000 epochs. The validation accuracy is used select the model. Each experiment repeated five times, and the average experiment results are reported in Figure 11.
Analysis. It is observed that the parameter p increases, the accuracy of pseudo labels selected into minority classes decreases, indicating influence of classifier bias is amplified. The end results demonstrate number of intriguing aspects. (1) Highly imbalanced scenarios render the pseudo labels generated by the classifier unreliable. Moreover, the addition of unlabeled nodes with pseudo labels corresponding to minority classes in the training set introduces excessive noise during the ST process. (2) It is important to highlight that the evaluation of ST focuses solely on the top 100 nodes selected based on Confidence Rankings (Section 4.2). So, we believe that even node's pseudo label is correct, the classifier's confidence is skewed, resulting in the inclusion of low-quality unlabeled nodes in the training set while overlooking high-quality unlabeled nodes. This factor likely contributes to the underperformance of ST in imbalanced scenarios. (3) Irrespective of selecting majority class nodes or minority class nodes, our approach consistently outperforms ST. A thorough examination of Figure 11 clearly indicates the significant superiority of ours over ST, with the gap in Fl scores between the two methods widening as the imbalance ratio increases.
J.6.2 F1 Score Performance for Self-Training and Our Method
Experimental Setup. We utilized three citation datasets, Cora, CiteSeer, and PubMed, to construct scenarios representing varying degrees of class imbalance. Specifically, we select half of the classes as minority classes and convert a randomly selected subset of labeled nodes into unlabeled nodes until the training set achieved the desired imbalance ratio (p). The architecture we employed consisted of a 2-layer graph neural network (GNN) with 128 hidden dimensions, using GCN [25], GAT [54], or GraphSAGE [12]. The models were trained for 2000 epochs. As for the Self-Training (ST) method, the size of added nodes for each class was treated as hyperparameter, which we fine-tune based on the accuracy of the validation set. Each experiment was repeated five times, and the average results reported for different imbalance ratios, depicted in Figure 12.
46

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Analysis. The findings presented in Figure 12 demonstrate the efficacy of ST in improving imbalanced learning across the Cora, CiteSeer, and PubMed datasets under three GNN architecture.. The results consistently reveal that ST surpasses the performance of the vanilla model across different ratios and datasets, indicating the usefulness of unlabeled nodes. Nevertheless, the study uncovers a gradual decline in ST performance in heavily imbalanced scenarios, particularly for Cora and CiteSeer. We posit that this diminished performance in highly imbalanced data stems from the biased and unreliable predictions of classifiers, leading the inclusion of low-quality nodes in the training set at an early stage.
J.6.3 Discussion on the Choice of Clustering Method in Our Approach
Although embeddings in model are approximately distributed on a hypersphere, we employ K-Means clustering for its computational efficiency and robustness. While vMF mixture models other manifold-awan aware approaches (e.g.. spectral clustering cosine-based K-Means) oretically well-suited for such data, they involve expensive iterative optimization steps (such as DoneotatiionMaxxmizzzttii
the-
with Bessel function evaluations) and are less scalable to large graphs. In practice, we observe that normalized embeddings already exhibit meaningful Euclidean structure, allowing K-Means perform effectively without significant overhead. Our framework remains modular, and replacing K-Means with vMF or cosine-based variants can be considered as future work.
47

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100
100
100
90
90
Accuracy (%)
Accuracy (%)
Accuracy (%)
80
80
70
70
ST
70
60
ST
60
UNREAL Kajor
Minor
UNREAL AL Major
Minor
50
UNREAL Major
Minor
UNREAL_Kior
UNREAL_Kinor
UNREAL_Minrr
50
20
100
50
40
20
100
Imbalance Ratio
Imbalance Ratio
Imbalance Ratio
(a) Cora-GCN
(b) Cora-GAT
(c) Cora-SAGE
85
80
80
75
60
Accuracy (%)
75
Accuracy (%)
70
Accuracy (%)
70
70
65
60
65
60
60
50
ST
55
ST_ Minor
Si_Minor
40
ST. Minor
UNREAL Major
UNREAI, Major
UNREAL KNReAL_Kajrr
45
Winor
UNREAL Miner
30
UNnor
20
100
20
100
20
100
Imbalance Ratio
Imbalance Ratio
Imbalance Ratio
(d) CiteSeer-GCN
(c) CiteSeer-GAT
(f) CiteScer-SAGE
100
100
100
95
95
Accuracy (%)
90
Accuracy (%)
90
Accuracy (%)
90
85
85
80
75
75
70
ST Minor
MinM
ST Minor
65
UNREAL Kater
60
UNREAL Major
65
UNREAL Majer
Minor
UNREAL Minor
60
Minor
60
20
60
100
20
60
100
20
60
100
Imbalance Ratio
Imbalance Ratio
Imbalance Ratio
(g) PubMed-GCN
(h) PubMed-GAN
(i) PubMed-
100
100
100
90
90
Accuracy (%)
80
Accuracy (%)
Accuracy (%)
80
70
70
60
60
50
Major
ST_Major
Sajoojor
ST. Minor
Minar
ST. Minor
40
UNREALNajor
UNREAL Kajor
UNREAL
UNREAL Minor
UNREAL
40
UNREAL Minor
30
20
60 80 100
20 40
80 100
20
60 100
Imbalance Ratio
Imbalance Ratio
Imbalance Ratio
(i) Computers-GCN
(k) Computers-GAT
(1) Computers-SAGE
Figure 1I: Here, we present the experimental results from four benchmark datasets under various imbalance scenarios. We select top 100 unlabeled nodes newly added to the training set via ST & Ours, and evaluate the performance of ST & Ours based on three GNN architectures by testing the accuracy with the standard errors of these nodes' pseudo labels. Minor means that we only test unlabeled nodes which are selected into the minority classes, and Major means that we only test unlabeled nodes which selected into the majority classes.
48

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Varila
Varilo
Varila
80
LIBREAL
LINREAL
80
LIAREA
F1-score (6)
60
F1-score( 9%)
60
(6)
60
40
40
20 Imbalance Ratio
50
100
20
50
100
10
20 Imbalance Ratio
50
100
Imbalance
Ratio
(a) Cora-GCN
(b) Cora-GAT
(c) Cora-SAGE
Vaiila
Varilo
Vaiila
80
UIREAA
LEREAAL
80
LEREAL
F1-score (K))
60
F1-score (%)
60
F1-score (%)
60
40
40
20
20
10
20
100
10
20
50
100
10
20
100
Imbalance Ratio
Imbalance Ratio
Imbalance Ratio
(d) CiteSecr-GCN
(c) CiteSeer-GAY
(f) CiteScer-SAGE
Varilla
Varilla
Varill
80
URREAL
URREAL
80
UEREAL
Fl-scoretgo (6)
60
F1-score (6)
60
F1-score(%0
60
-0
30
20
20
10
20
50
100
10
20
50
100
10
20
50
100
Imbalance Ratio
Imbalance Ratio
Imbalance Ratio
(g) PubMed-GCN
(h) PubMed-GAT
(i) PubMedd-MSae SAGE
Figure 12: The experimental results on the three citation datasets under different imbalance scenarios
(p 10, 20, 50, 100). We report the F1-score (%) with the standard errors of Vanilla, ST, and Our Method.
49

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K
Experimental Setup
In this section, we introduce the method of imbalanced datasets construction, evaluation protocol, and the details of our algorithm and baseline methods.
K.1 Imbalanced Datasets Construction
Table 33: Summary of the datasets used in our experiments.
Nodes
Edges
Features
Cora
2,708
5,429
1,433
7
Citeseer
3.327
4,732
3,703
Pubmed
19,717
44,338
500
3
Amazon- Computers
767
Connthor-CS
18,335
163,788
6805
15
Flickr
89,250
899.756
500
7
Obgn-Arxiv
169343
1166243
128
40
Table 34: Label distributions on the whole graphs
Co
C,
C,
C,
C
C1
Cp
Gis
C
Cura (p 4.54)
351
217
418
818
426
CiteSeer (p 2.00)
264
590
068
701
696
308
PrlhMad (n a 1.91)
4103
7730
7835
Amazo--Cmmjneer
17.73)
436
2142
1414
542
5158
308
487
818
2156
201
Couthor CS (p 35.05)
708
462
2050
429
1394
2193
371
924
775
118
1444
2033
420
4136
876
Fückr 10.88)
22966
Table 35: Ogbn-arxiv label statistics: node counts per class in the full graph, training set, and test set.
Class
Whole
Train
Test
Class
Whole
Train
Test
Class
Whole
Train
Test
Class
Whole
Train
Test
Co
565
437
C1o
7869
5182
1455
Cs:
2076
1495
313
Con
11814
4334
4631
C1
68/7
382
187
C1:
750
391
239
Cz:
393
304
51
Cy
2828
1350
892
799
386
Can
270
83
Cg
2080
1014
654
C15
2358
1290
Cza
2834
1539
808
Css
1271
926
220
C4
5832
2864
1869
Cys
597
433
71
Cas
22187
6989
10740
Csi
7867
5436
1414
1246
248
1257
475
Cas
127
25
Co
1618
703
622
Ci6
27321
9948
10471
Coi
4605
2834
1041
Coi
3524
2506
627
C7
589
380
134
C17
515
202
203
C27
4801
1661
2066
Cer
2369
1615
481
4056
1250
749
402
21406
16284
2849
1507
1100
214
Co
2820
2245
345
C19
2877
1873
Cz
416
239
120
Con
2009
1551
260
The detailed descriptions of the datasets are shown in Table 33. For each citation dataset, for 10, 20, follow the "public" split, and randomly convert minority class nodes unlabeled nodes until the dataset reaches an imbalanced ratio p. For p 50, 100, since there are not enough nodes per class in the public split training set, choose randomly selected nodes training samples, and for validation and test sets we still follow the public split. For the co-purchased networks Amazon-Computers,
randomly select nodes as training in each replicated experiment, construct a random validation set with 30 nodes in each class and treat the remaining nodes as the testing set. For Flickr, we follow the dataset split from [69]. For Cemputers-Randone a training set of equal proportions based on the label distribution of the entire graph (Amazon- Computers). The label distribution the training set for Computers Random is summarized in Table 36. The details of label distribution in the training set of the five imbalanced benchmark datasets are Table 36, and the label distribution of the full graph is provided Table 34.
we build
Saaputers-Radom
K.2 Details of GNNs
We evaluate our method with three classic GNN architectures, namely GCN [25]. GAT [54], and
GraphSAGE [12]. GNN consists of = 1, 2, 3, layers, and each GNN layer is followed by BatchNorm layer (momentum = 00.9) and PRelu activation [15]. For GAT, we adopt multi-head attention with heads. We search for the best model on the validation set. The choices of the hidden unit size for each layer 64, 128, and 256.
50

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K.3 Evaluation Protocol
We adopt Adam [24] optimizer with an initial learning rate of 0.01 or 0.005. We follow [47] devise a scheduler, which cuts the learning rate by half if there is no decrease in validation loss for 100 consecutive epochs. All learnable parameters in the model adopt weight decay with a rate of 0.0005. For the first training iteration, we train the model for 200 epochs using the original training set for Cora, CiteSeer, PubMed, or Comput-C.mputers. For Flickr, we train for 2000 epochs in the first iteration. We train models for 2000 epochs in the rest of the iteration with the above optimizer and scheduler. The best models are selected based on validation accuracy. Early stopping is used with patience set to 300.
K.4 Implementation Details
In UNREAL, we employ the vanilla K-means algorithm as the unsupervised clustering method. The number of clusters K is chosen from (100, 200, 250, 300, 450, 450, 500} for Cora, CiteSeer, PubMed, mmaon-Computers, Computers-Randon and CS-Random. For Flickr, is selected among (1000, 2000, 3000, 5000}. For Cora, CiteSeer, PubMed, and Amazon-Computes and CS-Random, T
the number of
training round 7 are tuned among {40, 60, 80, 100). For Computers- selected from {4, 8, 12, 16, 20, 24). For Flickr, 7 is tuned among {40, 50, 60, 70}. We also introduce hyperparameter a, which is the upper bound on the number of nodes being added per class per round. The tuning range of is {4, 6, 8, 10) for Cora, CiteSeer, Amazon-Componn (64, 72, 80} for PubMed. For Computers-Raune Random and DS-Random, the value of a is chosen among {2, 4. 6, 8, 10, 12, 14, 16}. For Flickr, the value of is selected among {30, 40, 50, 60}. The weight parameters p in RBO is selected among {0.5, 0.75, 0.98], and the threshold in DGIN is tuned among (0.25, 0.5, 0.75, 1.00). For Flickr, only add minority nodes the training in all iterations, which means that we set a = 0 for majority classes in Flickr.
and
K.S Baselines
For GraphSMOTE [73], we use the branched algorithms whose edge predictions are discrete- valued, which have achieved superior performance over other variants in most experiments. For the ReNode method [6]. we search hyperparameters among lower bound of cosine annealing
Wmin {00.55, 0.5, 0.75] and upper bound of the cosine annealing Wmax E {1.25, 1.5, 1.75) fol- 0.15, which is the default setting in the released codes. For TAM [47], we search the best hyperparameters among the coefficient of ACM term {1.25, 1.5, 1.75). the coefficient of ADM term 3 {0.125, 0.25, 0.5). and the minimum temperature of class-wise temperature á E {0.8, 1.2] following [47]. The sensitivity to imbalance ratio of class-wise temperature 5 is fixed as 0.4 for all main experiments. Following [47], adopt warmup for iterations since we utilize model prediction for unlabeled nodes. For BIM [71] and GraphSR [75], follow the hyperparameter settings and network architectures provided in their official code repositories.
lowing [6]. PageRank teleport probability is fixed
K.6 Configuration
All the algorithms and models are implemented in Python and PyTorch Geometric. Experiments conducted on a server with an NVIDÍA 3090 GPU (24 GB memory) and an Intel(R) Xeon(R) Silver 4210R CPU @ 2.40GHz.
51

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Table 36: Label distributions in the training sets.
4.55%
C14
23.30%
C13
2.27%
C12
11.36%
C11
7.95%
Ci0
1.82%
0.95%
0.39%
0.20%
1.50%
0.57%
Co
0.95%
1.99%
0.20%
15.79%
3 3.98%
Co
1.82%
0.95%
0.39%
0.20%
6.02%
5.11%
C7
2.32%
1.19%
0.49%
0.25%
1.82%
0.95%
0.39%
0.20%
3.01%
1.70%
42. 16%
Ce
2.32%
1.19%
0.49%
0.25%
3.03%
1.59%
0.65%
0.33%
0.95%
0.39%
0.20%
2.26%
11.93%
3.90%
Cs
C₁
2.32%
19%
0.49%
0.25%
3.03%
0.65%
0.33%
18.184
19.05%
19.61% 1%
80%
38.35%
7.39%
25.83 83%
23.26%
24. 10%
24.63%
24.81%
3.03%
1.59%
0.655
0.33%
99.05%
19.61%
19 80%
3.76%
2.27%
5.45%
18.
23 26%
24. 10%
24. 24.63%
24.81 %
31.75%
32. 62.68%
33 30.00%
4.76%
2.44%
0.99%
0.50%
19.05%
19.61%
19 80%
10.53%
11.36%
7.09%
C2
30
23.26%
24. 10%
24.63%
24.81 1
31.75%
32. 63.68%
33.00 000%
47.62 77.6%%
48. 78%
49. 50%
49.75%
19.05%
19.61%
15.79%
2.27%
9.68%
C1
30.
19.
23.26 %
70%
24.63%
24.81 %
30. %0%
31.75%
32.68%
33.00%
47. 62%
48 78%
49 50%
49. 75%
19 05%
19.61%
80%
3.01%
3.98%
89%
Co
18.
Computers- Random (p = 25. 505
Amazon Computers (p= = 100)
Amazon Computers (p=
Amazon- Computers (p
Amazon- Computers (p
IS-Random (p= 41.00)
CheSSer (p= 10)
CheeSer (p= 20)
CieSser (p= 50)
CieeSer (p= 100)
PubMed (p= = 10)
PubMed (p = 20)
PubMed (p = 50)
PubMed =p= 100)
Flickr (p 22 (0.80)
Cora (p= 100)
Cora (0= 10)
Cora (0= =0) 20)
Cora (p=50) 50)
Dalaset
52

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L Pseudocode of Our Algorithm
Algorithm 1 Our Algorithm
Input: Graph G (V,E). labeled set Co, feature matrix X, adjacency matrix A, unlabeled
u = V Lo. max rounds T, per-class selection threshold a, RBO weight parameter GI
threshold 7, cluster size k, learning rate 77, GNN model fg. clustering function feluster mean function (.).
for 1 T do
2: Train GNN fg on current labeled set Li-1-
3: Obtain embeddings H and H for labeled and unlabeled nodes.
4: Predict class logits i and confidence scores for U.
5:
// Step 1: Dual Pseudo-lahel Alignment (DPAM)
6:
Cluster H into k clusters: fclusterH' - {K PX, 1.
f=11
7:
Compute class centroids from labeled data: P4c... = M(f ye = =m]).
8:
Assign pseudo-labels to clusters: Dj = arg mimm distance( Mc..)
9:
Construct cluster-based pseudo-label sets Um and classifier-based sets Um.
10:
Obtain consistent node sets: 4 = Lm nm
11:
// Step 2:
(NR)
12:
for each class m = 1 to C' do Compute geometric distances: 6, distance(hu, MC,.) for Ufinall
13:
14:
Build geometric ranking Sn, (ascending by 6.) and confidence ranking Tm (descending by ru).
15:
Compute RBO score Tm = RBO(S Tm).
16:
Fuse rankings: A/New max (' T' S,n min 1 rm Tm- Select top-o: nodes from /Now as candidates Ceand
17:
18:
end for
// Step 3: Discarding Geometrically Imbalanced Nodes (DGIS) for each selected node u E Um Ccand do
20:
Let 8, be distance to closest centroid, S. be distance second closest. Compute GI index: GI, Bu=6__
66
GI, < then Discard node u.
24:
else
26:
Add node u to labeled set L, with its pseudo-label.
27:
end if
28:
end for
end for
53

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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
The main claims in the abstract and introduction are consistent with the theoretical contributions, method design, and extensive experimental results presented throughout the paper, including the identification and mitigation of geometric imbalance in semi-supervised imbalanced node classification.
Justification:
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
is fine to include aspirational goals as motivation as long as is clear that these goals not attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: The limitations of the proposed method are discussed in the paper, such as the reliance on clustering quality and the potential computational cost in certain components. The paper also mentions potential directions for future work and acknowledges that the approach is mainly evaluated on node classification tasks (see Conclusion).
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model vell-specifiiattin, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested few datasets with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and faimess.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't acknowledged the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
54

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3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
Justification: All theoretical results are stated with explicit assumptions, and proofs or proof sketches provided either in the main text referenced in the appendix. The mathematical formulation geometric imbalance and theorems are clearly presented (see Section 3).
Guidelines:
The answer NA means that the paper does not include theoretical results. All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should properly referenced.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided not)?
Question:
Answer: [Yes]
Justification: The paper provides sufficient details on datasets, model architectures, evalua- tion metrics, and experimental setups to allow reproduction of the main results. The code implementation also provided in the supplementary material (see Abstract and Section
5.1).
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
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Reviews:
[{"review_id": "JdrHXgRwFs", "reviewer": "Reviewer_gHPB", "summary": "This work formally explains the concept of geometric imbalance in the embedding space within the context of semi-supervised node classification. The authors theoretically analyze the geometric imbalance problem in a hyperspherical embedding space and propose \n frameworks - pseudo-labeling method and a node reordering strategy that accounts for geometric proximity informed by classifier confidence. Experiments are conducted on imbalanced homophilous graph datasets, including naturally imbalanced graphs, using three representative GNN architectures.", "questions": "- Does this method also work on heterophilous graphs?\n\n- Can this problem be extended beyond GNN architectures, such as to graph transformers or graph-specific MLPs?", "limitations": "The manuscript lacks a limitations section. The authors should discuss the limitations of their approach in the main paper.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### Strengths\n- The authors make an effort to position their work within the existing literature by explicitly discussing differences from closely related methods, such as ReNode and GraphENS.\n- The experimental design is well-designed. The chosen baselines span both general-domain and graph-domain methods, and the proposed approach shows promising improvements on graphs with high imbalance ratios.\n- The paper offers theoretical insights into the geometric imbalance problem, enhancing understanding of its impact in semi-supervised node classification.\n\n---\n\n### Weaknesses\n- While the authors attempt to establish the novelty of their work through comparisons with prior methods (e.g., ReNode and GraphENS) in both the main text and appendix, the originality of the findings remains unconvincing for two reasons. First, the notion that class imbalance introduces bias in the embedding space - and that minor-class instances are susceptible to pseudo-labeling noise - is already well-established in the broader domain, even if not explicitly analyzed via the von Mises-Fisher (vMF) distribution. Second, although the authors emphasize that their focus is on the embedding space and on the ambiguity of unlabeled nodes, it is unclear whether this constitutes a significant departure from prior work, given that GNN embeddings are inherently shaped by the graph structure. For these reasons, the primary contribution appears to lie more in the theoretical interpretation of geometric imbalance rather than in the identification of a novel problem.\n- The paper lacks comparisons with existing pseudo-labeling strategies. Numerous recent pseudo-labeling approaches have been proposed in both semi-supervised learning and unsupervised domain adaptation. A discussion or empirical comparison with these methods would strengthen the paper by clarifying its relative advantages.\n- The improvements on naturally imbalanced graphs appear marginal, which is arguably more critical than performance on synthetically imbalanced datasets. It would be helpful if the authors could elaborate the reasons.\n- There is a weak connection between the theoretical findings and the proposed methods, even though the authors acknowledge computational challenges and the absence of ground-truth labels.", "was_revised": true, "final_justification": "Thank the authors for their detailed response and the additional experiments provided during the rebuttal period.\n\nAfter carefully reading the response, I find that some of my concerns have been addressed - particularly regarding the comparison with existing pseudo-labeling approaches [W2] and the experiments on heterophilous graphs [Q1] (assuming the baselines were fairl tuned in the heterophilous setting). The results are clear.\n\nTwo points remain unconvincing. For [W1: Questions about the novelty of the work], the authors argue that their work differs from general-domain studies by focusing on how message passing behaves under class imbalance. However, my concern is that the known issues from the general domain, such as unreliable pseudo-labels for minority classes, combined with prior graph-specific studies (*e.g., * GraphENS, ReNode, TAM), which already address unreliable representations for minority or topological boundary nodes, make the contribution of this work limited. The authors’ claim that *their perspective is different because it operates in the embedding space rather than on the raw graph itself* is quite unconvincing, as embeddings are inherently derived from graph structure, and structure-embedding relations on inter/intra-class embeddings have been explored (*e.g.,* [1]).\n\nThe authors also attribute the stronger results on synthetic data to factors such as sparse labels, long-tail distributions, and structural noise. However, I find it hard to understand that the method does not perform well under these conditions, and in my experience, such types of noise are also observed in synthesized imbalanced graphs. These are not minor issues - being able to handle real-world imbalance is essential for the method to go beyond being merely publication-oriented and become applicable in practice. Also, improved results rely on additional hyperparameter tuning gives the impression that the work is not yet fully ready at this stage.\n\nAs mentioned, some concerns have been addressed while others remain.Since the authors have responded to certain concerns, if all other reviewers strongly support acceptance, I will respect the consensus decision.\n\n\n[1] IS HOMOPHILY A NECESSITY FOR GRAPH NEURAL NETWORKS? (ICLR 2022)", "rebuttal": "We sincerely thank Reviewer gHPB for acknowledging the strengths of our theoretical formulation and experimental design, as well as for the detailed and constructive feedback. Your thoughtful comments and suggestions are invaluable in helping us clarify our contributions and improve the rigor of our work. Below, we respond point-by-point to address each of your concerns.\n\n**W1: Questions about the novelty of the work.**\n\n**R1:** Thank you for the thoughtful and constructive feedback. We acknowledge the reviewer’s concern regarding the novelty of our concept. However, we respectfully argue that our work presents a distinct and previously uncharacterized phenomenon—*geometric imbalance induced by message passing in GNNs*, which cannot be reduced to general notions of pseudo-labeling noise.\n\nSpecifically, while pseudo-label uncertainty has been widely explored in non-graph settings, our formulation is the first to **theoretically and empirically identify how message passing uniquely amplifies embedding ambiguity for minority classes** in the hyperspherical space. This effect is not observed in models without relational structure (e.g., MLPs), and we demonstrate this through a controlled comparative analysis.\n\nOur Section 3  has clarified this contribution and emphasize the distinction from prior work. As shown in Figure 1(a–e) of the revised manuscript:\n\n- In **Cases 1 and 2** (GNNs), class imbalance increases intra-class dispersion ($D_{\\text{intra}} \\uparrow$) and reduces inter-class separation ($D_{\\text{inter}} \\downarrow$) for minority-class nodes after fine-tuning;\n- In contrast, in **Cases 3 and 4** (MLPs), the embeddings remain geometrically stable and well-separated, showing little effect from imbalance.\n\nFurthermore, Figures 1(f–g) provide quantitative validation that **geometric imbalance correlates strongly with prediction entropy and imbalance ratio only under graph-based models**, underscoring the unique role of message passing.\n\nOur approach is also theoretically grounded: by modeling class distributions on the unit hypersphere via a von Mises-Fisher (vMF) mixture, we derive new measures of angular ambiguity and formally connect them to uncertainty in pseudo-labeling. This perspective is fundamentally different from prior notions of topology imbalance, as it operates in the embedding space, not the raw graph.\n\nWe hope these clarifications help convey that our work provides both **a novel theoretical lens and a practical mitigation strategy** for a GNN-specific challenge that has not been sufficiently addressed in prior studies.\n\n---\n---\n---\n**W2: Lacks comparisons with existing pseudo-labeling strategies.**\n\n**R2:** We thank the reviewer for raising this important point. We realize that our presentation of baseline methods and related supplementary experiments may not have been sufficiently clear in the original manuscript. To improve clarity and transparency, we have revised both the main text and appendix to make relevant comparisons more explicit. Please see below for a summary:\n\n1. **State-of-the-art pseudo-labeling baselines for imbalance node classification:** As discussed in the main text (Lines 325--326), we have used GSR and BIM—two representative pseudo-labeling models specifically tailored for imbalanced node classification—as key baselines. Their results are comprehensively reported in Tables 1–7 and 12–15, covering a variety of imbalance ratios, synthetic and real-world datasets, and GNN architectures. Our method consistently outperforms both baselines on balanced accuracy and Macro-F1.\n\n2. **General pseudo-labeling methods:** We also included results on general pseudo-labeling frameworks such as Self-Training, Co-Training, and M3S, summarized in Table 11 (Appendix I.6). Our approach shows clear advantages by explicitly addressing hard samples in the representation space.\n\n3. **Soft pseudo-labeling strategies (new supplementary experiments):** In direct response to your suggestion, we additionally evaluated our method against recent soft pseudo-labeling approaches—SCR (ICLR 2021) and ConsisGAD (ICLR 2024)—under two most challenging settings. For fairness, we performed thorough hyperparameter searches using Wandb  for both baselines. The results (see Table 1, 2) demonstrate that while SCR and ConsisGAD struggle under severe class imbalance (especially in Macro-F1), our approach remains robust.\n\nWe hope that these clearer explanations and new supplementary results adequately address your concerns. \n\nTable 1. Results with $\\rho=100$ (SAGE) on three  severe class-imbalanced node classification benchmark datasets.\n|Method| Cora bAcc.| Cora F1|CiteSeer bAcc.|CiteSeer F1|PubMed bAcc.|PubMed F1|\n|-|-|-|-|-|-|-|\n|**Ours** |**73.47±2.31** |**68.30±2.11**|**59.77±2.98** | **58.92±3.07** | **77.11±0.59**|**74.03±0.81**|\n| SCR| 59.31±2.91|55.34±4.32|36.51±1.29|27.19±2.84|63.41±0.76|54.97±1.64|\n|ConsisGAD|59.63±2.89|55.18±4.43|36.39±1.28|27.28±2.87|63.27±0.74|55.33±1.60|\n\nTable 2: Experimental results on Ogbn-arxiv ($\\rho\\approx 775.4$)\n| Model|GCN bAcc.|GCN F1|GAT bAcc.|GAT F1| SAGE bAcc.| SAGE F1|\n|-|-|-|-|-|-|-|\n| **Ours**|**51.21 ± 0.32**|**50.65 ± 0.32**|**51.84 ± 0.87**|**51.28 ± 0.42**|**51.34 ± 0.32**|**51.36 ± 0.27**|\n|SCR| 50.23 ± 0.41| 49.69 ± 0.13|51.10 ± 0.81| 49.67 ± 0.26| 50.83 ± 0.20|49.80 ± 0.23|\n|ConsisGAD | 50.25 ± 0.39| 49.72 ± 0.14| 51.13 ± 0.79| 49.63 ± 0.27| 50.80 ± 0.19|49.77±0.22|\n\n\n---\n**W3: Improvements on naturally imbalanced graphs appear marginal.**\n\n**R3:**  We thank the reviewer for raising this important point about naturally imbalanced graphs. Please see our clarifications below: \n- Naturally imbalanced datasets (e.g., ogbn-arxiv) involve not just class imbalance, but also additional challenges such as sparse labels, long-tail distributions, and structural noise. As a result, even marginal performance gains on these datasets are often hard to achieve. Despite this, as shown in Table 4–7, our method consistently achieves strong or state-of-the-art performance across different GNN backbones, confirming its robustness in practical settings.\n- Role of Synthetically Benchmarks: Synthetic datasets with controlled imbalance (e.g., Cora, CiteSeer, Pubmed) allow us to isolate and analyze specific algorithmic behaviors more clearly. The effects of geometric imbalance are more easily observed, which facilitates ablation studies and theoretical validation of each component (DPAM, Node Reordering, DGIS).\n- Further Improvements: The previous results on natural graphs were based on early-stage hyperparameter tuning. We have now performed more thorough parameter and architecture searches. Updated results on ogbn-arxiv (see below) show more significant performance gains, which will be reflected in the revised manuscript.\n|Model|GCN bAcc.| GCN F1|GAT bAcc.|GAT F1|SAGE bAcc.|SAGE F1|\n|-|-|-|-|-|-|-|\n| BIM| 50.33 ± 0.42| 49.71 ± 0.19| 51.35 ± 0.60|49.36 ± 0.28| 50.87 ± 0.18| 49.56 ± 0.23|\n|**Ours**|**51.21 ± 0.32** |**50.65 ± 0.32** |**51.84 ± 0.87** |**51.28 ± 0.42** |**51.34 ± 0.32** |**51.36 ± 0.27**|\n---\n**W4: Weak connection between the theoretical findings and the methods.**\n\nWe would like to clarify that our theoretical analysis directly motivates and informs the design of our proposed framework. In particular, our formal definition of **geometric imbalance** (Definition 1) and the accompanying theorems (Theorem 1 and 2) establish that minority-class nodes with high intra-class dispersion and low inter-class separation on the hypersphere are more susceptible to pseudo-labeling errors. This insight forms the foundation for all three components of our framework:\n\n- **DualPath PseudoLabeler (DPAM):** Designed to mitigate ambiguity by aligning clustering and classification views, directly addressing the high-entropy regions described in Theorem 1.\n- **Node Reordering (NR):** Fuses classifier confidence with geometric distance to class centroids, in line with our formal definition of embedding proximity.\n- **Discarding Geometric Imbalanced Samples (DGIS):** Implements a lightweight approximation of the theoretical geometric imbalance $G(u_j)$ score (Definition 1), using distances to the first and second nearest class centers.\n\nFurthermore, we validate the theoretical connections empirically in **Figure 1(f)(g) and Figure 3**, where we observe a strong correlation between geometric imbalance, prediction entropy, and class imbalance ratio, as predicted by our analysis. We have revised Section 4 of the manuscript to explicitly highlight these connections and better bridge the theory with our algorithmic design.\n\n---\n**Q1: Results on heterophilous graphs.**\n\n**R1:** Thank you for the question. While our main experiments focus on homophilic graphs, we are also evaluating our method on heterophilic benchmarks (Chameleon and Squirrel with imbalance ratio 5, Wisconsin with 11.63). Preliminary results show our method achieves SOTA performance. We will include these results in the revised manuscript.\n|Method|Chameleon bAcc.|Chameleon F1|Squirrel bAcc.|Squirrel F1|Wisconsin bAcc.| Wisconsin F1|\n|-|-|-|-|-|-|-|\n|GSR(GraphSR)|38.45±0.75|37.67±0.72|27.94±0.44|27.01±0.32|31.54±2.30|28.91±2.05|\n|BIM|38.26±0.68|37.48±0.66|27.83±0.46|27.16±0.37|31.59±2.25|28.79±2.01|\n|Ours|**43.59±0.60**|**42.24±0.63**|**30.04±0.48**|**29.12±0.46**|**46.22±3.51**|**41.52±4.05**|\n\n---\n\n**Q2: Extend GI to graph transformers or graph-specific MLPs.**\n\n**R2:** Thank you for the comment. Any model with message passing or structural aggregation (e.g., GNNs, graph Transformers) may suffer from similar geometric imbalance. Our three modules are based on embedding distributions and can be easily applied to any architecture with such mechanisms. We plan to explore this in future work.\n\n---\n\n**We sincerely hope that our additional results and clarifications resolve your concerns. It would mean a great deal to us if you could kindly reconsider your rating of our submission. Thanks.**"}, {"review_id": "e9urQXTdVS", "reviewer": "Reviewer_JjGt", "summary": "This paper introduces the concept of geometric imbalance, characterizing how class imbalance induces geometric ambiguity among minority nodes in the embedding space. Then, it provides a rigorous theoretical analysis and proposes a self-training framework to address geometric imbalance through pseudo-label alignment, node reordering, and ambiguity filtering.", "questions": "- Have you considered evaluating the proposed method on heterophilic graphs? It would be interesting to see how the approach performs in settings where connected nodes tend to have different labels.\n    \n- The results on the ogbn-arxiv dataset, as shown in Table 7, are relatively weak compared to other benchmarks. Could you provide an explanation for this performance drop?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n- The paper formally introduces the novel concept of geometric imbalance for the first time, which is a meaningful contribution to the field. By linking this concept with prediction entropy, the authors demonstrate its practical value and relevance to model uncertainty.\n    \n- The theoretical analysis is thorough and well-developed, providing solid justification for the proposed ideas.\n    \n- The experimental evaluation is comprehensive, covering multiple datasets and scenarios to validate the effectiveness and generalizability of the proposed method.\n    \n- This paper is well-written and easy to follow.\n    \n\nWeaknesses:\n\n- Expression issue in line 119: “introduced by class imbalance induced by class imbalance”.\n    \n- This paper lacks an analysis of the time complexity of the proposed method.", "was_revised": true, "final_justification": "Thank the authors for the detailed rebuttal. The response addressed most of my concerns. In particular, the time complexity analysis and the additional empirical results on heterophilic and large-scale graphs (e.g., ogbn-products) demonstrate the scalability and robustness of the proposed method. The updated experiments on ogbn-arxiv also show improved performance. Based on these clarifications and the additional evidence provided, I have increased my score to 5.", "rebuttal": "We sincerely thank Reviewer JjGt for carefully reading our paper and for recognizing **the novelty and significance of our proposed concept and theoretical analysis.** We appreciate your constructive feedback and thoughtful suggestions, which are invaluable for improving the quality and clarity of our work. **Below, we address each of your comments and concerns point-by-point.**\n\n---\n\n**W1: Expression issue in line 119: \"introduced by class imbalance induced by class imbalance\".**\n\n**R1:**   We thank the reviewer for pointing out this phrasing issue. In our revised manuscript (Line~119), we have rephrased the sentence to avoid the redundant wording. The new version reads:\n\n```\n... introduced by class imbalance in the embedding space ...\n```\n\nWe have also carefully reviewed the surrounding paragraph and the entire manuscript, including the appendix, to ensure there are no similar typos elsewhere and to maintain the rigor and clarity of our writing.\n\n---\n\n**W2: This paper lacks an analysis of the time complexity of the proposed method.**\n\n**R2:** Thanks for highlighting the need for an analysis of computational complexity and scalability. We address your concern in three aspects:\n\n**1. Time Complexity Analysis**. Our framework, as described in Algorithm 1 (see Appendix L), consists of three main modules per round: Dual-Path Pseudo-label Alignment (DPAM), Node-Reordering (NR), and Discarding Geometrically Imbalanced Nodes (DGIS). (1) The dominant computational cost arises from the DPAM step, which clusters the embeddings of unlabeled nodes using k-means (complexity $O(n_u k' d T_{iter})$, with $n_u$ unlabeled nodes, $k'$ clusters, $d$-dimensional embeddings, and $T_{iter}$ k-means iterations). Computing class centroids and intersecting with classifier-predicted labels are both linear-time operations. (2) NR involves ranking nodes by geometric distance and classifier confidence (each $O(n_u \\log n_u)$), with ranking fusion being negligible. (3) DGIS computes distances from candidate nodes to their closest and second-closest centroids ($O(n_u C d)$ per round, where $C$ is the number of classes). (4) Overall, the per-round complexity is primarily determined by k-means, but since $k' \\ll n_u$ and k-means can be efficiently implemented (e.g., on GPU), **so, the total complexity remains comparable to standard self-training pipelines. Memory usage is linear in the number of nodes, as we only store embeddings and label predictions.** *Importantly, our method is significantly more efficient than prior over-sampling or augmentation approaches such as GraphENS and GraphSMOTE, which may have $O(n^2)$ or worse complexity on large graphs. In summary, the only notable overhead beyond vanilla GNN training comes from clustering and ranking, both of which are scalable and can be accelerated by modern hardware.* \n\n\n\n**2. Empirical Scalability and Runtime**. To address practical efficiency, we conducted comprehensive scalability analyses on both medium- and large-scale datasets, comparing our method (\"Ours\") to vanilla GNN backbones and representative baselines. (1) On **Cora**, **CiteSeer**, and **PubMed**, the training time per epoch for our method is almost identical to the backbone GNNs (see Table 1), confirming the negligible overhead of our extra modules (clustering and ranking). (2) On large-scale graphs (e.g., **Flickr**, ogbn-products), our method remains tractable and runs smoothly on a single RTX 3090 GPU (Tables 6 and 7 in our submission). By contrast, prior data augmentation and oversampling baselines (e.g., **GraphSMOTE**, **GraphENS**) either become prohibitively slow or run out-of-memory, as synthesizing large numbers of new nodes/edges is expensive.\n\nTable 1: Refined Running Time Analysis of Vanilla Model vs. Ours: (a) Training Over 1000 Epochs, and (b) Selecting 100 Nodes to the Training Set.\n\n\n\n| Model/Time (1000 epochs)   | Cora     | CiteSeer  | PubMed    |   | Model/Time (Selecting 100 Nodes) | Cora     | CiteSeer | PubMed   |\n|----------------------------|----------|-----------|-----------|---|----------------------------------|----------|----------|----------|\n| Vanilla (GCN)              | 6.221(s) | 7.212(s)  | 10.771(s) |   | Vanilla Self-Training (GCN)      | 2.244(s) | 2.234(s) | 2.675(s) |\n| Ours (GCN)                 | 6.208(s) | 7.191(s)  | 10.858(s) |   | Ours (GCN)                       | 2.367(s) | 2.305(s) | 2.790(s) |\n| Vanilla (GAT)              | 8.201(s) | 9.574(s)  | 14.443(s) |   | Vanilla Self-Training (GAT)      | 2.624(s) | 2.520(s) | 2.431(s) |\n| Ours (GAT)                 | 8.218(s) | 9.517(s)  | 15.088(s) |   | Ours (GAT)                       | 2.697(s) | 2.670(s) | 2.411(s) |\n| Vanilla (SAGE)             | 6.522(s) | 11.170(s) | 15.475(s) |   | Vanilla Self-Training (SAGE)     | 2.424(s) | 2.780(s) | 2.032(s) |\n| Ours (SAGE)                | 6.452(s) | 11.256(s) | 14.658(s) |   | Ours (SAGE)                      | 2.461(s) | 2.827(s) | 2.115(s) |\n\n\n**3. Implementation Efficiency**. We employ GPU-accelerated k-means for clustering (via NVIDIA cuML), which substantially reduces the per-round clustering overhead. **Moreover, by filtering candidates after dual-path alignment, the more expensive geometric checks are only performed on a small subset of reliable nodes, further improving efficiency.**\n\n\n---\n**Q1: Results on heterophilous graphs.**\n\n**R1:** Thank you for the question. While our main experiments focus on homophilic graphs, we are also evaluating our method on heterophilic benchmarks (Chameleon and Squirrel with imbalance ratio 5, Wisconsin with 11.63). Preliminary results show our method achieves SOTA performance. We will include these results in the revised manuscript.\n|Method|Chameleon bAcc.|Chameleon F1|Squirrel bAcc.|Squirrel F1|Wisconsin bAcc.| Wisconsin F1|\n|-|-|-|-|-|-|-|\n|GSR(GraphSR)|38.45±0.75|37.67±0.72|27.94±0.44|27.01±0.32|31.54±2.30|28.91±2.05|\n|BIM|38.26±0.68|37.48±0.66|27.83±0.46|27.16±0.37|31.59±2.25|28.79±2.01|\n|Ours|**43.59±0.60**|**42.24±0.63**|**30.04±0.48**|**29.12±0.46**|**46.22±3.51**|**41.52±4.05**|\n\n---\n\n**Q2: The results on the ogbn-arxiv dataset, as shown in Table 7, are relatively weak compared to other benchmarks. Could you provide an explanation for this performance drop?**\n\n**R2:**  We thank the reviewer for this helpful observation. We agree that our performance on the *ogbn-arxiv* dataset (Table 7) is relatively less impressive compared to smaller citation networks like Cora, CiteSeer, and Pubmed, and we would like to provide the following clarification.\n    \n\nThe ogbn-arxiv dataset presents several unique challenges:\n\n- It is a large-scale and naturally imbalanced graph with a mild imbalance ratio, where the advantage of our method in severe imbalance scenarios is less pronounced.\n- Its temporal split (based on publication years) effectively imposes a distribution shift between training and test nodes, making pseudo-labeling more sensitive and less reliable.\n- The dataset exhibits complex global dependencies and limited local homophily, which may reduce the benefit of local embedding refinement techniques.\n\n**Also, the previous results were based on early-stage hyperparameter tuning. We have since conducted a more thorough parameter search and architecture calibration after the paper submission, and the updated results on ogbn-arxiv show more significant performance gains. These improvements will be reflected in the revised manuscript.**\n\n| Model     | GCN bAcc.        | GCN F1           | GAT bAcc.        | GAT F1           | SAGE bAcc.       | SAGE F1          |\n| --------- | ---------------- | ---------------- | ---------------- | ---------------- | ---------------- | ---------------- |\n| Vanilla   | 50.21 ± 0.65     | 49.60 ± 0.14     | 51.21 ± 0.87     | 49.23 ± 0.33     | 50.76 ± 0.21     | 49.43 ± 0.29     |\n| RW        | 50.24 ± 0.40     | 49.71 ± 0.12     | 51.12 ± 0.80     | 49.65 ± 0.25     | 50.81 ± 0.19     | 49.78 ± 0.22     |\n| PS        | 50.20 ± 0.58     | 49.64 ± 0.12     | 51.18 ± 0.77     | 49.16 ± 0.28     | 50.82 ± 0.19     | 49.65 ± 0.24     |\n| GS        | OOM              | OOM              | OOM              | OOM              | OOM              | OOM              |\n| BS        | 50.34 ± 0.41     | 49.73 ± 0.13     | 51.35 ± 0.69     | 49.36 ± 0.22     | 50.89 ± 0.19     | 49.56 ± 0.18     |\n| BS(w TAM) | 50.34 ± 0.48     | 49.72 ± 0.10     | 51.36 ± 0.72     | 49.98 ± 0.26     | 50.94 ± 0.17     | 49.95 ± 0.22     |\n| RN        | OOM              | OOM              | OOM              | OOM              | OOM              | OOM              |\n| RN(w TAM) | OOM              | OOM              | OOM              | OOM              | OOM              | OOM              |\n| GS        | OOM              | OOM              | OOM              | OOM              | OOM              | OOM              |\n| GS(w TAM) | OOM              | OOM              | OOM              | OOM              | OOM              | OOM              |\n| GSR       | 50.31 ± 0.24     | 49.70 ± 0.17     | 51.31 ± 0.41     | 49.33 ± 0.26     | 50.86 ± 0.30     | 49.53 ± 0.20     |\n| BIM       | 50.33 ± 0.42     | 49.71 ± 0.19     | 51.35 ± 0.60     | 49.36 ± 0.28     | 50.87 ± 0.18     | 49.56 ± 0.23     |\n| **Ours**  | **51.21 ± 0.32** | **50.65 ± 0.32** | **51.84 ± 0.87** | **51.28 ± 0.42** | **51.34 ± 0.32** | **51.36 ± 0.27** |\n\n---\n**We hope that these additional results and clarifications effectively address your concerns, and we sincerely hope you will reconsider your rating of our submission. Thank you again for your valuable feedback and thoughtful review.**"}, {"review_id": "yZbiDRoe1Q", "reviewer": "Reviewer_uqpA", "summary": "This paper studies imbalanced graph learning, and proposes a novel geometric imbalance for node classification. The theoretical analysis reveals the superiority of geometric imbalance for imbalanced GNN learning, motivated by which, several components including pseudo-label alignment, node reordering, and DGIS are proposed. Experiment results demonstrate the effectiveness.", "questions": "- Section 4.1 clustering: Why using K-Means clustering? K-Means uses Euclidean distance which may not be suitable for samples distributed on the sphere. Why not continuing using vMF mixture model for clustering?\n- Line 257: seems that geometric ranking is more important than confidence ranking, is there any evidence support this? Any experiments validating the necessity of (fusing) both rankings? In other words, what would happen if only CR or GR is functioning?\n- Line 259: “which is more reliable in early stages.”: any experiment validating this claim?\n- Writing\n    - Line 117: notations $u^i$ and $u^j$ are not consistent\n    - Figure 1: need better scatter plot, the current figures b-e are not clear.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n- Extensive analysis on geometric imbalance provides insight on the challenges of GNN imbalanced learning.\n- Neatly designed modules address the challenges.\n\n**Weaknesses**\n- Some claims need more justifications.\n- More studies needed to better understand the functionality of each module.", "was_revised": true, "final_justification": "The authors have addressed most of my concerns during the rebuttal. The additional experiments on K-Means clustering methods empirically solve my concerns on the choice of clustering methods. I will maintain my original rating.", "rebuttal": "We sincerely thank Reviewer uqpA for carefully reading our paper and for **recognizing the strengths of our analysis and module design.** We greatly appreciate your constructive feedback and insightful questions, which are very helpful for clarifying our work and improving the manuscript. **Below, we respond to each of your comments point-by-point.**\n\n---\n\n**Q1: Section 4.1 clustering: Why using K-Means clustering? K-Means uses Euclidean distance which may not be suitable for samples distributed on the sphere. Why not continuing using vMF mixture model for clustering?**\n\n**R1:**  We appreciate the insightful suggestion. Indeed, when data is distributed on a hypersphere , Euclidean-based clustering such as K-Means may not always be the optimal choice. Methods like von Mises–Fisher (vMF) mixture models or other manifold-aware clustering algorithms (e.g., spectral clustering on Riemannian manifolds, or angular K-Means using cosine similarity) are theoretically more suitable for such settings.\n\nHowever, in our case, we opted for K-Means due to the following practical considerations:\n\n- **Computational Complexity**: vMF mixture models require iterative Expectation-Maximization (EM) steps, and the computation of normalization constants involves Bessel functions, which are expensive and less scalable to large graphs. In contrast, K-Means has linear complexity in the number of samples and clusters.\n- **Robustness and Implementation Simplicity**: K-Means is widely used and numerically stable. In our experiments, we found that normalized embeddings already exhibit meaningful Euclidean structure, possibly due to the inductive bias of GNN training. Thus, K-Means works effectively in practice without introducing significant overhead.\n\n- **Modular Flexibility**: Our pipeline can be extended to alternative clustering methods. We will include a note in the revision to clarify that replacing K-Means with vMF or cosine-based variants is straightforward, and we leave this exploration as future work.\n\nWe thank the reviewer again for highlighting this aspect and will mention this discussion explicitly in Section 4.1 of the revised manuscript.\n\n---\n\n**Q2:  Line 257: seems that geometric ranking is more important than confidence ranking, is there any evidence support this? Any experiments validating the necessity of (fusing) both rankings? In other words, what would happen if only CR or GR is functioning?**\n\n**R2:** Thank you for your question. We agree that it is important to validate the necessity of combining Geometric Ranking (GR) and Confidence Ranking (CR).\n\nTo address this, we have conducted ablation studies that isolate the effects of CR and GR. Specifically, we compare three configurations: (1) using CR only, (2) using GR only, and (3) using the fused CR+GR ranking. The results are provided in Appendix J.3 (see Table 22, 23, 24), which clearly show that:\n\n- GR typically contributes more to performance than CR, especially on datasets with noisy neighborhoods or irregular topology.\n- However, combining CR and GR consistently yields better or comparable results than either one alone, demonstrating the complementary nature of these rankings.\n\nWe will make this finding more prominent in the revised manuscript and thank the reviewer again for highlighting this point.\n\n---\n\n**Q3:  Line 259: “which is more reliable in early stages.”: any experiment validating this claim?**\n\n**R3:** We appreciate the reviewer’s attention to this point. To validate the claim that geometric ranking (GR) is more reliable in the early stages, we conducted experiments that compare the effectiveness of CR and GR across training iterations.\n\nAs shown in Appendix J.3 (see Figure 10, Table 22, 23, 24), GR maintains higher consistency and informativeness during early training epochs, especially before model convergence. This supports our statement in Line 259. We have now clarified this observation in the revised manuscript and pointed explicitly to the corresponding experimental evidence in the appendix.\n\n---\n\n**Q4: Line 117: notations $\\mu_i$ and $\\mu_j$ are not consistent.  Figure 1: need better scatter plot, the current figures b-e are not clear.**\n\n**R3**: Thank you for the helpful feedback. We have corrected the inconsistent notations around Line 117 to ensure terminological clarity and coherence throughout the manuscript.\n\nRegarding Figure 1, we appreciate the suggestion. We have updated subfigures (b)--(e) with higher-resolution scatter plots and adjusted the color contrast and point size to improve visual clarity. The revised figure is now more readable and better illustrates the intended embedding structures.\n\n---\n\n**We hope that our additional results and clarifications satisfactorily address your concerns. We sincerely appreciate your valuable feedback and hope you will kindly reconsider your rating of our submission.**"}, {"review_id": "KgfP8WKAF8", "reviewer": "Reviewer_T2hF", "summary": "This paper introduces geometric imbalance, a phenomenon where class imbalance in graph data leads to ambiguous node embeddings for minority classes, hindering semi-supervised node classification. To address this, the authors propose a unified framework combining pseudo-label alignment, node reordering, and ambiguity filtering. Extensive experiments demonstrate that the approach consistently outperforms existing methods, especially under severe class imbalance.", "questions": "See Weaknesses.", "limitations": "See Weaknesses.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n-\tThe paper introduces the novel concept of geometric imbalance, offering a new perspective on how class imbalance affects node embeddings in graph data.\n-\tIt proposes a unified framework that effectively mitigates geometric ambiguity using pseudo-label alignment, node reordering, and ambiguity filtering.\n-\tThe method achieves strong empirical performance, consistently outperforming existing baselines under varying and severe imbalance scenarios.\n\nWeaknesses:\n-\tThe evaluation is limited to relatively small or moderately sized datasets, omitting large-scale benchmarks like ogbn-products, which is used by the baseline BIM.\n-\tThe paper lacks a theoretical or empirical analysis of computational complexity, leaving the scalability of the proposed method unclear; the absence of evaluation on large-scale datasets like ogbn-products further limits understanding of its practical efficiency.\n-\tThe paper focuses on hard pseudo-label selection for a subset of nodes. However, methods such as SCR [2] and ConsisGAD [3] do not require pseudo-label selection, and instead use a soft pseudo-labeling strategy with consistency regularization across all nodes, showing strong performance on semi-supervised node classification. Compared to pseudo-label selection methods, these approaches are more convenient as they avoid the complexity of confidence-based filtering or threshold tuning. Including a comparison with such approaches would strengthen the empirical evaluation and help justify the necessity of selective pseudo-labeling under class imbalance.\n\n[1] Wentao Zhang, Xinyi Gao, Ling Yang, Meng Cao, Ping Huang, Jiulong Shan, Hongzhi Yin, and Bin Cui. \"BIM: improving graph neural networks with balanced influence maximization.\" In 2024 IEEE 40th International Conference on Data Engineering (ICDE), pp. 2931-2944. IEEE, 2024.\n[2] Chenhui  Zhang, Yufei He, Yukuo Cen, Zhenyu Hou, Wenzheng Feng, Yuxiao Dong, Xu Cheng, Hongyun Cai, Feng He, and Jie Tang. \"SCR: Training graph neural networks with consistency regularization.\" arXiv preprint arXiv:2112.04319 (2021).\n[3] Nan Chen, Zemin Liu, Bryan Hooi, Bingsheng He, Rizal Fathony, Jun Hu, and Jia Chen. \"Consistency training with learnable data augmentation for graph anomaly detection with limited supervision.\" In The Twelfth International Conference on Learning Representations. 2024.", "was_revised": true, "final_justification": "The rebuttal is good with detailed results. I have adjusted my rating.", "rebuttal": "We sincerely thank Reviewer T2hF for recognizing **the novelty of our geometric imbalance formulation** and the strong performance of our method. We greatly appreciate your constructive and detailed feedback, which helps us improve the clarity and rigor of our work. **Below, we provide point-by-point responses to address each of your comments.**\n\n---\n\n**W1: The evaluation is limited to relatively small or moderately sized datasets, omitting large-scale benchmarks like ogbn-products.**\n\n\n**R1:** Thanks for raising this important point about large-scale evaluation. To address your concern, we have conducted additional experiments on the ogbn-products dataset, **strictly following the same experimental settings as BIM.** As shown in the following tables, our method achieves clear improvements over BIM and all other baselines on this large-scale benchmark, further demonstrating the robustness and effectiveness of our approach.\n\nWe **will incorporate these new results into the revised version of the paper.** We appreciate your comment, which helped us strengthen the empirical validation of our method.\n\nTable 1: Performance  on large-scale dataset Ogbn-product with GraphSAGE as the base model.\n\n|Methods|F1 score|ACC|AUC-ROC|\n|-|-|-|-|\n| Vanilla GraphSAGE|58.18±1.21|61.03±0.82|94.12±0.12|\n|(RN) ReNode|67.89±0.35|68.19±0.41|94.86±0.14|\n|BIM|69.36±0.18|69.55±0.19|95.51±0.12|\n|**Ours**|**71.85±0.10**|**71.62±0.11**|**96.59±0.14**|\n\n\n\nTable 2: Performance  on large-scale dataset Ogbn-product with GAT as the base model. \n\n|Methods|F1 score|ACC|AUC-ROC|\n|-|-|-|-|\n|Vanilla GAT|59.27±1.08|62.10±0.73|94.25±0.14|\n|(RN) ReNode|68.55±0.28|69.27±0.36|95.01±0.10|\n|BIM|69.88±0.16|70.05±0.21|95.63±0.11|\n|**Ours**|**72.04±0.09**|**71.95±0.13**|**96.67±0.10**|\n\n\n\nTable 3: Performance on large-scale dataset Ogbn-product with GCN as the base model. \n\n|Methods|F1 score|ACC|AUC-ROC|\n|-|-|-|-|\n|Vanilla (GCN)|57.41±1.18| 60.12±0.85| 93.96±0.15|\n|(RN) ReNode|66.82±0.32| 67.80±0.44| 94.75±0.13|\n|BIM|68.05±0.21| 68.47±0.23| 95.41±0.13|\n|**Ours**|**70.71±0.13**|**70.33±0.10**|**96.45±0.12**|\n\n\n\n---\n\n **W2: The paper lacks a theoretical or empirical analysis of computational complexity, leaving the scalability of the proposed method unclear.**\n\n**R2:** Thanks for highlighting the need for an analysis of computational complexity and scalability. As mentioned in our previous response, we have already included experiments on the large-scale ogbn-products dataset to demonstrate the practical efficiency of our method. In addition, we address your concern in three aspects:\n\n**1. Time Complexity Analysis**. Our framework, as described in Algorithm 1 (see Appendix L), consists of three main modules per round: Dual-Path Pseudo-label Alignment (DPAM), Node-Reordering (NR), and Discarding Geometrically Imbalanced Nodes (DGIS). (1) The dominant computational cost arises from the DPAM step, which clusters the embeddings of unlabeled nodes using k-means (complexity $O(n_u k' d T_{iter})$, with $n_u$ unlabeled nodes, $k'$ clusters, $d$-dimensional embeddings, and $T_{iter}$ k-means iterations). Computing class centroids and intersecting with classifier-predicted labels are both linear-time operations. (2) NR involves ranking nodes by geometric distance and classifier confidence (each $O(n_u \\log n_u)$), with ranking fusion being negligible. (3) DGIS computes distances from candidate nodes to their closest and second-closest centroids ($O(n_u C d)$ per round, where $C$ is the number of classes). (4) Overall, the per-round complexity is primarily determined by k-means, but since $k' \\ll n_u$ and k-means can be efficiently implemented (e.g., on GPU), **so, the total complexity remains comparable to standard self-training pipelines. Memory usage is linear in the number of nodes, as we only store embeddings and label predictions.** *Importantly, our method is significantly more efficient than prior over-sampling or augmentation approaches such as GraphENS and GraphSMOTE, which may have $O(n^2)$ or worse complexity on large graphs. In summary, the only notable overhead beyond vanilla GNN training comes from clustering and ranking, both of which are scalable and can be accelerated by modern hardware.* \n\n\n\n**2. Empirical Scalability and Runtime**. To address practical efficiency, we conducted comprehensive scalability analyses on both medium- and large-scale datasets, comparing our method (\"Ours\") to vanilla GNN backbones and representative baselines. (1) On **Cora**, **CiteSeer**, and **PubMed**, the training time per epoch for our method is almost identical to the backbone GNNs (see Table 4), confirming the negligible overhead of our extra modules (clustering and ranking). (2) On large-scale graphs (e.g., **Flickr**, ogbn-products), our method remains tractable and runs smoothly on a single RTX 3090 GPU (Tables 6 and 7 in our submission). By contrast, prior data augmentation and oversampling baselines (e.g., **GraphSMOTE**, **GraphENS**) either become prohibitively slow or run out-of-memory, as synthesizing large numbers of new nodes/edges is expensive.\n\nTable 4: Refined Running Time Analysis of Vanilla Model vs. Ours: (a) Training Over 1000 Epochs, and (b) Selecting 100 Nodes to the Training Set.\n\n\n\n| Model/Time (1000 epochs)   | Cora     | CiteSeer  | PubMed    |   | Model/Time (Selecting 100 Nodes) | Cora     | CiteSeer | PubMed   |\n|----------------------------|----------|-----------|-----------|---|----------------------------------|----------|----------|----------|\n| Vanilla (GCN)              | 6.221(s) | 7.212(s)  | 10.771(s) |   | Vanilla Self-Training (GCN)      | 2.244(s) | 2.234(s) | 2.675(s) |\n| Ours (GCN)                 | 6.208(s) | 7.191(s)  | 10.858(s) |   | Ours (GCN)                       | 2.367(s) | 2.305(s) | 2.790(s) |\n| Vanilla (GAT)              | 8.201(s) | 9.574(s)  | 14.443(s) |   | Vanilla Self-Training (GAT)      | 2.624(s) | 2.520(s) | 2.431(s) |\n| Ours (GAT)                 | 8.218(s) | 9.517(s)  | 15.088(s) |   | Ours (GAT)                       | 2.697(s) | 2.670(s) | 2.411(s) |\n| Vanilla (SAGE)             | 6.522(s) | 11.170(s) | 15.475(s) |   | Vanilla Self-Training (SAGE)     | 2.424(s) | 2.780(s) | 2.032(s) |\n| Ours (SAGE)                | 6.452(s) | 11.256(s) | 14.658(s) |   | Ours (SAGE)                      | 2.461(s) | 2.827(s) | 2.115(s) |\n\n\n**3. Implementation Efficiency**. We employ GPU-accelerated k-means for clustering (via NVIDIA cuML), which substantially reduces the per-round clustering overhead. **Moreover, by filtering candidates after dual-path alignment, the more expensive geometric checks are only performed on a small subset of reliable nodes, further improving efficiency.**\n\n\n---\n---\n**W3: To compare the method with recent soft pseudo-labeling approaches (e.g., SCR, ConsisGAD) to demonstrate the necessity and effectiveness of the strategy in imbalanced node classification.**\n\nWe thank the reviewer for emphasizing the need to compare with soft pseudo-labeling methods such as SCR and ConsisGAD. In response, we **conducted extensive experiments under two most challenging settings**, carefully tuning both methods via Wandb based on their published hyperparameter ranges. As shown in Table 3-4, they substantial performance drops under severe class imbalance, while our method consistently outperforms them. This is because soft pseudo-labeling with global consistency often lets majority classes dominate, suppressing minority signals. In contrast, our method directly addresses geometric imbalance by selectively assigning pseudo-labels to ambiguous nodes, improving minority class representation. We have incorporated these results and discussions into the revised manuscript.\n\nTable 3. Results with $\\rho=100$ (SAGE) on three  severe class-imbalanced node classification benchmark datasets.\n| Method    | Cora bAcc.     | Cora F1        | CiteSeer bAcc. | CiteSeer F1    | PubMed bAcc.   | PubMed F1      |\n| --------- | -------------- | -------------- | -------------- | -------------- | -------------- | -------------- |\n| GSR       | 66.45±2.10     | 64.42±1.83     | 53.52±1.47     | 53.01±1.75     | 74.09±2.12     | 72.97±1.90     |\n| BIM       | 67.75±2.13     | 64.68±1.95     | 53.83±1.62     | 53.29±1.80     | 74.38±2.08     | 73.24±1.85     |\n| **Ours**  | **73.47±2.31** | **68.30±2.11** | **59.77±2.98** | **58.92±3.07** | **77.11±0.59** | **74.03±0.81** |\n| SCR       | 59.31±2.91     | 55.34±4.32     | 36.51±1.29     | 27.19±2.84     | 63.41±0.76     | 54.97±1.64     |\n| ConsisGAD | 59.63±2.89     | 55.18±4.43     | 36.39±1.28     | 27.28±2.87     | 63.27±0.74     | 55.33±1.60     |\n\nTable 4: Experimental results on Ogbn-arxiv ($\\rho \\approx 775.4$)\n| Model     | GCN bAcc.        | GCN F1           | GAT bAcc.        | GAT F1           | SAGE bAcc.       | SAGE F1          |\n| --------- | ---------------- | ---------------- | ---------------- | ---------------- | ---------------- | ---------------- |\n| GSR       | 50.31 ± 0.24     | 49.70 ± 0.17     | 51.31 ± 0.41     | 49.33 ± 0.26     | 50.86 ± 0.30     | 49.53 ± 0.20     |\n| BIM       | 50.33 ± 0.42     | 49.71 ± 0.19     | 51.35 ± 0.60     | 49.36 ± 0.28     | 50.87 ± 0.18     | 49.56 ± 0.23     |\n| **Ours**  | **51.21 ± 0.32** | **50.65 ± 0.32** | **51.84 ± 0.87** | **51.28 ± 0.42** | **51.34 ± 0.32** | **51.36 ± 0.27** |\n| SCR       | 50.23 ± 0.41     | 49.69 ± 0.13     | 51.10 ± 0.81     | 49.67 ± 0.26     | 50.83 ± 0.20     | 49.80 ± 0.23     |\n| ConsisGAD | 50.25 ± 0.39     | 49.72 ± 0.14     | 51.13 ± 0.79     | 49.63 ± 0.27     | 50.80 ± 0.19     | 49.77 ± 0.22     |\n\n---\n---\n---\n**We hope these additional results and clarifications address your concerns, and sincerely hope you can reconsider your rating of our submission. Thank you for your valuable feedback.**"}]

Author rebuttal:

Meta-review: The paper formally introduces and quantifies geometric imbalance—a GNN-specific phenomenon in which class imbalance, amplified by message passing, distorts node embeddings on the Riemannian manifold and increases pseudo-label errors for unlabeled minority and medium-class nodes. Extensive experiments on synthetic, real-world, heterophilous, and large-scale graphs (e.g., ogbn-products) demonstrate consistent SOTA performance, especially under severe imbalance. All reviewers ultimately acknowledged the novelty, theoretical depth, and empirical rigor. The remaining issues were satisfactorily resolved in the rebuttal. The work is therefore recommended for acceptance at NeurIPS 2025.
Decision: Accept (poster)