SpaceGNN: Multi-Space Graph Neural Network for Node Anomaly Detection with Extremely Limited Labels

Lei Chen; Mingxuan Yuan; Sibo Wang; Xiangyu Dong; Xingyi Zhang

REVIEW 4 major objections 6 minor 1 cited by

SpaceGNN: Multi-Space Graph Neural Network for Node Anomaly Detection with Extremely Limited Labels

T0 review · 4 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read SpaceGNN: multi-space graph learning detects anomalies with few labels

desk verdict A sensible multi-space GNN for scarce-label anomaly detection with a useful architectural story, but the headline claim of beating the best rival by 8.55% is computed against a single baseline on a single split and needs multi-seed verification. read the letter →

arxiv 2502.03201 v1 pith:EDOL5BR3 submitted 2025-02-05 cs.LG

Xiangyu Dong , Xingyi Zhang , Lei Chen , Mingxuan Yuan , Sibo Wang This is my paper · ORCID

classification cs.LG

keywords nodeanomalydetectiongraphneuralnetworkshyperbolicspacesphericallimitedsupervisionlearnablecurvatureweightedhomogeneitymodelensemble

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that node anomaly detection under extremely limited supervision is better served by projecting graph data into several curved spaces at once than by staying in Euclidean space or by generating synthetic labels. It argues that different anomaly patterns, such as hierarchical structures, circular money-laundering rings, and plain relational data, have different natural curvatures, so a fixed Euclidean embedding misses information. The proposed SpaceGNN learns per-layer curvatures, weights message-passing edges by distance-based similarity, and ensembles one Euclidean, one hyperbolic, and one spherical GNN. Across nine real datasets, the authors report average gains of 8.55% in AUC and 4.31% in F1 over the strongest rival, with additional gains under a semi-supervised benchmark setting.

What carries the argument

The load-bearing machinery is the kappa-stereographic model of spaces of constant curvature, together with the expansion rate $ER_\kappa = r_\kappa/r_0$, which quantifies whether projecting a node triple into curvature $\kappa$ enlarges the inter-class versus intra-class distance ratio. Weighted homogeneity $WH_\kappa$ uses distance-based similarities $1-\sigma(d_\kappa(X_i, X_j))$ to weight same-label neighbor information and is shown, under Gaussian feature assumptions, to raise the probability that propagation preserves a node's class. These feed two modules: Distance Aware Propagation computes edge weights $\omega^\kappa_{ij} = \text{MLP}(\text{CONCAT}(X_i, \hat{s}_{ij}X_j))$ using a Taylor-approximated distance, and Multiple Space Ensemble combines one Euclidean GNN with learnable-curvature hyperbolic and spherical GNNs. The theoretical anchors are Theorem 1, which says higher weighted homogeneity increases the chance that a node remains on its original distribution after propagation, and an ensemble bound showing that the ensemble cross-entropy loss is no worse than the weighted single-model loss.

What would settle it

Take a graph dataset with strongly anisotropic or non-Gaussian node features and compare SpaceGNN against a Euclidean-only variant with the distance-aware weighting removed. The proof of Theorem 1 predicts the ablation gap should track weighted homogeneity; if distance-aware propagation fails to help, or hurts, on such data despite high weighted homogeneity, then the Gaussian-covariance assumption is carrying the theoretical argument.

Watch

Extended reading notes

Core claim

The central claim is that a multi-space ensemble with learnable curvature is a more effective response to scarce labels in node anomaly detection than pseudo-label generation or data augmentation. Concretely, the paper introduces an expansion rate measuring how a curved projection enlarges the ratio of inter-class to intra-class distance, uses it to motivate learnable curvature in a kappa-stereographic model, defines weighted homogeneity to justify distance-aware edge weighting during propagation, and proves an ensemble bound showing that combining models from different spaces reduces expected cross-entropy loss. On nine benchmark datasets, SpaceGNN is reported to outperform the best competing model by 8.55% AUC and 4.31% F1 on average.

Load-bearing premise

The proof that weighted homogeneity helps propagation assumes normal and anomalous node features follow independent Gaussian distributions with roughly equal, near-scalar covariance $\Sigma_n \approx \Sigma_a \approx cI$; if real features are non-Gaussian or strongly anisotropic, that justification for distance-aware propagation is not established.

Editorial extensions

If this is right

If the central claim is correct, scarce-label NAD systems can be improved without generating synthetic labels: an ensemble of models spanning several geometries substitutes for data augmentation.
Because the learned curvatures stay close to zero, the reported gains suggest that even modest curvature flexibility plus distance weighting drives most of the improvement.
The distance-aware propagation component gives a principled reason to down-weight noisy inter-class edges during message passing rather than only pruning them.
The ensemble bound implies that combining additional independent space-based models could further reduce expected loss, provided the base models remain sufficiently independent.
The reported semi-supervised results indicate the approach transfers to a standard 20-positive and 80-negative label regime on the same benchmark datasets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct test of the expansion-rate rationale would regress the dataset-level performance gain of each curved branch against the empirical $ER_\kappa$ computed from labeled triples; the paper reports learned curvatures but does not systematically make that connection.
The argument that pseudo-labeling is harmful rests on the observed false anomalous rate of one trained CONSISGAD model, so a natural extension is to vary pseudo-label confidence thresholds and see whether the ensemble advantage persists against calibrated pseudo-labeling.
Theorem 2 is a small-curvature expansion, so a stress test could rewire graphs to force large effective curvature and check whether Distance Aware Propagation degrades outside the approximation regime.
The framework treats anomaly detection as binary classification with imbalanced priors, leaving open whether the same space-ensemble idea transfers to unsupervised or single-class anomaly detection settings.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: the architecture is real, the headline is misleading. If you take the abstract's 8.55% AUC claim at face value, you'll be over-reading what the tables actually show. The core method is sound and worth engaging with, but the empirical evidence needs another pass.

What's new and good: this is the first application of a learnable-curvature multi-space GNN (Euclidean/hyperbolic/spherical ensemble) to supervised node anomaly detection with extremely limited labels. The Expansion Rate and Weighted Homogeneity analyses are genuinely new diagnostic tools, and the Distance Aware Propagation module is a reasonable attention mechanism. The argument that ensembling multiple spaces beats pseudo-label data augmentation, backed by the CONSISGAD false-rate analysis, is a useful contrast. Code is public, and the ablations confirm that both the learnable projection and the distance-aware propagation contribute.

Soft spots, in proportion: the empirical foundation is thinner than the abstract implies. The 8.55% gain is against CONSISGAD, not the per-dataset best rival; Table 2 shows XGBGraph beating SpaceGNN on Amazon F1, and on several datasets the margin over the actual best baseline is much smaller. All main-table numbers come from one 50/50 random split with no standard deviation, no seed count, and no significance test. With 50 labeled nodes, split variance is high, especially on Reddit where the expected number of anomalous training nodes is about 1.7. Hyperparameters are grid-searched on the same 50-node validation set, compounding the selection noise. The theory is also conditional: Theorem 1 assumes independent Gaussian classes with nearly equal isotropic covariance, which is strong for real graph features, and Proposition 1 is essentially AM-GM. None of this is fatal. The GADBench semi-supervised setting with fixed 20 positive and 80 negative labels is more controlled and provides genuine supporting evidence, but it is a different protocol from the headline experiments.

Who this is for: GNN researchers working on anomaly detection or scarce-label graph learning, and anyone comparing multi-space embeddings. This paper deserves a serious referee. If I were handling it, I'd ask for multi-seed reruns, error bars, and a corrected comparison against the best per-dataset baseline before trusting the magnitude of the gains. The ideas, however, are solid enough to build on.

Referee Report

4 major / 6 minor

Summary. The paper proposes SpaceGNN, a multi-space graph neural network for node anomaly detection (NAD) with extremely limited labels. It combines a learnable space projection (LSP) with learnable curvature, a distance-aware propagation (DAP) module motivated by a new weighted homogeneity metric, and a multiple-space ensemble (MulSE). The authors provide two theorems and two propositions as theoretical support, and report experiments on 9 real-world NAD datasets against 16 baselines, claiming average improvements of 8.55% in AUC and 4.31% in F1 over the best rival. The code is released.

Significance. If the empirical claims were fully supported, the paper would make a meaningful contribution: it offers a fresh geometric perspective on NAD under scarce supervision, introduces measurable concepts (expansion rate and weighted homogeneity), and provides an extensive comparison with code release. The controlled GADBench semi-supervised experiments (Appendix L) with fixed 20 positive and 80 negative labels give credible supporting evidence, and the ablation study shows that the LSP and DAP components each contribute. I do not see a circularity problem: the ER/WH analyses use labels for motivation, while the model is trained with cross-entropy on labeled nodes and tested on held-out nodes. However, the headline result is currently overstated because the reported margin is computed only against CONSISGAD, and the main 50/50 single-split protocol has large variance and can produce degenerate F1 scores on low-anomaly datasets. The paper's significance depends on whether these experimental weaknesses can be corrected.

major comments (4)

[Abstract; Section 5.2, Tables 1-2] The abstract and Section 5.2 state that SpaceGNN 'outperforms the best rival by an average of 8.55% in AUC and 4.31% in F1 scores,' but these figures are the average margins over CONSISGAD only, not over the best competing method on each dataset. In Table 2, XGBGraph achieves F1 0.9231 on Amazon while SpaceGNN achieves 0.8935, and CONSISGAD achieves F1 0.5084 on DGraph-Fin while SpaceGNN achieves 0.5017; the per-dataset best-baseline margins are therefore smaller, and the headline claim as written is not supported. Please restate the claim as a comparison with CONSISGAD or, preferably, report averages against the per-dataset best baseline.
[Section 5.1; Tables 1-2; Appendix G] Tables 1 and 2 report a single run on a single random 50/50 split with no standard deviation or number of seeds. Because the anomaly rates are very low (e.g., Reddit 3.33%, DGraph-Fin 1.27%), a training set of 50 nodes contains on average fewer than two anomalous nodes for Reddit, and many splits will contain no anomalous training nodes at all; the F1 value of 0.4915 shared by nearly all methods on Reddit is consistent with predicting the majority class. The single-split protocol therefore cannot support the claim of consistent superiority. Please run multiple random splits (or use stratified/fixed labels as in Appendix L), report mean±std, and include a paired significance test; Appendix L's fixed 20-positive/80-negative setting is a better-controlled protocol and should be promoted in the evaluation.
[Algorithm 4; Table 4; Table 5] The ensemble hyperparameters α and β chosen by validation (Table 4) collapse the model to a single space for 7 of the 9 datasets: for Weibo, T-Finance, DGraph-Fin and T-Social, β=1 so Z=Z_0 (Euclidean only), and for Reddit, Amazon and YelpChi, α=β=0 so Z=Z_{κ^-} (hyperbolic only). Only Tolokers and Questions actually use a mixture of spaces. Consequently, the main experiments do not demonstrate the benefit of the Multiple Space Ensemble module, and the ablation in Table 5 removes LSP and DAP but does not remove MulSE. Please add an ablation 'w/o MulSE' and either report results with the full ensemble active over all datasets or discuss why the validation procedure selects degenerate hyperparameters.
[Appendix D; Section 5.1] Hyperparameters (including α, β, learning rate, hidden dimension, layer count, and dropout) are selected by grid search to maximize the F1 score on a validation set of only 50 nodes. With such a small validation set, F1 differences of one or two points are within sampling noise, so the selection procedure is likely to overfit the validation split; this compounds the variance problem raised above. The authors should use a larger validation set, nested cross-validation, or fixed hyperparameters per protocol, and show that the reported test results are stable across choices.

minor comments (6)

[Section 3, Eqs. (1)-(2)] The text says that when κ=0 the projection functions act as the identity, but substituting κ=0 into Equations (1)-(2) gives exp_o^0(x)=o (the origin) rather than x; the separate handling of κ=0 in Algorithm 1 should be stated in Section 3.
[Appendix A, proof of Theorem 2] In the proof of Theorem 2, the Taylor expansion of tan^{-1}_κ(t) is written as t + κ t^3/3 in the κ→0+ and κ→0− cases, but then as t − κ t^3/3 in the concluding line; if the minus sign is intended, the expansion steps need to be reconciled, and the final formula in Theorem 2 should be re-verified.
[Figure 3] The y-axis label '(Weighted) Hmogeneity' and the legend label 'Homogeniety' contain spelling errors; the figure would also benefit from a description of how the values are aggregated across nodes.
[Appendix B] In the baseline list, 'AMNet' is spelled 'AMMNet'; please make the spelling consistent.
[Section 5.1] The statement 'randomly divide each dataset into 50/50 for training/validation' is ambiguous because it could be read as a 50% split; please clarify that it means 50 labeled nodes for training and 50 for validation.
[Appendix A, Theorem 1] Theorem 1 assumes Σ_n ≈ Σ_a ≈ cI and independent Gaussian features; this assumption is not verified on the real datasets. Since the theorem is used to justify the distance-aware propagation, the authors should either provide empirical evidence for the assumption (e.g., per-class covariance estimates) or weaken the theoretical claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the design motivations are measured metrics and standard inequalities; the empirical claims rest on held-out test comparisons, with the protocol weaknesses being evaluation concerns rather than circular derivations.

full rationale

The claimed derivation chain is not circular. The Learnable Space Projection is motivated by the Expansion Rate, a label-based ratio metric measured on real node triplets in Section 4.2; the actual curvatures are trained by cross-entropy and reported in Appendix I, so no predicted quantity reduces to the ER values. The Distance Aware Propagation is motivated by Weighted Homogeneity, which is by definition the label-weighted fraction of same-class neighbor similarity, and Theorem 1 proves a monotonicity statement under a stated Gaussian and equal-covariance assumption; even if that statement is close to saying that more same-class information moves the propagated mean toward the original class mean, DAP's edge coefficients are produced by an MLP on approximate distances and are evaluated on held-out test nodes through the Table 5 ablation, not set equal to WH. The Multiple Space Ensemble is justified by Propositions 1 and 2, which are applications of the AM-GM/Jensen inequality and a cited KL bias-variance decomposition; these are external mathematical facts, not self-citations, and the empirical claim that the ensemble is more beneficial than data augmentation is a direct comparison with CONSISGAD under the same data split rather than a consequence of the propositions. The only self-citation overlap is the authors' own SmoothGNN paper (Dong et al., 2025), cited in the introduction for a generic statement about graph data complexity; it is not load-bearing and does not raise the circularity score. The headline average of 8.55% in AUC and 4.31% in F1 is computed against CONSISGAD on a single 50/50 split without reported variance, and Theorem 1 relies on an idealized Gaussian assumption; both are robustness or correctness concerns, not instances where a derivation reduces to its own input by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 2 invented entities

The central claim rests on per-layer learnable curvatures and per-dataset ensemble weights selected by validation, plus a strong Gaussian distribution assumption in Theorem 1 and the heuristic ER and WH metrics that motivate the architecture. These are reasonable for a machine learning paper but should be listed explicitly.

free parameters (3)

Learnable curvature per layer for hyperbolic and spherical branches (kappa^-, kappa^+) = e.g., kappa^- about -0.06 to -0.14, kappa^+ about 0.06 to 0.14 (Table 11)
Learned on each dataset during training; central to the LSP module and the multi-space claim.
Ensemble coefficients alpha and beta = per dataset from {0, 0.5, 1} (Table 4)
Selected by validation F1; control the contribution of Euclidean vs non-Euclidean branches, central to MulSE.
Architecture hyperparameters (learning rate, hidden dimension, layer count, dropout) = lr 0.001/0.0001; hidden 128; layers 1-6; dropout 0-0.1 (Table 4)
Grid-searched on validation; standard hyperparameters but part of the per-dataset fitting.

assumptions (4)

domain assumption Features of normal and anomalous nodes follow independent Gaussian distributions, and Sigma_n approx Sigma_a approx cI
Invoked in the proof of Theorem 1 (Appendix A) to conclude that weighted homogeneity improves propagation; strong and unverified on real graphs.
ad hoc to paper A projection with higher Expansion Rate (ER) benefits NAD
Motivates the Learnable Space Projection module based on Figure 2, a few illustrative triplets rather than a systematic test.
standard math Ensemble of independent models reduces cross-entropy loss (AM-GM inequality)
Proposition 1 is a direct application of the weighted AM-GM inequality; Proposition 2 invokes the KL bias-variance decomposition of Wood et al. (2023).
ad hoc to paper Weighted homogeneity with sigmoid-transformed distances is a good guide for propagation edge weights
Definition 2 introduces WH; Theorem 1 supports the general idea but the specific design of s_i = 1 - sigmoid(d(X_i, X_j)) and the DAP MLP in Eq. 5 are heuristic.

invented entities (2)

Expansion Rate (ER)
purpose: Quantifies how much projecting into curvature expands the inter-class vs intra-class distance ratio for a node triplet
A new metric defined by the authors (Definition 1) with no external falsifiable handle; used to motivate learnable curvature.
Weighted Homogeneity (WH)
purpose: Measures how much useful intra-class information a node receives from neighbors, weighting by distance-based similarity
A new metric (Definition 2) used to motivate the Distance Aware Propagation module; depends on labels and the chosen similarity transform.

how reviews work

0 comments

Cite this review

Pith. "Pith review of SpaceGNN: Multi-Space Graph Neural Network for Node Anomaly Detection with Extremely Limited Labels." pith.science (2026). https://pith.science/paper/EDOL5BR3

@misc{pith2026250203201,
  author       = {Pith},
  title        = {Pith review of: SpaceGNN: Multi-Space Graph Neural Network for Node Anomaly Detection with Extremely Limited Labels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EDOL5BR3}},
  note         = {Machine review of arXiv:2502.03201}
}

read the original abstract

Node Anomaly Detection (NAD) has gained significant attention in the deep learning community due to its diverse applications in real-world scenarios. Existing NAD methods primarily embed graphs within a single Euclidean space, while overlooking the potential of non-Euclidean spaces. Besides, to address the prevalent issue of limited supervision in real NAD tasks, previous methods tend to leverage synthetic data to collect auxiliary information, which is not an effective solution as shown in our experiments. To overcome these challenges, we introduce a novel SpaceGNN model designed for NAD tasks with extremely limited labels. Specifically, we provide deeper insights into a task-relevant framework by empirically analyzing the benefits of different spaces for node representations, based on which, we design a Learnable Space Projection function that effectively encodes nodes into suitable spaces. Besides, we introduce the concept of weighted homogeneity, which we empirically and theoretically validate as an effective coefficient during information propagation. This concept inspires the design of the Distance Aware Propagation module. Furthermore, we propose the Multiple Space Ensemble module, which extracts comprehensive information for NAD under conditions of extremely limited supervision. Our findings indicate that this module is more beneficial than data augmentation techniques for NAD. Extensive experiments conducted on 9 real datasets confirm the superiority of SpaceGNN, which outperforms the best rival by an average of 8.55% in AUC and 4.31% in F1 scores. Our code is available at https://github.com/xydong127/SpaceGNN.

Figures

Figures reproduced from arXiv: 2502.03201 by the authors.

**Figure 2.** ERκ for different node triplets, varying based on κ. (Opt. stands for optimal.) As for the ratio rκ(x0, x1, y), an effective projection into space with curvature κ should maximize the separation between data points from different classes, making their distances sufficiently distinct for accurate detection. Since data points are originally embedded in Euclidean space, it is natural to utilize the ratio in Euclidean … view at source ↗

**Figure 4.** True/False Anomalous Rate. In this section, we first provide an empirical analysis of CONSISGAD (Chen et al., 2024), the most recent work for the supervised NAD task aimed at tackling the issue of limited labels. CONSISGAD leverages a learnable framework to generate pseudo labels, thereby increasing the number of labeled nodes for training. However, the quality of these pseudo labels may not always be sufficien… view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: Varying the Hidden Dimension and Layer. α [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: Varying α and β on different datasets. We provide the detailed algorithm in this Section. In Algorithm 1, we calculate expκ o (·) and logκ o (·) based on the original point o of the space with curvature κ. Besides, to satisfy the range of logκ o (·), we utilized Algori…

**Figure 7.** Figure 7: Varying the training set size of Reddit. [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

**Figure 8.** Figure 8: Varying the training set size of Questions. [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

**Figure 9.** Figure 9: Varying the training set size of DGraph-Fin. [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Bridging Cognitive Neuroscience and Graph Intelligence: Hippocampus-Inspired Multi-View Hypergraph Learning for Web Finance Fraud
cs.LG 2026-01 conditional novelty 4.0 of 10

HIMVH, a hippocampus-inspired multi-view hypergraph model, reports new state-of-the-art results on six web-finance fraud datasets with average AUC/F1/AP gains of 6.42%/9.74%/39.14% over 15 baselines.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · cited by 1 Pith paper

[1]

Let X and Y denote the distribution of the normal node and the informati on over Rd, respectively

Let p denote W Hκ, then the information a normal node can gain within its neighborhood during a propagation process follows N (pµ n + (1− p)µ a, p2Σ n + (1− p)2Σ a) according to the linear properties of independent Gaussian variables. Let X and Y denote the distribution of the normal node and the informati on over Rd, respectively. We then use Fr´ echet i...

work page 2017
[2]

Hence, we conclude that near 0, tan−1 κ (t) = t− κ t3 3 +O(κ2)

First, we apply Taylor expansion on tan−1 κ (t) for a ﬁxed t when κ→ 0+: tan−1 κ (t) =κ− 1 2 tan(κ 1 2 t) =κ− 1 2 (κ 1 2 t + κ 3 2 t3 3 +O(κ 5 2 )) =t + κ t3 3 +O(κ2) When κ→ 0−: tan−1 κ (t) =(−κ)− 1 2 tanh((−κ) 1 2 t) =(−κ)− 1 2 ((−κ) 1 2 t− (−κ) 3 2 t3 3 +O(κ 5 2 )) =t + κ t3 3 +O(κ2) 13 Published as a conference paper at ICLR 2025 When κ → 0, we also h...

work page 2025
[3]

Baselines. The ﬁrst group is generalized models: • MLP (Rosenblatt, 1958): A type of neural network with multi ple layers of fully connected artiﬁ- cial neurons; • GCN (Kipf & Welling, 2017): A type of GNN that leverages conv olution function on a graph to propagate information within the neighborhood of each node ; • GraphSAGE (Hamilton et al., 2017): A ...

work page 1958
[5]

As shown in Table 5, SpaceGNN consistently outperforms w/o LSP and w/o DAP by a large margin, which demonstrates the beneﬁts of these two components

Speciﬁcally, we set the κ as ﬁxed values for different spaces during the w/o LSP experiment and set the ˆsi as 1 for each node i during the w/o DAP experiment. As shown in Table 5, SpaceGNN consistently outperforms w/o LSP and w/o DAP by a large margin, which demonstrates the beneﬁts of these two components. G A DDITIONAL EXPERIMENTAL RESULTS In addition ...

work page 2025
[7]

We take the expectation of the equation in Proposition 1 over entire graph G and apply KL bias-variance decomposition in previous study (Wood et al., 2023), then we have: EG[L(p, ¯q)] =EG[ m∑ i=1 αiL(p, qi)]− EG[Ω( p)] = m∑ i=1 αiL(p, ˆqi) + m∑ i=1 αiEG[DKL(ˆqi||qi)]− EG[Ω( p)] = m∑ i=1 αiL(p, ˆqi) + Θ( p), where Θ( p) is demonstrated as non-negative in p...

work page 2023
[9]

Notice, for each lay er l during the propagation, we assign a different learnable κl to capture comprehensive information from different space s

Speciﬁcally, we use the appr oximated distance to calculate the similarities between nodes and their neighbors, and then le verage them as the corresponding coefﬁ- cients during the propagation process. Notice, for each lay er l during the propagation, we assign a different learnable κl to capture comprehensive information from different space s. To simpl...

work page 2025
[12]

For κ − and κ +, we want the Hyperbolic GNN and Spherical GNN to search for the optimal curvatures for different 20 Published as a conference paper at ICLR 2025 Table 11: Learned κ Datasets κ − 1 κ − 2 κ − 3 κ − 4 κ − 5 κ − 6 κ + 1 κ + 2 κ + 3 κ + 4 κ + 5 κ + 6 Weibo -0.1272 -0.0766 -0.0795 -0.1176 -0.1239 -0.1103 0.0989 0.1233 0.0936 0.0961 0.0747 0.1116...

work page 2025
[13]

SEMI -SUPERVISED SETTING For a fair comparison, we also provide AUC, AUPRC, and Rec@K s cores on 9 datasets with data split of the semi-supervised setting in GADBench (Tang et al., 2023). Speciﬁcally, in this setting, we use 20 positive labels (anomalous nodes) and 80 negative lab els (normal nodes) for both the training set and the validation set in each...

work page 2023

Show all 13 references

[2017]

Und erstanding the detrimental class-level effects of data augmentation

11 Published as a conference paper at ICLR 2025 Polina Kirichenko, Mark Ibrahim, Randall Balestriero, Dia ne Bouchacourt, Shanmukha Ramakr- ishna V edantam, Hamed Firooz, and Andrew Gordon Wilson. Und erstanding the detrimental class-level effects of data augmentation. In Neur...

2025
[2019]

or the Lorentz model (Nickel & Kiela, 2018). We discover that the Poincar´ e Ball model can be a special form of κ-stereographic model when setting the κ to−1, which inspires us to investigate the general form of the Lorentz model. Following the deﬁniti on of the κ-stereograph...

2018
[2020]

Can abnormality be detected by graph neural networks? In IJCAI, pp

Ziwei Chai, Siqi Y ou, Y ang Y ang, Shiliang Pu, Jiarong Xu, Haoyang Cai, and Weihao Jiang. Can abnormality be detected by graph neural networks? In IJCAI, pp. 1945–1951,

1945
[2023]

12 Published as a conference paper at ICLR 2025 APPENDIX A P ROOFS Proof of Theorem

2025
[2024]

Rumor detection on twitte r with tree-structured recursive neural networks

Jing Ma, Wei Gao, and Kam-Fai Wong. Rumor detection on twitte r with tree-structured recursive neural networks. In ACL, pp. 1980–1989,

1980

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Let X and Y denote the distribution of the normal node and the informati on over Rd, respectively

Let p denote W Hκ, then the information a normal node can gain within its neighborhood during a propagation process follows N (pµ n + (1− p)µ a, p2Σ n + (1− p)2Σ a) according to the linear properties of independent Gaussian variables. Let X and Y denote the distribution of the normal node and the informati on over Rd, respectively. We then use Fr´ echet i...

work page 2017

[2] [2]

Hence, we conclude that near 0, tan−1 κ (t) = t− κ t3 3 +O(κ2)

First, we apply Taylor expansion on tan−1 κ (t) for a ﬁxed t when κ→ 0+: tan−1 κ (t) =κ− 1 2 tan(κ 1 2 t) =κ− 1 2 (κ 1 2 t + κ 3 2 t3 3 +O(κ 5 2 )) =t + κ t3 3 +O(κ2) When κ→ 0−: tan−1 κ (t) =(−κ)− 1 2 tanh((−κ) 1 2 t) =(−κ)− 1 2 ((−κ) 1 2 t− (−κ) 3 2 t3 3 +O(κ 5 2 )) =t + κ t3 3 +O(κ2) 13 Published as a conference paper at ICLR 2025 When κ → 0, we also h...

work page 2025

[3] [3]

Baselines. The ﬁrst group is generalized models: • MLP (Rosenblatt, 1958): A type of neural network with multi ple layers of fully connected artiﬁ- cial neurons; • GCN (Kipf & Welling, 2017): A type of GNN that leverages conv olution function on a graph to propagate information within the neighborhood of each node ; • GraphSAGE (Hamilton et al., 2017): A ...

work page 1958

[4] [5]

As shown in Table 5, SpaceGNN consistently outperforms w/o LSP and w/o DAP by a large margin, which demonstrates the beneﬁts of these two components

Speciﬁcally, we set the κ as ﬁxed values for different spaces during the w/o LSP experiment and set the ˆsi as 1 for each node i during the w/o DAP experiment. As shown in Table 5, SpaceGNN consistently outperforms w/o LSP and w/o DAP by a large margin, which demonstrates the beneﬁts of these two components. G A DDITIONAL EXPERIMENTAL RESULTS In addition ...

work page 2025

[5] [7]

We take the expectation of the equation in Proposition 1 over entire graph G and apply KL bias-variance decomposition in previous study (Wood et al., 2023), then we have: EG[L(p, ¯q)] =EG[ m∑ i=1 αiL(p, qi)]− EG[Ω( p)] = m∑ i=1 αiL(p, ˆqi) + m∑ i=1 αiEG[DKL(ˆqi||qi)]− EG[Ω( p)] = m∑ i=1 αiL(p, ˆqi) + Θ( p), where Θ( p) is demonstrated as non-negative in p...

work page 2023

[6] [9]

Notice, for each lay er l during the propagation, we assign a different learnable κl to capture comprehensive information from different space s

Speciﬁcally, we use the appr oximated distance to calculate the similarities between nodes and their neighbors, and then le verage them as the corresponding coefﬁ- cients during the propagation process. Notice, for each lay er l during the propagation, we assign a different learnable κl to capture comprehensive information from different space s. To simpl...

work page 2025

[7] [12]

For κ − and κ +, we want the Hyperbolic GNN and Spherical GNN to search for the optimal curvatures for different 20 Published as a conference paper at ICLR 2025 Table 11: Learned κ Datasets κ − 1 κ − 2 κ − 3 κ − 4 κ − 5 κ − 6 κ + 1 κ + 2 κ + 3 κ + 4 κ + 5 κ + 6 Weibo -0.1272 -0.0766 -0.0795 -0.1176 -0.1239 -0.1103 0.0989 0.1233 0.0936 0.0961 0.0747 0.1116...

work page 2025

[8] [13]

SEMI -SUPERVISED SETTING For a fair comparison, we also provide AUC, AUPRC, and Rec@K s cores on 9 datasets with data split of the semi-supervised setting in GADBench (Tang et al., 2023). Speciﬁcally, in this setting, we use 20 positive labels (anomalous nodes) and 80 negative lab els (normal nodes) for both the training set and the validation set in each...

work page 2023