Robust Explanations of Graph Neural Networks via Graph Curvatures
Authors: Yazheng Liu
Xi Zhang
Sihong Xie
Hui Xiong
Venue: NeurIPS 2025 poster ()
Keywords: robust; explanation; graph curvature; GNN
Primary area: social_and_economic_aspects_of_machine_learning

Abstract: Explaining graph neural networks (GNNs) is a key approach to improve the trustworthiness of GNN in high-stakes applications, such as finance and healthcare. However, existing methods are vulnerable to perturbations, raising concerns about explanation reliability. Prior methods enhance explanation robustness using model retraining or explanation ensemble, with certain weaknesses. Retraining leads to models that are different from the original target model and misleading explanations, while ensemble can produce contradictory results due to different inputs or models. To improve explanation robustness without the above weaknesses, we take an unexplored route and exploit the two edge geometry properties curvature and resistance to enhance explanation robustness. We are the first to prove that these geometric notions can be used to bound explanation robustness. We design a general optimization algorithm to incorporate these geometric properties into a wide spectrum of base GNN explanation methods to enhance the robustness of base explanations. We empirically show that our method outperforms six base explanation methods in robustness across nine datasets spanning node classification, link prediction, and graph classification tasks, improving fidelity in 80\% of the cases and achieving up to a 10\% relative improvement in robust performance. The code is available at [https://github.com/yazhengliu/Robust_explanation_curvature](https://github.com/yazhengliu/Robust_explanation_curvature).

Robust Explanations of Graph Neural Networks via Graph Curvatures
Yazheng Liu', Xi Zhang', Sihong Xie Hui Xiong' The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China 2 'Te Beijing University of Posts and Telecommunications, Beijing, China yl i1iu533coneect www.h-ukkeee gz. edu. en, zhangx@bupt.c edu. (sihongxie xionghui] Jonghhi)@hkustt Phhksst t-aest--z.... -gz. edu. cn.
Abstract
Explaining graph neural networks (GNNs) is key approach to improve the trust- worthiness of GNN in high-stakes applications, such as finance and healthcare. However, existing methods are vulnerable perturbations, raising concerns about explanation reliability. Prior methods enhance explanation robustness using model retraining or explanation ensemble, with certain weaknesses. Retraining leads models that are different from the original target model and misleading explana- tions, while ensemble can produce contradictory results due different inputs or models. To improve explanation robustness without the above weaknesses, take an unexplored route and exploit the two edge geometry properties curvature and resistance to enhance explanation robustness. We are the first prove that these geometric notions can be used to bound explanation robustness. We design general optimization algorithm incorporate these geometric properties into wide spectrum of base GNN explanation methods to enhance the robustness of base explanations. We empirically show that our method outperforms six base explanation methods in robustness across nine datasets spanning node classification, link prediction, and graph classification tasks, improving fidelity in 80% of the cases and achieving up to a 10% relative improvement in robust performance. The code is available at http:///iitbb..comyyyaaeggu/uuuu http:hhccccccucooooossccaaaaanoa explanation_ curvature
Introduction
Graph neural networks (GNNs) have proven their strengths in many real world applications, such as social network modeling [ and fraud detection (2). To establish human trust in GNN prediction, is essential to make GNN predictions transparent humans in risk-critical applications [3] 4. For example, biological researchers may seek to understand the reasoning behind GNN predictions for protein function verify whether the model is trustworthy. Many explanation methods have been designed to explain the behavior of GNNs.
However, recent studies have shown that these explanation methods are vulnerable to perturbations 7. The fragility explanations 8 can mislead users into making wrong decisions, raising security concerns in high-stakes domains such as finance, healthcare, and criminal justice (9] 10] For example, we consider a graph in which nodes represent customers and edges denote financial relationships such as borrowing or transaction flows. We use GNNs to predict personal loan approvals. a loan is denied, customers may feel the need understand the possible reasons and corrections to improve the chance of future approval. When small changes in the graph structure (adding or removing a transaction edge) lead to substantially different explanations, trust in the model may be undermined as customers begin to question the decisions of companies.
"Sionng Xie the the corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Most methods for robust GNN explanations focus on the robust evaluation of explanations [T] 12 and robust counterfactual explanations [T3 Robust evaluation methods introduce new metrics to compute the fidelity [15] of explanations while accounting for distribution shifts during the evaluation process. Robust counterfactual explanation methods aim to extract explanations similar to the input graph that flip the model prediction and keep stable under edge perturbations. Fewer works focus on the robustness of explanations [16]. One work define robustness of explanations as the expected change in predicted probabilities between the original and perturbed graphs, with the explanation subgraph fixed during perturbations [7].. We adopt this definition, formally stated in Equation (2) To improve the robustness of explanations, one approach is to retrain models with an extra regularization term [18). They theoretically and empirically illustrate that the retrained model has improved explanation robustness. However, these retrained models are different from the original model, so that the robust explanations in fact do not explain the original model (Figure T (b) top).
Alternatively, ensemble methods improve explanation robustness. For example, [99] propose to combine explanations from different algorithms, while others [16] generate perturbed graphs and aggregate explanations from these perturbed graphs. However, as target models can be sensitive to small perturbations, multiple explanations can disagree on graph element importance, leading to
uncertainty and sabotaging the trustworthiness of the aggregated explanations (Figure (bb) bottom).
To address these challenges, we leverage two geometric properties of graph edges to generate robust explanations without model retraining explanation ensembling. In particular, two important concepts in graph theory are Ricci curvature and effective resistance. The Ollivier-Ries curvature [20] and effective resistance 21 can measure the connectivity between the neighborhoods of nodes u and v. Positive curvature and low effective resistance indicate efficient message transmission between neighhorhoods, while negative curvature and high effective resistance suggest that edge (u, b) acts as a bottleneck of message transmission (22]. Recent works have shown that these properties are
related over-squashing in GNNs (3 24 25 Nevertheless, there has been no work exploring the
relationship between graph geometry and explanation robustness.
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Figure 1: Assuming a GNN model has 3 layers, use node classification example. (a): the explanations become fragile if the edge (0.3) is removed. (b): significant differences between the retrained and original models make the robust explanations not faithful to the original model. For ensemble explanations, contradictory results can lead to more uncertain explanations. (c): the prediction of node 0 on G2 is more robust than that on G1 as removing edge (0,3) in G. significantly affects the model's predictions. We prove that higher minimum Ricci curvature and lower average effective resistance contribute to greater robustness of model predictions. While edge (3,4) in G1 and edge (0,3) in G2 have the smallest Ricci curvatures in their respective graphs, the curvature of (0.3) in G2 is higher due to the presence of alternative paths for message passing between nodes O and 3. The Gi has fewer propagation paths from neighhorhoods to target node 0 than G2, resulting in higher average effective resistance. (d): we use Ricci curvature and effective resistance robust scores and combine them with importance scores from explanation methods to obtain robust explanations.
We first theoretically analyze that the Ollivier- Ricci curvature and effective resistance can be used to bound the robustness of model prediction and explanations, as shown in Theorems 4.3 4.4 4.7 and 4.8 Specifically, higher minimum Ollivier-Rice Ricci curvature and lower average effective resistance across all edges in the graph and explanation subgraphs are associated with greater robustness in model predictions (Figure (c)) and explanations. Based on these theoretical results, as shown in Figure [ (d), we leverage Ricci curvature and effective resistance as robust scores to enhance explanation reliability and our method is agnostic to explanation methods. We propose objective functions that improve the robustness of existing explanation methods without modifying the model or generating multiple potentially contradicting explanations. Empirically, on nine datasets across node classification, link prediction, and graph classification tasks, we demonstrate that our method improves robustness in six state-of-the--n GNN explanation methods, enhancing fidelity in 80% of the cases and achieving up to 10% relative improvement in robustness.
2

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2 Related Work
Explainability and explanation robustness. Recent post-hoc explanation methods for GNNs can be categorized into instance-level and model-level methods [T5]. Instance level explanation methods [3]
focus on explaining model predictions by identifying important subgraphs that strongly correlate with predictions. For example, GNNexplainer [3] and PGExplainer (77 learn edge masks to identify important edges by maximizing the mutual information to explain the prediction. Methods like GNN-LRP [4. DeepLIFT (26,, FlowX [88] and Convex 20 analyze the importance of graph walks. Model level explanation methods (30 produce a high-level explanation about the general behaviors of GNNs. However, these methods vulnerable small adversarial perturbations (7 Some studies aim to enhance explanation robustness through adversarial training [8] 31 32 The retrained model may be different from the original model, thus the explanations may faithfully reflect the model [17]. Unlike raining-based approaches, researchers have proposed explanation ensemble methods to enhance explanation robustness. For example, the explanation averaged across different methods improves robustness [19] Besides, some researchers obtain robust explanations by averaging explanations of perturbed graphs [16]. However, results from different explanations may be contradictory, making it difficult to integrate them and potentially leading to less stable explanations.
26 27 28 29]
Graph curvature. Efforts have heen made extend the geometric concept of curvature graphs [33 34 [21] Among these, the Ollivier curvature (20 and effective resistance are prominent approaches (35. Studies have demonstrated their relevance the issues of over-squashing and over-smoothing in GNNs [22 21 23 (25). For example, a rewiring technique has been proposed to alleviate over-squashing by increasing the curvature of edges in the graph [23]. Effective resistance has been utilized to quantify over-squashing in GNNs [2 Furthermore, a theoretical analysis has linked both over-smoothing and over-squashing Ollivier-Rice curvature (22). Moreover, they have been applied in complex networks [36 37 38) and deep clustering [39]. However, recent research
has not explored their potential for enhancing the robustness of explanations.
3 Preliminaries
3.1 Notation
Let G = (V,E) denote the connected, undirected, unweighted graph with the vertex set V and the edge set E. Let A and be the adjacency and degree matrices, respectively, with the Laplacian defined as L = D - A. The normalized adjacency matrix is Å = D-1/2 AD11/2 and the normalized Laplacian is = À D-1/2T,D-1/2 Let G' = (V', E') be the perturbed graph generated by adding and/or removing edges in G, where V C V. Let G. (V., E.) denote the explanation graph, where& E, EE V. V. We use (u, v) E E. Lc, denote the l-hop neighborhood of node u. The shortest path distance between two nodes u, is denoted by d(u, ((u, v). n, m, and n', m' denote the degrees of node u, in G and G',
and q to denote nodes in a graph. We denote u ~ v if edge
3.2 Graph Neural Networks
Let f(() denote the node feature matrix of graph G and (g) be the hidden vector at layer t (t = 1, ... IT), with the convention that (G) h. The hidden vector of node u at layer is denoted by h" (G), and is exactly the u-th row of ht' (g). The formulation for GNN is given by 22
(C) = Φt
( +
(h(((())) WEN
where v/2 is a message function (e.g., a linear function), and de is an update function (e.g., an activation function). + is an aggregating function. If ④ is the sum function, + ='EEN Aun. If + is the mean function,
EvEn. W
For node classification, no is mapped to the class distribution Pr. using the softmax or sigmoid function. In link prediction, we concatenate h (9) and (G) as the input linear layer to obtain the logits: h (G) < (Gh
(G): (T]( .LP)). where OLP is the parameter in linear layer. Then, it is mapped to the probability Pru. (9) of the edge (u, v) existing using the sigmoid function.
3

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For the graph classification task, average pooling of h across all nodes in graph 9 yields a single vector representation, which is used for classification and the probability denoted as Pr(g).
4 Relating robustness to curvature and resistance
We assume that the update function de is L-Lipschitz, i.e. ( or (222) S L|z 221, V 22.
The softmax function -1-Lisschitz The message function v/((=)) bounded, i.e. (x)))) S
M | Væ, 2 Additionally, | - C' for all p and 2 0. Let denote the L2 norm and N V These notations will be used in the following lemma, proposition, and theorem.
Definition 4.1. Given an input graph G and explanation G, (V., Es), we construct perturbed
graph G = (V', E²) by adding and/or deleting edges not &,. For node u, the approximate robustness of explanation on the node classification task is defined as 40]:
8 Eg (Pr() (ra()")el s.t. c = arg max Pr S
(2)
where denotes the predicted class of original graph G. (G), and Pr.. (G'); denote the probability of u on graph G and g' for a given class i. A lower value indicates a greater robustness of the explanation. The definitions for link prediction and graph classification are provided in Appendix
4.1 Robustness and Ollivier-Ried Curvature
Ollivier- Suliiee--Ricci Curvature on Graphs. The Ollivier-Riee curvature 20 considers random walks from nearby points. Let (k) represent the probability that random walker starting at u reaches node after a certain number of steps. On a graph, the random walk f is defined by [22
if u,
Pu (k)
deg(u)
otherwise.
Then, for any the 1- I-Wasserstein distance Wi Wi(Mss Mv) is given by
inf W Wi =ETT()
Σ
T(P (l,.))d(p.0)
(F.))
where IT Mb) is the family of joint probability distributions of pa and pea. The 1- Wassersteii distance measures the minimal cost required to transform Ph. into pa. The Ollivier-Rae Ricci curvature n(u, v) defined as
<(u,,) v) Wi
d(u, v)
Ricci curvature measures how easily information propagates from to based on topological connections of differentt lengths of paths around edge (u, v) 41 Negative curvature implies the edge is a "bottleneck", while positive curvature indicates the edge is present in a highly connected community. We use Ricci curvature bound the robustness of model predictions and explanations.
Proposition 4.2. Given any node in G. assume that there exists constant > 0 such that for edge v) E E, the curvature is bounded by k(u, v) Z a > 0. Let 3 = max(n, n'). If θ is the mean operation, (a)) 3(1oa) If is the sum operation, 91 (ax) = 2(11 a.). Then,
(G h(((") S BLCM (LLCM(++50) (1+)1(a)).
(5)
shows that for node u, the difference in the hidden representation vectors between G and g' is related to the minimum value of k(u, v) for any such that u ~ a is specific to node u. Intuitively, a large a implies high curvature for all edges, suggesting efficient information flow between u and its neighbors. Even with random perturbations, multiple alternative paths remain, leading to minimal change in representation have low curvature s(u, v), implying sparse connectivity. If edge (u, is removed, information flow may severely disrupted, increasing the change node representation h (9). The proof is in Appendix C.I We can use this proposition to bound the robustness of model predictions.
Proposition 4.2
u. However, when a is small, some neighbor v may

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Theorem 4.3 (Ricci curvature bounds the robustness of model prediction), In a GNN with T layers, let 9,, (Vu, En) denote the Thhop subgraph of target node u, where V, {uv Vid(u,o) S T). Let ( be the muximum degree of nodes in the T T-hop subgraph of node for both the original graph 9 and the perturbed graph g'. Assume that there exists a constant a' > such that for edges (u, v) E Ew the curvature is bounded by s(u, u) 2 a' > O. Let 17 I, NT M C. Then, the following inequality holds for the node classification task,
If + is the sum operation,
Prru(9) - Pra ( (1
+ Σ 210 + --
If + is the mean operation,
(9) - Pra(g')) 5 2T + 3(1 3(1-a') + 4(T - -1)3')
a'
Theorem 4.3 states that for a target node u, the robustness of model prediction is bounded by minimum k(u, v). V(u,v) v) E e.. A high a' indicates efficient message passing across all edges in G.. leading to robust predictions under perturbations. In contrast, a low a' shows that the presence of an edge (u, v) with small s(u, v), which may bottleneck message propagation. Deleting such an edge can cause a significant change in predictions. The proof is provided in Appendix C.2 The link prediction task and the graph classification task can be seen in Appendix [B.] and B.2
Theorem 4.4 (Ricci curvature bounds the robustness of model explanations). For node u, let
-= E.) be the explanation subgraph of Pr, Let L'rg'= =V", ,E'),E, S. E' denote perturbed graph by adding and/or deleting edges not in &.. Let a, denote the minimum k(u, v), such that for edges (u, v) E Eg, the curvature bounded by k(uu, v) 2 a > O. 17 L,L" M C. Let 3' be the maximum degree of nodes in the T-hop subgraph of node for both the original graph G and the perturbed graph G', and 3g be the maximum degree of nodes in explanation graph 9 Then, for node classification task,
If θ is the sum operation.
T
Pr.,(g) Pra( s (1+ (2.,(1 a.) + 45.) +
(288 + +2) + B.)) (b) Σ 1=2
If + is the mean operation,
(P) (S. S 2T-1 ( +
2T-1 3(1 3(1-aw)) + 4(T 1)3' +2( +1) -1)
as
Theorem 4.4 shows that the robustness of explanations is bounded by the minimum Ollivier-Riee value of edges in the explanation subgraphs. A larger 0g allows messages to be transmitted efficiently across all edges, resulting in robust explanations. In this case, graph perturbations lead minimal changes in the model's explanation. The proof is in Appendix C.33 The link prediction and graph classification tasks are in Appendix [B.3 and B.4 According the Theorem and 4.4 a relationship between Ollivier-Ricci curvature and robustness.
we establish
4.2 Robustness and Effective Resistance
Effective resistance quantifies how well two nodes are connected [24]. While many studies have explored its relationship with over-squashing, we aim to investigate its connection to the robustness of model predictions and explanations. Let u and v be vertices of G. The effective resistance between u and v is defined
(6) where 1,, 1, are the indicator vector of the vertices u, v and I+ is the pseudoinverse of L. The effective resistance can also be computed using the normalized Laplacian f. When multiple paths exist between two nodes, the effective resistance Ru, is small, indicating higher connectivity. Conversely, when available paths between two nodes are limited, Ryy becomes larger, reflecting lower connectivity.
1w), Ra = (1, (L. 11),
5

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Lemma 4.5. Let G be a connected graph. Let u and v be two vertices. Then
Ba = (v,.11- i+
(Vinl
1
vdu" 1,
where i is pseudoinverse of the normalized Laplacian L.
Lemma 4.6. Let G be a connected, non-bipartite graph. Then 1+ = -ドーo Af 121).
Theorem 4.7 (Effective resistance bounds the robustness of model prediction). Given a GNN with T layers, for u, let Gu = (Vus Eu) denote the T T-hop subgraph of u. Let R. = Liawee Ry denote Eal
the average effective resistance of edges in Gu- Let "i denote the maximum eigenvalue of A, and dmin min deg(v). Let n LLL'M"C Whether + is the mean sum operation, the following inequality holds for the node classification task,
2
[Pri(g) (r.(( S 2N + Ru + dmin 7)
Theorem 4.7
shows that the robustness of the model is related to Ry A smaller R. indicates greater robustness. The R. becomes small if there are many short paths between two vertices in the graph [24]. In such cases, removing an edge has smaller impact on connectivity or message propagation due to the availability alternative paths. The proof is provided Appendix [C.4 The [.
cases for link prediction and graph classification are shown in Appendices Bnd B.6
Theorem 4.8 (Effective resistance bounds the robustness of explanations). For node u, ler R.
Dia.reE, Rq.". denote the average effective resistance of edges in Gr, and dmin = minvev, deg(v). [xx] Let G' = "Y),",, S E' denote perturbed graph by adding and/or deleting edges not in & Let y denote the maximum eigenvalue of Å. Whether θ is the sum mean operation, the following inequality holds for the node classification task,
2
(r. Pra(g')) L
2/ R.
+ 2
dmin
Theorem 4.8 shows that smaller R, implies greater robustness of the model explanations under input perturbations. The proof is provided Appendix C.3 The cases for link prediction and graph
classification tasks are shown in Appendices B.7 and B.8
5
Method
Existing explanation methods compute the importance score F(u, v) for each edge (u, v) to faithfully explain model predictions. However, F(u, v) fails to capture the sensitivity of edges to perturbations, resulting in potentially non-robust explanations. Thus, we incorporate Ricci curvature s(u, v) and effective resistance R(u, e) as the robust scores of edges. Let 9. = (V., E,) denote the explanation subgraph and æuu, v) denote whether edge (u, v) is in E,. balance fidelity and robustness, we define the following objective functions select important edges,
max (u, (F(u,)) +C(u,))) where C(u,v) [k(u, v), (0.. [s(
R(u,)), v).
the parameter to control the trade-off between fidelity and robustness, Due negative comelation between Ricci curvature and effective resistance (see Appendix D.5). we construct robust explanations using each metric separately. For curvature-based explanations, we select edges with higher F(u. v) and s(u,, v). For resistance-based explanations, we select edges with higher F(u, v) and lower R(u, v). more robust explanation is obtained using Equation
Computational complexity. The complexity of Ollivier-Ricci curvature is -1))) Computing effective resistance requires the pseudoinverse of i. which is computationally expensive. To address this, we adopt an efficient approximation method based on solving linear systems of the form Lax 43 with complexity of O((E) log([V))).

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Experiments
Datasets and tasks. We evaluate our method on three tasks across nine datasets: Cora, Citeseer, and PubMed for node classification; BC-OTC, BC-Al,ha, and UCI for link prediction; and MUTAG, PROTEINS, and IMDB-BINARY for graph classification. Details are provided in Appendix D.1
Explanation methods. We consider six explanation methods: GNNExplainer, PGEXxplinner, GNN- LRP, Iee, FlowX, and Convex. Details of these methods are provided in Appendix D.4
Experimental setup. For each dataset, train GNN on the training set according the task. Two GNN architectures are evaluated, with implementation details provided in Appendix D.2 We apply six classical explanation methods compute edge importance scores F(u, 8) and caleulate Ricci curvature and effective resistance for all edges. To determine the optimal in Equation the explanation targets into training and test subsets. Optuna is used tune À on the training subset, which is then fixed on the test subset used in Equation (7). The selected À values for Ricci curvature
we split
and effective resistance across methods and datasets shown in Tables I1 12 13 and 14
Table 1:
Relative error performance for robust explanations based on Ricci curvature. The + in GNN is sum operation. A higher value (1) indicates better performance. Blue highlights the best result.
Methods
Cora
BC-OTC
BC-Alpha
MUTAG
PROTEINS
INDE-RINARY
GNNExplainer
4,5%
-6.0%
0.5%
54.4%
70.1%
73.3%
2.2%
4.0%
-0.2%
GNNEEplainer+Curvaauue
16.0%
2.6%
55.2%
70.5%
74.7%
1.5%
PGExplainer
2.7%
6.1%
2.6%
3,8%
-0.3%
-0.8%
-2.8%
0.2%
-1.2%
PGEsplainer-CCrraauue
4.5%
8.4%
5.6%
6,4%
1.9%
1.9%
3.9%
2.6%
1.3%
Convex
32.5%
60.2%
-4.3%
Convex+Curvature
15.9%
60.6%
37.7%
$6.3%
71.2%
58.2%
5.6%
1.1%
12.3%:
DeepLIFFT
11.6%
15.8%
13.4%
0.3%
-2.1%
11.1%
-2.5%
0.5%
-0.5%
DeepLIFT-Carvature
16.7%
23.9%
14.6%
1.3%
1.9%
11.5%
1.4%
2.1%
1.9
GNNLRP
16.4%
-2.0%
19.4%
7,5%
5.1%
19.8%
-1.1%
0%
-2.7%
GNNLRP-CCrvature
22.7%
6.2%
23.3%
9.15
5.7%)
24.4%
2.5%
1.5%
2.7%
FlowX
2.7%
30.6%
8.6%
18.3 %
3.5%
7.0%
-1.8%
-3.1%
0.5%
FlovX-Curvature
7.8%
38.
17.5%
19.3%
4.3%
11.9%
1.3%
1.6%
1.8%
Table 2: Fidelity k performance for robust explanations based on Ricci curvature. The + in GNN is sum operation. A lower value (L) indicates better performance, with blue highlighting the best result.
Methods
Cora
Citeseer
Pubmed BC-OTC BC-Alpha
UCI
MUTAG
PROTEINS IMDB-BINARY
GNNExplainer
0,961
0.578
0.549
0.184
0.245
0.093
0.177
0.068
0.284
GNNEsplainer--Crrvaure
0.924
0.518
0.566
0.175
0.242
0.093
0.170
0.067
0.282
PGExplainer
1.762
0.537
0,447
0.227
0.351
PGEsslainer-Curraaure
1.754
1.339
0.726
0.537
0.447
0.399
0.225
0.048
0.344
Convex
0.353
0.277
0.075
0.132
0.009
0.011
0.010
0.001
0.017
Convex-Curvature
0,340
0.273
0.073
0.053
0.008
0.010
0.008
0.001
0.021
DeepLIFT
1.020
0.481
0,277
0.178
0.134
0.202
0.206
0.054
0.315
DeepLIFT+Curvatue
0.956
0.473
0.304
0.177
0.133
0.202
0.180
0.054
0.321
GNNLRP
0.472
0.589
0,293
0.174
0.202
0.234
0.116
0.033
0.303
GNNLRP+Curvature
0.442
0.572
0.276
0.173
0.201
0.234
0.153
0.036
0.321
FlowX
0.495
0.499
0.370
0.184
0.086
0.244
0.174
0.051
0.298
FlowxeCurvature
0.489
0.470
0.365
0.182
0.065
0.244
0.171
0.051
0.301
Quantitative evaluation metrics. We adopt the robustness metric 5" proposed in [40] defined in Equation (2). where lower values indicate more robust explanations. However, in some cases, 8* is small and cannot significantly show difference across methods. We report the relative error, defined as Surenn - where 6mnodm is the robustness of randomly sampled explanation subgraph. A negative mexin relative error indicates worse robustness than the random baseline, while higher value implies stronger robustness by reflecting greater improvement over randomness. We also evaluate fidelity, which quantifies how well the explanation subgraph preserves the model's predictive distribution.
6
7

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29 we use the KL-based metrie: Fidelity KL = (L(()(())l Pr(G.)). where lower Fidelity KL values suggest that 9, is important for model. More details can be seen Appendix D.3
Following [5
Table Relative error performance robust explanations based on effective resistance. The + GNN sum operation. A higher value (t) indicates better performance. Blue highlights the best result.
Methods
Cora
Citeseer
Pubmed
BC-OTC BC-Alpha
UCI
MUTAG
PROTEINS IMDE-BINARY
GNNExplainer
13.6%
32.0%
5.8%
19.8%
14.6%
41.6%
1.1%
0.9%
CuuNEspaanee-Curvauuru
14.4%
36.5%
7.7%
25.6%
19.2%
50.7%
5.3%
2.3%
3.1%
PGExplainer
-1.5%
0.5%
0.6%
1.7%
3.9%
11.3%
-3.9%
1.7%
-1.8%
PGExplainer-
1.0%
1.5%
3.2%
11.6%
14.2%
25.4%
3,5%
2.2%
2.0%
Convex
40.9%
32.6%
21.2%
63.7%
56.9%
33.7%
1.7%
-3.1%
4.3%
Convex-Curaauure
29.1%
2.1%
DeepLIFT
14.9%
5,5%
14.7%
-3.4%
0.1%
-3.8%
-1.1%
0.9%
-1.0%
DeepIFTTCurvatun
1.8%
GNN-LRP
57.9%
24.3%
43.7%
2.6 96
0.25%
1.9%
1.7%
-0.9%
-1.2%
GNN-IRP-Curvature
60.5%
25.5%
45,4%
6.0%
1.4%
2.3%
3.8%
0.7%
1,3%
FlowX
12.3%
21.5%
-0.8%
Flowx-Curvature
19.0%
13.3%
22.7%
25.5%
24.1%
5.3%
3.6%
0.2%
0.3%
Table 4: Fidelity performance for robust explanations based effective resistance. The + in GNN is sum operation. A lower value (L) indicates better performance, with blue highlighting the best result.
Methods
Cora
Citeseer
Pubmed
BC-OTC BC-Alpha
UCI
MUTAG
PROTEINS
IMDB-BINARY
GNNExplainer
1.163
0,557
0,516
0.219
0.335
0,200
0.189
0.050
0.293
GNNExplainer-Curvatm
0.983
0.546
04855
0,212
0.145
0.
0.049
0.276
PGExplainer
1.573
1.521
0.695
0.254
0.433
0.392
0.199
0.054
0.297
PGExplainer-Cun Curvattur
0.404
0.258
Convex
0.195
0.165
0.071
0.011
0.001
0,046
0.108
0.001
0.012
Conves-Curvature
0.159
0.113
0.060
0,006
0.001
0.017
0.143
0.001
0.011
DeepLIFFT
0.877
0.435
0.357
0.439
0.318
Deep.JFT-Curratuu
0.833
0.457
0.361
0.314
0.326
0,449
0.258
0.044
0.320
GNN-LRP
0.096
0.257
0.397
0.304
0.317
0.290
0.152
0.059
0.255
GNN-LRP-Curvatte
0,073
0.228
0.367
0.302
0.315
0,340
0.153
0.058
0.262
FlowX
0.941
0.631
0.519
0,245
0.321
0.254
0.188
0.047
0.308
Flovx-Curvvaure
0.869
0.632
0.518
0.202
0.357
0.269
0.178
0.046
0.307
Table 5: Relative error performance for robust explanations based on Ricci curvature. The ① GNN is mean operation. A higher value (1) indicates better performance. Blue highlights the best result.
Methods
Cora
Citeseer
Pubmed
BC-OTC BC-Alpha
UCI
MUTAG
PROTEINS
IMDB-BINARY
GNNExplainer
-1.1%
1.1%
2.0%
24.9%
12.2%
2.1%
-2.5% 1.3%
-0.3%
0.2%
GNNEsplainerrCCuraatue
4.8%
6.8%
5.5%
27.9%
13.5%
5.3%
0.5%
0.6%
PGExplainer
..66%
-0.5%
-1.7%
.0.3%
PGEsplainer-Currature
2.4%
2.4%
1.2%
1.8%
0.5%
0.9%
0.9%
0.5%
0.3%
Convex
23.4%
.3.6%
4.6%
14.0%
2.3%
9.5%
0.5%
-43%
-0.5%
Convex-Curvature
37.9%
10.8%
11.3%
16.8%
3.5%
16.7%
3.2%
1.1%
0.2%
DecpLIFT
-0.1%
55.3%
14.9%
-0.9%
-2.3%
0.8%
-0.2%
0%
-1.8%
DeepL.FT+Curran
1.7%
67.9%
19.8%
0.4%
0.5%
1.7%
0.3%
2.2%
1.0%
GNNLRP
20.29
8.9%
10.3%
22.9%
22.8%
-..2%
.0.5%
-0.7%
0.3%
GNNIRPPCCrvatuue
26.6%
FlowX
7.0%
6.8%
6.8%
0.4%
-0.6%
-1.6%
0.6 %
-0.2%
0.1%
HoovaaCure
0.2%
1.3%
Performance evaluation and comparison. We report the relative error and fidelity of explanations under different aggregation functions in GNNs. If the + is sum operation, the corresponding results are presented in Tables I 3 relative error, and Tables 2 4 for fidelity. If the ① is mean operation, results are shown in Tables Jnd relative error, and Tables and for fidelity. Across all datasets and methods, curvature- and resistance-basee explanations consistently improve robustness. In terms

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Table 6: Fidelity K performance for robust explanations based Ricci curvature. The in GNN is mean operation. A lower value (L) indicates better performance, with blue highlighting the best result.
Methods
Cora
Citeseer
Pubmed
BC-OTC
BC-Alpha
UCI
MUTAG
PROTEINS
AMDB-BINAKU
GNNExplainer
1,458
13361
0.162
0.176
0.099
0,063
0.153
0,034
0.241
1,393
1.372
0.149
0.175
0.098
0.063
0.150
0.032
0.235
PGExppainer
1.647
1.34
0.581
0.209
0,110
0.054
0.156
0.025
0.249
PGExplainer+Curratuee
0.206
0.109
0.053
0.146
0.243
Convex
0.086
0.005
0.002
0.033
0.007
0.001
0.015
0.001
0.011
Convex+Curvature
0.029
0.030
0.004
0.001
0.019
0.011
DeepLIFT
1.012
0.291
0.270
0.193
0.089
0.050
0.143
0,040
0.302
Deepl.FT-Currature
0,992
0.260
0,270
0.192
0.089
0.049
0.126
0.040
0.301
GNNLRP
0.022
0.011
0.162
0.231
GNNLRP-Curvature
0,470
0.529
0,176
0.104
0.023
0.013
0.173
0.034
0.229
FlowX
1,276
1,164
0.466
0.190
0.081
0.054
0.161
0.039
0.253
FlowX-Curvature
1,267
1.164
0.466
0.190
0.081
0.054
0.160
0.038
0.249
Table 7: Relative error performance for robust explanations based on effective resistance. The + in GNN is mean operation. higher value (1) indicates better performance. Blue highlights the best result.
Methods
Cora
Citeseer
Pubmed
BC-OTC BC-Alpha
MUTAG
PROTEINS IMDB-BINARY
GNNExplainer
0.8%
0.6%
1.1%
22.0%
7.4%
-0.7%
0.3%
-0.2%
-0.1%
GNNExplainer-Curan
2.3%
0.2%
0.2%
PGEsplainer
0.6%
2.3%
5.0%
0.8%
-0.%%
1.1%
.43%%
0.1%
-0.2%
PGExplainer-Curvati
3.7%
4.1%
5.6%
2.1%
3.3%
2.1%
1.4%
0.5%
0.2%
Convex
10.7%
0.7%
.0.%%
Conves-Curvature
22.8%
28.9%
2.5%
19.9%
15.5%
3.1%
1.7%
2.9%
2.3%
DeepLIFT
16.3%
19.8%
23.5%
-0.5%
-4.3%
-0.6%
-1.3%
-0.6%
0.0%
DeepLIFT-Curatture
25.9%
21.5%
27.6%
0.8%
0.6%
1.5%
1.2%
0.3%
0.4%
GNN-LRP
19.9%
12.8%
11.8%
25.8%
14.1%
1.1%
3.7%
-0.3%
0.3%
GNN-IRP--Curvature
25.5%
18.2%
17.6%
30.7%
15.55%
2.7%
4.9%
0.4%
0.6%%
FlowX
7.1%
0,2%
6.3%
1.0%
0.8%
-1.9%
0.3%
0.1%
0.7%
FlowX-Curvature
8,5%
2,9%
7.4%
2.1%
2.6%
1.7%
2.9%
0.3%
0.9%
of fidelity, Ricci- and resistance- explanations match or exceed the base explanations in 85% and 80% of cases, respectively, under the mean aggregator, and in 83% and 80% of cases under the sum aggregator: The impact of robustness enhancement on fidelity varies by method. For GNNExplainer, robustness improvements do not clearly reduce fidelity. However, for GNN-LRP and Flow fidelity may decline, especially when importance scores are uniform across edges, causing curvature or resistance to dominate selection and shift the explanation away from the model's original behavior. While improving robustness may slightly affect fidelity, the trade-offs are generally acceptable.
Table 8: Fidelityxs performance for robust explanations based on effective resistance. The D in GNN is mean operation. lower value (L) indicates better performance, with bluc highlighting the best result.
Methods
Cora
Citeseer
Pubmed
BC-OTC BC-Alpha
UCI
MUTAG
PROTEINS IMDB-BINARY
GNNExplainer
1.565
1.349
0.477
0.205
0.125
0.065
0.145
0.034
0.263
GNNEEplainer-Cuuraat
1.422
1.263
0.466
0.203
0.065
0.135
0.035
0.262
PGExplainer
L6066
1.455
0.560
0.343
0.158
0.065
0.181
0.034
0.235
PGExplainer- Curvattre
1.602
0.065
0.233
Convex
0.049
0.054
0.014
0.020
0.007
0.015
0.004
0.001
0.002
Conves-Curvature
0.052
0.051
0.011
0.017
0.006
0.016
0.004
0.001
0.002
DeepLIFF
0.825
0.846
0.387
0.199
0.055
DeepI.IF T-Curratrueeeeeeeee
0.806
0.849
0.397
0.199
0.084
0.055
0.235
0.035
0.226
GNN-LRP
0.568
0.802
0.317
0.034
0.027
0.101
0.133
0.034
0.202
GNN-LRP-Curvane
0.499
0.610
0.277
0.033
0.026
0.101
0.139
0.035
0.205
FlowX
1.325
1.193
0.423
0,288
0.141
0.068
0.162
0.037
0.246
Flov--CCurvature
1.298
1.178
0.413
0.296
0.143
0.068
0.157
0.037
0.249

---

Ablation analysis. To evaluate the impact of À on explanation robustness and fidelity, we compute the differences in relative error and fidelity between the curvature- based and original methods across various datasets, methods, GNN architectures, and À values. A positive difference in relative error suggests enhanced robustness, while a negative difference in fidelity implies an improvement in explanation fidelity relative to the original method. The results are in Figures 5 6 7 and 8 (in Appendix). Results show that on some datasets, regardless of whether the GNN uses mean or sum aggregation and across different values of X, leveraging Ricci curvature and effective resistance yields more robust and faithful explanations.
GNNExplainer
PGExplainer
Pursature-PEExpaanr
Canvex
Riodi Sarvature-Covver
DeepLIFT
Ricoi curveture+ DeepLIFT
GNN-LRP
Riodi curvature+GNN-LLP
Ricdi curveture-Flowx
FlowX
Figure 2:
The green node denotes the target node to be explained, and yellow edges indicate the selected important edges. Titles with "Ricci curvature#" indicate that the explanations are curvature- explanation.
Running time and case study. We report the runtime for computing Ricci curvature and effective resistance on three large datasets (details in Table Results are shown in Figures 3 and 4 Ricci curvature takes 3.5 seconds for over 15,000 edges, while effective resistance takes about 80 seconds. These times are acceptable. Visualization results are shown in Figure 2 Explanations considering Ricci curvature tend to select edges forming triangular structures, leading to greater robustness.
7 Conclusions and Limitations
We are the first to explore the relationship between the robustness explanations and graph curvatures. We theoretically prove that minimum Ollivier-Ricei curvature and averagr effective resistance can bound the robustness of model predictions and explanations. We use Ricci curvature and effective resistance as robustness scores obtain roust explanations. Our method agnostic to specific GNN explanations. Empirically, it consistently improves robustness over six baselines across nine datasets spanning node classification, link prediction, and graph classification, while also improving fidelity in 80% of cases and achieving up to 10% relative gain in robustness. As a limitation, our analysis focuses the robustness of explanations under structural perturbations, extending feature-space promising direction for future work.
10

---

Acknowledgements
Hui Xiong was supported in part by the National Key R&D Program of China (Grant No. N0.2023YFF0725001) in part by the National Natural Science Foundation of China (Grant No. No.92370204), in part by the guangdong Basic and Applied Basic Research Foundation (Grant No.2023B 1515120057), in part by the Education Bureau of Guangzhou. Sihong Xie was supported by the Department of Science and Technology of Guangdong Province (2033XX1XX079) National the Guangzhou-HKUUS
Key R&D Program of China (Grant No. 20.22223FFF072.. Funding Program (Grant N0.2023A03)0008 and Education Bureau Guangzhou Municipality. Xi Zhang was supported by the Natural Science Foundation of China (No. 62372057).
Joint
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13

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NeurIPS Paper Checklist
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Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification:
The abstract and introduction outline the primary contributions. We investi- gate the connection between two geometric edge properties, curvature and resistance, and explanation robustness.
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Technical Appendices and Supplementary Material
A The definition of robustness explanations on link prediction and graph classification tasks
Definition A.1. Given an input graph G and explanation G3 (V.,&,) we construct a perturbed graph G' (V', E²) by adding and/or deleting edges not in & The robustness of explanation on graph classification task is defined as:
8 = (r((), - Pr(g")d s.t. c = arg max Er E
where denotes the predicted class of original graph G. Pr(g); and Pr(g"); denote the probability of graph G and g' for given class i. Lower value indicates greater robustness of the explanation.
Definition A.2. Given an input graph G and explanation G3 (V., E.), we construct a perturbed graph 9 = (V", (") by adding and/or deleting edges not in &,. For target edge (u, u)), the robustness of explanation on link prediction task defined as:
8" Eg: (9)
Pruv (9') s.t. arg max Pruo 3
where denotes the predicted class of original graph G. Pru. (9); and (9') denote the probability of edge (u, v) on graph G and G' for given class Lower value indicates greater robustness of the explanation.
B
Theorem
B.1
The Ricci curvature and model prediction robustness link prediction task.
Theorem B.1 (Curvature bounds the robustness of model predictions in link prediction). Given G -V, E) G' = (V', "") and the T layer model. For u, u, let Gu, (Vu,y; Eu,y) denote the denote the union of T Thop subgraphs of u and v. Assume that there exists constant a' >> O such that for edges (u, v) E Ev the curvature bounded by s(u, v) 2 a' >> Let 3 be the maximum degree of nodes in the T-hop subgraph of node u for both the original graph & and the perturbed graph G'.
Let 1) = L,LL M C. Then, the following inequality holds
If + is the sum operation,
T PPr (9) Prav((g)) (g')l V2 Σ 2(3))(++)))
(11+ - a²) + +38)
-2)
If ① is the mean operation,
Pr(g) - Pr(g') V2n 2T 3(1 aa') + 4(T - -1)8')
a'
B.2 The Ricci curvature and model prediction robustness graph classification task.
Theorem B.2 (Curvature bounds the robustness model predictions in graph classification). Given E') and the T layer model. Let V=|V"). Assume that there exists constant
G (V,
= (V' g'= a' > such that for all edges (u, v) E E, the curvature is bounded by k(u, v) 2 a' > O. Let 3 be the maximum degree of nodes in the T-hop subgraph of node for both the original graph and the perturbed graph G' Let 17 = L I. MT C. Then, the following inequality holds for the graph classification task,
If θ is the sum operation,
T [Pr(g) Pr(g") S 2(())(++))) (1+3)))))+++))-+ (2 (1-a)) - 418))
If + is the mean operation,
3(1 - a')
[Pr(g) - Pr(g') S 2T -T 1
+ +4T - 1),3)
a'
21

---

Proof.
Eoe,, Pra(g)
EvEN Pra(g')
[Pr(g) Pr(g")
[
[V]
Pra(()) Pra(gg)
max wENu(()
PPr.(G) (r))
[V]
(e)
According the Theorem 4.3, and for all edges (u, E, the curvature is bounded by s(u, v) 2 a'.
thus, the above inequality holds.
B.3 The Ricci curvature and model explanation robustness on link prediction task.
Theorem B.3 (Ricci curvature bounds the robustness of explanations on link prediction task). For edge (u,v), suppose that G3 = (Va,E, Eg) ) is the explanation subgraph of (g) Let g' S E' denote perturbed graph by adding and/or deleting edges not in Eg. Let a2 denote the minimum su(., v), such that for edges (u, u) E &s, the curvature is bounded by k(u, v) 2 0s > O. ? == L, I." M C. Let S' be the maximum degree of nodes in the T T-hop subgraph of node for both the original graph G and the perturbed graph G', and Bg be the maximum degree of nodes in explanation graph Gs. Then,
If + is the sum operation,
Pru (G) Pr,()')) V2n (1+
(28, -.,) + 48.)
128 +2) 8,) 8.)T-1).
i=2
If ① is the mean operation.
PPru(g) - Prr(')) v2n ( 22--1(48, + 3(1 3()=0s) + 4(T - + + 12)
B.4 The Ricci curvature and model explanation robustness on graph classification task.
Theorem B.4 (Ricci curvature bounds the robustness of explanations on graph classification task). Suppose that G- = (V.. E,) is the explanation subgraph of Pr(g). Let G' = (V',E" ,'),,, & C E' denote perturbed graph by adding and/or deleting edges in Eg. Let ag denote the minimum k(u, v), such that for edges (u, v) E Ex, the curvature is bounded by k(u, v) 2 a, > O. 1) =- MT C. Let 3 be the maximum degree of nodes in the T T-hop subgraph of node u. for both the original graph and the perturbed graph G', and Ba be the maximum degree of nodes in explanation graph G Then, If ④ is the sum operation,
T
(Pr,(g) (r.((")) S (1+.8))) (2.3(1 - a..) + 43.) + Σ (23 + +1)((1 (.) T-))
1=2
If ④ is the mean operation,
S 2 (2(0))1)22( ( + 3(1 -as) + 4(T 1)3² + +(T - -1)
(9r,(g) Pra()))
as
B.5 The Effective Resistance and model prediction robustness on link prediction task.
Theorem B.5, Given G = (V,E) and g' = =(V,,") and the T layer model. For (1, ler Gu.v
VV En,)) denote the union of T -hop subgraphs of and v. Let Rt, Eca.heeu. Ry denote the [uuel average effective resistance of edges in Gu,v- Let N \Vul, and y denote the maximum eigenvalue of Å. Let dmin = minvey., deg(v). Let 1 = L, L M C. Whether Q is the mean or sum operation, the following inequality holds for the link prediction task
(g) - Prav (g')l v2n 2N + R. + dmin (1 - -7))
22

---

B.6 The Effective Resistance and model prediction robustness on graph classification task.
Theorem B.6. Given G = (V,8) and g' = (V', ,E) and the T layer model. Let R. = Elarhee Bay
[E]
denote the average effective resistance of edges in Gu- Let N = Dul. and denote the maximum eigenvalue of A. Let dmin min deg(v). Let L1 LLLTM Whether + is the mean operation, the following inequality holds for the graph classification task
Pr(g) Pr(g") S 1) 2N + Ry + dmii (1 -
B.7 The Effective Resistance and explanation robustness on link prediction task.
Theorem B.7 (Robustness of explanations on link prediction task), For edge (u.,) suppose that
9, (Vs, E.) is the explanation subgraph of Pr (9). Let G' = (V' "',,'), S E' denote perturbed graph by adding and/or deleting edges not Eg. Let R. Luuvce Ru. denote the average effective resistance of edges in 9s, and dmni min deg(v). Let y denote the maximum eigenvalue of Å. Whether + is the sum or mean operation, then,
[E,l
2 Prree (G.) - (g')) 22n 2N + R + dmin (1 -7)
+ +2)
B.8 The Effective Resistance and explanation robustness graph classification task.
Theorem B.8 (Robustness of explanations on graph classification task). Suppose that G. (Ve, E.) is the explanation subgraph of Pr(g). Let G' = (V" ,E', C denote perturbed graph by adding and/or deleting edges not in e,, Let Rg = Eiamer, Bg." denote the average effective resistance of [.,
edges in Gs, and denin = min.ev, degree(v). Let y denote the maximum eigenvalue of A. Whether + is the sum or mean operation, then,
2 - S 2N R. dmin (1 7) +2)
C Proofs
C.1 Proof of Proposition 4.2
Proof.
[h((9) - h1((9)) S [L(99) - (i(9) + ( (9)
(8)
+ the sum operation,
(9(()) - (9) L
Σ
Σ
(hh
PEN,
gEN.
=
Σ
Ne(h
Σ G.A.
(a (G))
PEN.AN.
S
Σ
PEN.AN.
In 22 the curvature holds the following inequality
N. max (m.
2 k(u, v). Then,
W.o.Wll & [M V... = n + 1 - N. - n - nn(u, v).
The symmetric difference N₁ N, satisfies
W. A ACI. (OGWW)
2(1 --(( u))n.
(9)
23

---

Thus, (0) 1h) & 2m(1 k(u, L.M. The second part of Equation is
(N((9)) (9)
SL
Σ 11-1
11 (1))
Σ
44-1
(
h/
(9))
(EN:(19)
<7
E
14 1/ (00100))
E
42-1
h h (g)
(EN.(P))V.19)
(EN.CO)WG
1/-1 ( hi (0))
-v2-((1) (9)
E
1/2-1
(EN
S WW' A LCM + L <An'TCM
mar(n, n'), thus obtain the following equation:
["'('") (9) < 2n(1 x(u, DCM + 4n'LCM S 23(1 - + (10)
if ① is the mean operation. in we can obtain the following equation:
-
Σ
1
Σ
-
n
m
PEN.
QEN,
L Σ
(--1
1
1
(h (9))
m
n2
PEN
+L Σ
oh 4- (G)) (())
Σ
I v. (G))
PEN.WN
JEN.WA,
We have m 2 W,, NN, 2 n((, v)n.
1 1
1 1 k(u, v)
m
n k(u, v)n
n
<(u, v)n
Then,
1-)(u,
11+1
h
L Σ
+L
Σ
klu, v)n (ve(h))
k(u, v)n
PEN.AAA
PEN.
< Ln - -((( v)
1
CM + + v)lu,
CM
k(u,v)n
k(u,v)n
3
< - n(u,
k(u, v)
24

---

The second part of Equation (
can be rewritten as
(9')
(9)
SL
Σ
1 n'
91-1
h1 (1)1(0)
E m
V/-
n (9)
PENU(())
(EN).(G)
<7
E
1
41-1 11 (1:-1(01)
E
- -
(g)
h1
n"
PENUL0))NN.OO)
(9)
E
( ( (n)))0))
1/
11 (9)
(FNN(())
(PENN1TTA N
LCM
LCM + 2LCM
m
(i((S)) A No(9))
LCM + 2LCM
m
<4LCM
Thus,
3
(g'g') - (9 S(1 x(u, U)LMMM (((,,))
+ 4LCM
(11)
(1 a33CCM +43LCM
<LCMB(4-91())
(12)
C.2 Proof of Theorem 433
Proof. If + is the sum operation, and suppose that the GNN has two layer, then,
(9) Pr.((g)) L, - (9')
SLLLM
(l( - (''))) +
Σ (9(
Σ
vEN.(G)
vEN_(9")
SLLLM -h((9)-)h((')
Σ
h2(9)
Σ
(EN
(PN
Σ
(h!(G) - (h!(G)-hh m!(GT)
(FN
SLLLM
-h+(')++
Σ
Σ ()
(h!(9) h: n!(3T) )
Σ Σ
Σ
+
a' is the minimum curvature of all edges (u, v) E E,, and 3 is the maximum degree of nodes in the T' T-hop subgraph of node u for both the original graph G and the perturbed graph G'. According to
25

---

Equation 10
(Pra(9) - Pr
p + 8r + 3 28 (88(11 - -d' + 444)
SLILMC
(o'1 +
(1+ (1+ SLLLMMC
(1 + 43') + +
Similarly, If the GNN has the T layer, then,
PPr((g) - PPr((g)) - L1 1h (g)
n (g')
SLLM [I-T-1(9) no (g')())))++ +
E
h (9)
E
(ON,(1))
+ Σ (h) 12- no-)((0))
T 23th (1 8Y) T-)-
IL (1+ (2 - a') + 43) + Σ
1=2
If + is the mean operation, and suppose that the GNN has two layer, and n' 2 n 2 m. then,
(g) - (ra)) < L1 (g) (G')
1 n6' (l') + E LN((0) - E n
SL,LM
- (g)
(9)
VENU19")
SLIM
{1 h (g)
1 _n)(("))
+
_
+
Σ
(9)
n
n
n
(EN.16))W(000
Σ
h!((") n
Σ
(9)
1
=n((g))
+ -
1 =
+
+
(P'
(EN.((')
n
n
SL,LM (1 (( (1(09) ((10)())-+11)))))))0)))
+ +112
Σ
Σ
m 1)(
1
(F)) n
PEN.
Σ n'
Σ
m (3')")
1
(EN_( pEN.(0")
- =mm00o))
Σ
1 -,))))) + n
(ON
Let a' be the minimum curvature of all edges (u, v) E Vu, and 1 be the maximum degree of nodes in the hhp subgraph of node u for both the original graph G and the perturbed graph G'. According Equation [12
PPra(G) - (r))
3(1- a')
+ 3'' 23' + 28' + 3(1 a))
SL,L2M²C
43'
+ 8¹
a'
a'
a') 3(1 3(1- ((1-a))
SLILMMC 2 43'
+ 4,8
a'
26

---

Similarly, If the GNN has the T layer, then,
(Pr - Pr,((g)) s L1 (C (9')
SL,L 22'-1(43' 3(1 3(1-a') + 4(T 1),8¹)
a'
C,3 Proof of Theorem 4.4
Proof. G, = (V..e,) E.) is the explanation graph. G' (V",E") is the perturbed graph, where & C E obtained by adding removing edges not in &,. Then,
(g) Pru( $ (g) Pr.((..) (Pr.)l+PP((..))) Pr,(
① the sum operation, suppose that the GNN has two layer, and let a be the minimum curvature of all edges (u, v) E &s, and 3' is the maximum degree of nodes in the Thhopp subgraph of node for both the original graph 9 and the perturbed graph g'. Ag is the maximum degree of nodes in explanation graph G,, Then,
(Pr.(G.) Pra((")) L1 (9.) (9')
LL,LM (G.)) - h(()))+ +
Σ
E
(g')
(E.i.(9.)
vEN..(9)
SLILM (b)( - (1'0'))
Σ
h),))))
Σ
PEN.(O.JW..0))
(EN.(O')W..1.)
(hL(G)) - m1(1T)
+ Σ
PEN.I6")NWNIG.))
SLILM
(H.)))) - -9
Σ
E hy(G))
PEN.
Σ
Σ hL(0")
Σ
(N(().)-(((C')))
+
+
(o. (9))
VEN.I9''NW......
SLILMMCC
23.(1 - -a.) + 48, + BaB' + + B, 23, a.) + 43.)
SLIPMC
(1+ B.) 23,(1 a.) +48, + 303 + gr
LLLLMMC
(1+ (1+ 8.) 28,(1 -a.) +.)+48,
+ 2
Thus.
PPr((g) Pru(g')
P(rr(()) (r.( +(Pr)) Pr.(9))
SLLLMMCC (1 (1+13) Ba) 23 a.) + +33, + 232 +2
27

---

Similarly, If the GNN has the T layer, then,
(gu Pr LLLL (1+ -2) 48,)
+ Σ (28 + 2) 3B) 7-1
a.) +
+ is the mean operation, and suppose that the GNN has two layer, n' 2 n 2 m, and let dg be the minimum curvature of all edges (u, v) Eg, and 3 is the maximum degree of nodes in the T -hhop subgraph of node u for both the original graph & and the perturbed graph g'. Bg is the maximum degree of nodes in explanation graph Gs- Then,
(G.) (g)) Ta (G.) (g')
LL,LM
(),) n
1 (0') + E (l(9)))
E VEAU.(S))
n h3
n
vEN.(G.))
SLILM
1 (G.)
1 h
- ))
(=
1
1
+
E
(G.)
n2
ni
(o..(C))W..09)))
E
LG())) n
Σ
(L. - 1 (G' + ( -- 1 (D-)(()))
+
(EN,((7)nWW(O))
SLILM (M) - (1()))
+
1/2 72
-
1
+
Σ
Σ
1 (gg))))
m
10N.10.)W.00))
PEN.
E
n Σ
m (9))
PEN.(0)
(h-(9)) - (g')))) + +( - -===1)(()))
Σ
7
3(1 (.))
3(1 -as)
SLIPMC
48, +
+ +3, + 2,' 2,3 +
+ Bg
as
0.
3(1 025)
SLLDMCC
2 43,
+ 48'
as
Thus,
(g) Pr,(
< (r) (r,( + (Prr(g.) Pra(
MLIDM2C 2 4/3 + 3(1 ass)
4,8 +
0₂
Similarly, If the GNN has the T layer, then,
(Pr,(g) Pra()))
5..
SLILMC 2T-1 (43, 1 + 3(1 3(1-as)) + 4(T - 1)8' +(T
as
C.4 Proof of Theorem 4.7
Proof. let (g) = Vv r) S(). R., = Eiavice Ra.s
[8]
28

---

If + is the sum operation, then,
Pgr(()) - (ra(())) & L1 |h (g) n (g))
SLILM
Σ
Aouhe A
Σ
Aou an
(G)
vEN)(G0)
SLL'M
Σ
Σ
Apj ... Avch)(9)
E
Σ
A/p--. A'anh? 3
(9NG(9)
(9)..
PEN,(9))
vENu(99)
SLEMCC
Σ
AT
Σ
A
(w)((
DEBr(w)(O))
SLLLTMC
Σ
AT
Σ
A R. Ru
(E)r(
(w)
AT
Σ
A + P...
PEBT (M)(C)
(E))(
T
Eignjev
1 (Vd
i+
111,,
vd."" 1,
ddg
+
If + is the mean operation, then,
(G) - Pr,((")) S Ta |h (G) h (g')
SL,LM
Σ
Σ
A Ku
(9)
vEN.(99)
L
Σ
Σ
AnArt
Σ Σ
(E). vEN
VEN.(9") VENU(G")
S
SLLL M
Σ
Σ
App- ... Ago
Σ
Σ
Aps A'u
(EN
NEN. (G)
PEN,(99)
HENa(0)
SLIMMC
Σ
AT
Σ (EBr(
AT pu
pu
pEBr(u)(S)
SLIMMC
Σ
AT
Σ
År.
R., R.
pu
(EBr(u)(
(EBr(W)(O))
AT
Σ
ATT R.
PEB, (w)(g)
pEBr(u)(O)
Eignjev
(V - 1 11
vo.
it L
-
Vág
+
|
29

---

Let denote the maximum eigenvalue of A. We can bound the last term in the above equation using the Ceurant--Fssche Theorem, which says for symmetric matrix B with maximum eigenvalue ? and any vector one has that x.' Br S xx x lr). Then, we have that
E E(g,yev
1/1
vit vo.
i+
1 Vgg
vd. 1y
1V1
Elanyev ( + du, Ei=l 7d
|V]
Eignyev d + d I-9
(as y E (-1,1)
M
2
dmin 1 y
Thus, whether + is the mean or sum operation, we can obtain the following inequality,
2 1
PPru(9) (r,( s L,L" MT L,L"
2|11+ Ru
dmin 1 -
C.5 Proof of Theorem 4.8
Proof. Let (e( = E dg(u, v) S S1. R. Diaviee Rg.. [Ca
(r. Pr,, (g') - (g) Pr..(g.] + (Pra(9.) (r))
If I is the sum operation, then,
(G.) Pr,)(')| < Li (G.) (G')
LL.LM
Σ
Anuh (G.) Σ
A A
(G.)
vENu(9")
SL,L?M²
Σ
Σ
AnvAn -G.)) - Σ Σ
(9.) (9.).
(EN+(9") vENu(9")
E.......
SL,L-M² MT
Σ
Api ... Agch)(g.)
Σ
Apj... A (C')
(G.)
(..).
PEN,(G")
SLITMC
Σ
1
Σ
A
pu
(EBr(u)(G.)
PEBr(u)(G")
SLLLMCC
Σ
AT
Σ
AT R. R.
pu
pEBr(u)(S.)
pEBr(u)(S")
AT
Σ
ATT + R
(EBT(u)())
PEBr(W)())
EigN)ev
1 N v2,
i+
1 1g
v
1
/
dg
30

---

If + is the mean operation, then,
(P.)(G.) - Pr.(())) S L1 (g.) - -
SL1LM
Avnhy (9.)) - Σ
A
(O.)
vEN..99)
SLILiM²
Σ
Σ
Amnhn (G.))
E Σ
(G,) vENu(G.)
S......
ShIT M
Σ
Σ (E.
Apj... Aio (g
- E
E
Kzs A Ao
PEN)(G.)
PENŞ(0)
vENu(9)
SLL'MCC
E
AT pu
Σ
Apu
pEBr(u)(S.)
pEBr(u)(9))
SLL'MC
Σ
AT
Σ
ATT Rg R.
(E
pEBr(u)(O))
AT
Σ
ATT + R.
PEBT (u)(
(w)
T
( Ly
1 Van
it
(11
1g
vd Jo
Viyy
Vdg
Let y denote the maximum eigenvalue of A. We can bound the last term the above equation using the Courant-Fischa Theorem, which says for a symmetric matrix B with maximum eigenvalue and any vector æ. one has that xx Br S xx' x In]. Then, we have that,
(vag11 - To. - î+
Eigorev
1 dg
1
1
Jd.
Elgneev d + + Eee Der'" 72
|
E(gn)ee + 1-
(as 7 E (-1,1)
[
2
dmin
Then,
Pr. Prr,(g)) LL 222 + R. + dmin -
1
Thus,
(Pr,(9)- Pra(g')) L L,'" M' 2/ + + 2 1
+
dmin
1 -
31

---

D Experiments
D.1 Datasets
Node classification datasets: Cora, Citeseer, and PubMed [ are citation networks. Each node represents paper with bag-of-words feature vector, connected by citation relationships. The task is to predict each paper's s research area. Link prediction datasets: BC-OTC and BC-Alpha are trust networks of Bitcoin users trading platform. UCI is social network of University of California, Irvine students, where links represent sent messages between users, Graph classification datasets: MUTAG 44 represents atom graph, with edges between bounding atoms. PROTEINS represents protein structures, where nodes are amino acids, and edges form if the distance between nodes is is
less than 6 Á apart. The graph label indicates whether the protein is an enzyme. movie collaboration dataset. Each graph corresponds an ego-network for each actor/actress, where nodes correspond to actors/actresses and an edges indicate two actors/actresses co-appearances in movies. The details of data are in Table g
Table 9; The details of datasets
Datasets
Nodes(Avg, Nodes)
Edges(Avg, Edges)
task
Cora
2.708
10,556
node classification
Citeseer
3,321
9,196
made classitication
PubMed
19,717
44,324
nade classification
BC-OTC
35,588
link prediction
BC-Alpha
3,777
24,173
link prediction
UCI
1,899
59,835
link prediction
MUTAG
17.93
39.6
graph classification
PROTEINS
145.6
graph classification
BIDB-BINARY
19.8
193.1
graph classification
D.2 GNN models
We trained two 2-layer GNNs, with element-wist sum and the other with mean as the aggregation function. For node classification, (,, is mapped to the class distribution through the softmax function. For the link prediction, we concatenate hG) and h,,(G) the input to linear layer to obtain the logits, which are then mapped to the probability of the existence of the edge (u, v). For the graph classification task, the average pooling of (G. across all nodes in G can produce single vector representation h(G) for classification. It is mapped to the class probability distribution through the softmax function. During training, we the learning rate 0.01, the dropout rate 0.5 and the hidden size to 16. The model is trained and then fixed during the explanation stages. Our experiments were done on a CPU with kernel size of 32GB.
D.3 Quantitative evaluation metries
For the night datasets, we randomly select nodes, edges, and graphs as target sets. sets. For each target, given sparsity level of 0.9, we apply our method to generate robust explanations. As baselines, we include random explanations (via randomly selected edges) and base explanations produced by six existing GNN explanation methods. We then create 300 perturbed graphs evaluate 5, for four types of explanations: base, random, Ricci cervaturr-based and effective resistance-based For each target, we compute the relative error and Fidelity K and report their averages over all target sets.
D.4 Explanation Methods
GNNExplainer learns edges masks by maximizing the mutual information to explain GNN pre- dictions. PGExplainer learns approximated discrete masks for edges to explain the predictions. GNN-LRP utilizes the LRP back-propagation attribution method to GNN, attributing class prob- ability to input neurons. DeepLIFT attributes the log-odd between two probabilities and uses summation function to obtain contributions of edges. FlowX applies the Shapley value to derive initial contributions of message flows, then trains these scores with loss functions and maps them to edges. Convex designs a convex objective function to approximate the KI. divergence and obtains the important edges by solving convex optimization.
32

---

D.5
The details of three large datasets
For the Pubmed, Coauthor- Coauthor-Computer and Coauthor- Prautsor-Physics datasets, the statistics are summarized in Table [0]
Table 10: Three large graph datasets
Datasets
Classes
Nodes
Edges
Edge/Node
Features
PubMed
3
19.717
44,324
2.24
500
Coautoo-Computte
13
18,333
327,576
17.87
6,805
Coauthor-Physies
2
34,493
991,848
28.76
8,415
D.6 The values across methods and datasets
When the + is the mean function, the selected values for Ricci curvature and effective resistance across methods and datasets reported in Tables 11 and 12 respectively. When is the
function, the corresponding values are shown in Tables 13 and
Table 11: Selected À values for robust explanations based on Ricci curvature. The + in GNN is mean operation.
Methods
Citeseer
Puhmed
BC-OTC BC-Alpha
MUTAG
PROTEINS
IMDB-BINARY
GNNExplainer
0.782
0.487
0.035
0.024
0.067
0.009
0.004
0.223
0.784
PGExplainer
0.879
Convex
0.748
0.982
0.652
0.095
0,510
0.798
0.499
0,813
0.397
DeepLIFT
GNN-LRP 0.291
0.182
0.261
0.001
0,143
0.816
0,481
0.749
0.225
FlowX
0.030
0.001
0.002
0.176
0.323
0.704
0.077
0.491
0.062
Table 12: Selected values for robust explanations based on effective resistance. + GNN is mean operation.
Methods
Cora
Citeseer
Pubmed
BC-OTC
BC-Alpha
UCI
MUTAG
PROTEINS
INDE-BINARY
GNNExplalner
0.964
0.925
0.05
0.035
0.6
0.889
PGExplainer
0.101
0.074
0.059
0.424
0.8
0.523
0.983
0.483
0.529
Convex
0.581
0,226
0.524
0.052
0.129
0.984
0.082
0,226
0.547
DeepLIFT
0.011
0.010
0.006
0.096
0,002
0.021
0.462
0.739
0,148
GNN-LRP
0.056
0.052
0.037
0.001
0.162
0.111
0.432
0.490
0.620
FlowX
0.091
0.072
0.013
0.011
0.062
0.112
0.387
0.040
0.200
D.7 Running time
We report the running time for computing Ricci curvature and effective resistance on the Pubmed, Seautho-CCoppute and Coauthor-Physte datasets (see Table[0] Results are shown in Figures and As illustrated Figure 3 For 15,000 edges, the computation of Ricci curvature very time fime-efiicient, taking only 3.5 seconds. In contrast, computing effective resistance is more time- consuming, taking approximately 60 seconds for the same number of edges. Nevertheless, the overall computation time remains acceptable for practical use.
D.8 Ablation analysis
According the Figures 6 and g
can that even if À changes, on these datasets, using Ricci curvature and effective resistance can achieve better robustness and fidelity.
33

---

Table 13: Selected values for robust explanations based on Ricci curvature. The + in GNN is sum operation.
Methods
Citeseer
Pubmed
BC-OTC
BC-Alpha
MUTAG
PROTEINS
IMDE-BINARY
GNNExpliier
0.017
0.035
0.048
0.080
0.017
0.022
0.654
0.001
0.147
PGExplainer
0.871
0.188
0.194
0.687
0.583
0.538
0.654
0.92
0.292
Convex
0.022
0.04
0.084
0.822
0.295
0.470
0.513
0,813
0.792
DeepLIFT
0.154
0.589
GNNLRP
0.049
0.042
0.032
0.001
0.357
0.001
0.708
0.022
0.086
FlowX
0.01
0.05
0.017
0.001
0.001
0.007
0.023
0.939
0.660
Table 14: Selected values for robust explanations based effective resistance. The + in GNN is
operation.
Methods
Cora
Citeseer
Pabmed
BC-OTC BC-Alpha
UCI
MUTAG
PROTEINS
IMDB-BINARY
GNNEsplainer
0,200
0.008
0.677
0.033
0.011
0.066
0.294
0.007
0.644
PGExplainer
0.069
0.005
0.149
0.001
0.003
0.001
0.982
0,262
0.005
Convex
0.35
0.795
DeepLIFT
0.002
0.002
0.025
0.072
0.029
0.028
0.785
0,032
0.650
GNN-LRP
0.009
0,142
0.037
0.012
0.086
0.442
0.859
0,050
0.990
FlowX
0.25
0.005
0.005
0.095
0,070
0.472
0.629
0.015
0.291
Pubmed
Physics
The sf
Thm number adger
The nurbe edgas
Figure 3: The running time of calculating the Ricci curvature of edges.
Pubmed
ComuthonComuuee
CoouthoePryscss
11/8
201
Figure 4: running time calculating effective resistance of edges.
34

---

PROT ENNEG
0.000
3.11
00015
0.05
-1.00
04000
Barra
0.005
01000
-2.0
aelected
Cons-PG Popparr
Chneeer- PGEEspaane
UCI-PGGNEEsee
Flaiit
Realey
Fiailyy
6:31
-0:31
1.90
5065
-301
1.00
The wabalii witer 3
Cora- Conves
DC-Algh Convec
MUTAAG-Comvw
3.005
6:30
2.000
3.00
6:34
5.733
Resarve
2.99
6:32
5065
031
5.00
0.3
04
055
36
37
3.5
1.5
08
3.
044
05
05
0T
3.0
LIFT
LCI-DeepLIFT
Deepp
Raldboo Oror
Reliho Dror
0.55
0:35
3.5
631
0:30
-0.01
177
0.20
Cora-GNi-LRP
BC-OTC-GWW-LL
MCD- NARV--NVV--NNNN
6-15
300
6:35
5.00
-110
-2.02
6:30
-2004
Litrudi-Pook
LCI-PlowX
0.22
0.005
6415
5.005
2.000
021e
0.08
-5 1330
5.05
608
5.00
+0 2330
0.15
The VileII 10000 o7/
Figure 5: The + in GNN is mean operation. Rows represent explanation methods, columns denote datasets, and the x-axis indicates the selected value of X. The y-axis denotes the difference in relative error or fidelity between the Ricci curvature based and original explanations.
35

---

Chesset-G GWWhaiie
SC-Agpa Che FAT
PNCTEINS-GNNEEEO biner
2313
3.022
302
1.00
3.020
30015
1004
30010
-3004
3000
1000
3000
-0.00
-00010
salided
The sdsclic volue
M. M.ITAG-PGE:NNN
0.24
BVD SXX--BNAEE-PPGTTIE.E
Fiaeity
0.55
0.00
Fiaily
3.14
0.64
210
010
Fine
409
001
0.0
-0000
0:02
-008
6004
0.4
08
675
02
2.4
The libiaiil
Cleesen-Corwwe
BC-OTG-Ccmmer
PROTE No-
Fdbcly
3.08
Pliudly
3.5
0.000
0.015
210
0400
-0005
0.500
0,75
2480
2405
5-433
CATS
0OX
-0082
0.2
0.04
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6.30
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6.66
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C.16
218 022
LIFE
Poliive
Fuliliio Dine
Talakio Droe
0212
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Easser-
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CC-OTC--A GXPNLLRPP
3.08
3112
2312
3.04
2.312
300
1.00
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The LMMOII
wine
Cora-PlowX
Putomed-Piook loxX
ML TAG--Powk
0.700
310
0.025
-301
-1000
0.22
The relicie min 2
The
The selecie 18/00 of
Figure 6: The + in GNN is mean operation. Rows represent explanation methods, columns denote datasets, and the x-axis indicates the selected value of X. The y-axis denotes the difference in relative error or fidelity between the effective resistance based and original explanations.
36

---

Chamset-G CWWhhaaner
MUTAG- EAENN FHtr
3.04
004
0.02
-3.00
-1.0
seedli vilu
sdlided
Cor-- PGCxptare
DC-Appra-P PCEsspaner
3.04
Realyy
3.03
0.04
3.00
0.02
0.00
-0.00
-0.4
-900
-100
-126
-134
0:30
The salillie
Cora-Cotwes
BC-OTC- Conve-c
UCI-Carver
3.02
Faccly
3.2
Pliddyy
0.8
0.03
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3.00
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0.5
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0.20
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140
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04
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67
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2302
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PROTEINS GANALUP
0.20
2312
3.04
302
1.00
20010
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1.000
00.
1
The selicde
Coru-Piow/
-C-OTG-PPow
ML. TLLTA--PowW
1225
Pak-bb
3.02
0.2
3.04
Relitin Ert
3.00
0112
-304
0.05
-308
308
3.55
0.500
The seledie P01M
The snfected
The selecie
Figure 7: The + in GNN is sum operation. Rows represent explanation methods, columns denote datasets, and the x-axis indicates the selected value of A. The y-axis denotes the difference in relative error or fidelity between the effective resistance based and original explanations.
37

---

Chamset-GGMM GoNooe--GNEEceeeee Exchaneer
BC-NNma-- noT
MNCTEING-GNNEESOEE baneer
0.50
0222
0313
6002
1.00
0004
0002
seledted mile
sileded ril
susiii value
Cor-- Porparaeeeee
DC-Appa- PCExppaner
BULTAG-PGEEEAAAE
0.15
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0.00
Fodeliy
2010
3.00
2012
015
3.00
300K
0.00
3000
3.11
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8:0
-00
0.3
043
455
642
2:4
5:00
The
Clessen-Corae
BC-OTC-Comee
UCI-Carvee
0.25
022
3.00
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0.520
0.21
011
0.05
3.00
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000
0450 2.05 0,50 3.55 0258 3,75 EB0C 9625 Ca5E
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6.5
3.00
Talulio Dror
204
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0.0
8.25
Cora-GWN-LRP
Clessn-Ghhi-LPP
Pubmas-- DNNALIPP
3.14
3022
0.05
3011
3000
0.00
-205
-026
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-204
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-200
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3011
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01060
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20.00
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neme wgw..ee.........................
10003
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0.75
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Figure 8: The ① in GNN is sum operation. Rows represent explanation methods, columns denote datasets, and the x-axis indicates the selected value of A. The y-axis denotes the difference in relative error or fidelity between the Ricci curvature based and original explanations.
38


Reviews:
[{"review_id": "kUJ152WlnI", "reviewer": "Reviewer_VzwU", "summary": "The paper “Robust Explanations of Graph Neural Networks via Graph Curvatures” tackles the problems of non-faithfulness explanation of retraining method and the contradiction and uncertainty explanations of ensemble method, by introducing Ricci curvature and effective resistance in graph theory as robust score to obtain more robust explanation. It presents theoretical results on the relation of the explanation robustness to Ricci curvature and effective resistance, and an objective combining Ricci curvature and effective as robust score is proposed to achieving the trade-off between fidelity and robustness. The experimental results on nine datasets demonstrates the effectiveness of the proposed method.", "questions": "Please refer to weaknesses", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "**Strengths**\n1. The motivation of using Ricci curvature and effective resistance as robust score is clear, and the theorems is interesting and novel\n\n2. The theoretical analysis of the relation of robust explanation to Ricci curvature and effective resistance gives valuable insight\nThe ablation experiment results are convincing\n\n**Weaknesses**\n1. Some presentations remain unclear, e.g., what is the size of the explanation sub-edges E_s, how the determine this size\n\n2. Is there any connection between Ricci curvature and effective resistance, can we combine both of the score in the final objective to further boost performance, it is better to provide such discussion\n\n2. The baseline only includes the basic GNN-explaining methods, it is better to evaluate the performance on methods like ensemble method and retraining method\n\n3. What does the “+Curvature” mean in the effective resistance based results on Table 3 and 4, is that a typo? And the base result (e.g. GNNExplainer) in Table 1 cannot align with Table 3, what are the differences here", "was_revised": true, "final_justification": "Since the authors address the clarity concerns by the answers of questions 1 and 4, I raise the clarity score to 4. However, I still think the relations and differences between Ricci curvature and effective resistance should be more discussed and investigated.", "rebuttal": "Thank you for taking the time to review our paper!\n>**Q1: what is the size of the explanation sub-edges E_s**\n\n**A1:** Thank you for the question. In our experiments, the size of the explanation subgraph E_s is set to 10% of the total number of edges in the input graph. This fixed ratio allows for fair comparison across different methods and ensures the explanation remains concise.\n\n>**Q2: Is there any connection between Ricci curvature and effective resistance, can we combine both of the score in the final objective to further boost performance?**\n\n**A2:** Thank you for the insightful suggestion. We conduct a statistical analysis on three datasets using both Spearman and Pearson correlation coefficients, and find no significant correlation between Ricci curvature and effective resistance. The results are shown  in the table.\n\nTable 1: The correlation coefficient and p-value\n| Datasets            | Pearson | p-value | Spearman | p-value |\n|---------------------|---------|---------|----------|---------|\n| PubMed              | -0.033  | 0       | -0.023   | 0       |\n| Coauthor-Computer   | 0.055   | 0       | 0.067    | 0       |\n| Coauthor-Physics    | 0.022   | 0       | 0.028    | 0       |\n\nIn the theory section, we analyze how Ricci curvature and effective resistance are individually related to model robustness and explanation robustness. However, we do not analyze their joint effect on robustness. Therefore, in our final objective, we treat these two metrics separately.\n\nWe think that combining both scores into a unified objective does not always lead to better performance. For example, in some cases, selecting edges A, B, and C based on effective resistance provides more robust explanations, while in the same cases, Ricci curvature favors edges D, E, and F. These conflicting preferences may reduce overall robustness. We acknowledge this as an interesting direction, and plan to explore it further in future work.\n\n>**Q3:The baseline only includes the basic GNN-explaining methods, it is better to evaluate the performance on methods like ensemble method and retraining method.**\n\n**A3:** Thank you for the helpful suggestion. Regarding the robustness of graph model explanations, few works use ensemble or retraining methods to improve explanation robustness. In other fields, such as computer vision, such methods have been used to improve robustness, but they are not commonly applied in the graph domain.  \n\nIn this work, our proposed method is both model-agnostic and explanation-method-agnostic. It identifies robust edges based on the geometric properties of the graph data. For this reason, we focus our evaluation on standard GNN explanation methods and show that our approach can enhance basic GNN-explaining methods robustness. We appreciate the suggestion and consider evaluating the performance on methods like ensemble method and retraining method for future work.\n\n>**Q4: What does the “+Curvature” mean in the effective resistance based results on Table 3 and 4, is that a typo? And the base result (e.g. GNNExplainer) in Table 1 cannot align with Table 3, what are the differences here?**\n\n**A4:** Thank you for pointing this out. The “+Curvature” in Tables 3 and 4 is indeed a typo. We apologize for the confusion and the poor reading experience it may have caused. Regarding the base results, although the same dataset and method are used in Table 1 and Table 3, the target nodes/edges/subgraphs to be explained are different. As a result, the reported results are not directly aligned."}, {"review_id": "hrB5olEdKJ", "reviewer": "Reviewer_nZXN", "summary": "This paper defines the GNN explanation robustness problem in a fidelity view, and firstly utilizes the graph curvature property to make theorical analysis and  build a robust explanation method. Under the defined metric, the proposed method is validated to be effective under different graph tasks and on various GNN explainers.", "questions": "**Q1.** Following **W2**, how Euclidean distance $|x-y|$ is replaced with a CDF difference which is built only on possibility without any distance information? And how the CDFs of $\\mu_{u}, \\mu_{v}$, are calculated as *Figure 3*? Please make clear clarification on this part. Since later proofs are all based on this simplified calculation, due to its importance, I could only give the rejection score first. If my confusion on its correctness is properly addressed, I will raise my score to **positive at least.** \n\n**Q2.** How the explanation subgraphs are chosen in original base explainers in *Section 6* and *D.4*? Are them selected using the same way in *Section 5*, or in a fixed amount (or fraction)?", "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**S1.** This paper firstly applies the graph curvature to analyze the GNN explanation robustness, which offers a new perspective to utilize a potential graph property to better understand GNN behavior in face of perturbation.\n\n**S2.** With detailed theorical proof on modeling the GNN, the proposed method could be applied to different graph classification tasks and various base explainers and all achieve good performance in acceptable time cost.\n\n**S3.** Abundant ablation studies on different experiment choices are provided in appendix, and implementation details are also provided. \n\n**Weaknesses**\n\n**W1.** The proof on *Lemma 4.2* is confusing (*C.1*): the first equation on $|x-y|$ introduced by **[1]** is defined on 2-D points with $\\mathbb{R}\\times\\mathbb{R}$ coordinates so $|x-y|$ could be used to express Euclidean distance; while in this paper the graph problem is defined under a topological form, so how such mapping--\"*from discrete to continuous*\"-- is done should be explained. However instead, later in *C.1*. the proof directly applies $\\mu_{u}$, $\\mu_{v}$ from *Eq. 3*, which (1) are defined on graph node set rather than $\\mathbb{R}\\times\\mathbb{R}$ so it’s unclear how could be used to replace on $F_X$ and $F_Y$; (2) seem not directly cumulable on a single $X$ axis as *Fig.3* since input space is not a continuous number. This leads to my further doubt on *Lemma 4.2*, which only have the node pair’s degrees as input without information from the specific structure of the graph. It seems that a denser graph (excluding edges between the node pair with their neighborhood) naturally leads to a smaller W1 distance, while this is not reflected under *Lemma 4.2*. \n\n**[1]** Why the 1-Wasserstein distance is the area between the two marginal CDFs.\n\n**W2.** The theoretical proof is not tightly related to the robustness definition. In the *Definition 4.1* the explanation robustness is defined as $\\mathbb{E} |Pr(G)-Pr(G’)|, \\mathcal{E}s \\subseteq \\mathcal{E}’$. Although in *Theorem 4.4* this expression is analyzed, after going through the proof in *C.2,3*, I found that the constraint $\\mathcal{E}{s} \\subseteq \\mathcal{E}’$ is not used. In other words, this theorem holds for any perturbed graph $G’$ with no relation to explanation aspect. Later it is also found that indeed serves as a preliminary proof for *Theorem 4.5*. Instead, the *Theorem 4.5* truly contributes to the explanation robustness, motivated by line 179 “*explanation subgraph should continue to reflect the model’s behavior under perturbations to the input graph*”, with the robustness expressed as $|Pr(G_{s})-Pr(G’)|$. Though such motivation also makes sense, it’s a pity that this analysis is not truly related to the defined problem itself. I suggest in the future version, the author could consider to refine the theoretical proof to directly solve the *Definition 4.1*, maybe with some proper approximation or relaxation, or simply change the definition and metric to $|Pr(G_{s})-Pr(G’)|$, which seems also reasonable. Therefore, the scientific reasoning process of the paper could be better.\n\n**W3.** The main performance contains few baselines, with only metric results on models with and without Curvature provided. If there’s similar defense baseline exist, it’s better to test and compare with them. If not, then I suggest the author could add some adversarial attack methods and test the performance after attack to further validate the practical effectiveness (at lease random graph noise as attack could be easily applied).", "was_revised": true, "final_justification": "Since (1) My main confusion is solved with the acknowledgement of the approximation taken in the Lemma 4.2 from the authors; (2) The experimental results on precise Ricci Curvature is additionally given, which makes the evaluation process and the underlying scientific reasoning sounder; (3) The gap between the original theoretical results and problem definition is bridged by a new analysis result, I think this paper has achieved the acceptance standard. Therefore I recommend to accept with a rating of 5.", "rebuttal": ">**Q1:The proof on Lemma 4.2 is confusing. The lemma 4.2 only has the node pair’s degrees as input without information from the specific structure of the graph.**\n\n**A1:** We thank the reviewer for carefully pointing out the concerns regarding Lemma 4.2. Lemma 4.2 was originally derived based on an approximate view, inspired by the formulation introduced in [1]. To adapt this approach for discrete graph topologies, we implicitly utilized node-degree distributions as a simplified approximation of the underlying graph structure. This approximation does not explicitly incorporate detailed graph distances. However, our results show that this method of selecting explanation subgraphs based on the approximately calculated Ricci curvature can also improve the robustness of the explanation.\n\nThe Lemma 4.2 is used in lines 507-508 and 513-514 of our proofs. Motivated by the reviewer’s valuable comment, we revised these proofs by using the exact Ricci curvature, updated Theorem 4.4 accordingly. When we use exact Ricci curvature, theorem 4.4 changes only in a **constant** term, and the conclusion remains **unchanged**. The theorem 4.4 become as \n\n**If ⊕ is the sum operation**,  \n$|\\Pr_u(\\mathcal{G}) - \\Pr_u(\\mathcal{G}')| \\leq \\eta \\left( (1 + \\beta')^{T-1} \\left(2 \\beta'(1 - \\alpha') + 4 \\beta' \\right) + \\sum_{i=2}^{T} 2 (\\beta')^i (1 + \\beta')^{T-i} \\right)$\n\n**If ⊕ is the mean operation**,  \n$|\\Pr_u(\\mathcal{G}) - \\Pr_u(\\mathcal{G}')| \\leq \\eta \\left( 2^{T-1} \\left(4 \\beta' + \\frac{3(1 - \\alpha')}{\\alpha'} \\right) + 4(T - 1)\\beta' \\right)$\n\nAdditionally, we have conducted new experiments based on this precise Ricci curvature. Our updated experiments also support our theoretical claims. Considering Ricci curvature can improve the robustness of the explanation\n\n**Table 1: Relative Error (↑) with exact Ricci Curvature (%). The ⊕ in GNN is mean operation. (Best values in bold)**\n\n| Method| Cora | Citeeseer | Pubmed | BC-OTC | BC-Alpha  | UCI  | MUTAG  | PROTEINS | IMDB-BINARY |\n|-|-|-|-|-|-|-|-|-|-|\n| GNNExplainer|1.1 |1.1 | 2.0  | 24.9  | 12.2 | 2.1  | 2.5  | 0.3  | 0.2  |\n| GNNExplainer+Curvature   |**4.8** | **6.8** | **5.5** | **27.9** | **13.5** | **5.3** | **1.3** | **0.5**  | **0.6**  |\n| PGExplainer | -0.6  | -0.2 |-0.7 |-0.5 |-1.7 |-1. |-0.3|-0.2| -0.9|\n| PGExplainer+Curvature | **2.4**| **2.4**  | **1.2** | **1.8** | **0.5** | **0.9**   | **0.9**   | **0.5**   | **0.3**  |\n| Convex | 23.4 | -3.6  | 4.6  | 14.0  | 2.3  | 9.5  |0.5| -4.3  | -0.5  |\n| Convex+Curvature  | **37.9** | **10.8** | **11.3** | **16.8** | **3.5** | **16.7** | **3.2** | **1.1** | **0.2** |\n| DeepLIFT   | -0.1  | 55.3 | 14.9 | -0.9   | -2.3  | 0.8  | -0.2  | 0.0  | -1.8  |\n| DeepLIFT+Curvature | **1.7** | **67.9** | **19.8** | **0.4** | **0.5** | **1.7** | **0.3** | **2.2** | **1.0** |\n| GNNLRP | 20.2 | 8.9  | 10.3 | 22.9  | 22.8 | -2.2  | -0.5  | -0.7  | 0.3  |\n| GNNLRP+Curvature  | **22.4** | **13.7** | **14.2** | **23.5** | **26.6** | **3.6** | **1.7** | **0.2** | **0.8** |\n| FlowX | 7.0  | 6.8  | 6.8  | 0.4   | -0.6  | -1.6  | 0.6  | -0.2  | 0.1  |\n| FlowX+Curvature | **7.6** | **7.9** | **8.1** | **1.7** | **0.2** | **0.7** | **1.1** | **0.3** | **1.3** |\n\n**Table 2: $Fidelity_{KL}$ (↓) with exact Ricci Curvature. The ⊕ in GNN is mean operation. (Best values in bold)**\n\n| Method                 | Cora | Citeeseer | Pubmed | BC-OTC | BC-Alpha  | UCI  | MUTAG  | PROTEINS | IMDB-BINARY |\n|-|-|-|-|-|-|-|-|-|-|\n| GNNExplainer    | 1.458| 1.361| 0.162| 0.176 | 0.099| 0.063| 0.153| 0.034| 0.241|\n| GNNExplainer+Curvature      | **1.393**| **1.372**| **0.149**| **0.175** |**0.098**| 0.063| **0.150**| **0.032**| **0.235**|\n| PGExplainer | **1.647**| 1.534| **0.581**| 0.209 | 0.110| 0.054| 0.156| 0.025| 0.249|\n| PGExplainer+Curvature  |1.664| **1.483**| 0.582| **0.206** | **0.109**| **0.053**| **0.146**| **0.024**| **0.243**|\n| Convex  | 0.086| 0.005| 0.002| 0.033 | 0.007| 0.001| **0.015**| 0.001| 0.011|\n| Convex+Curvature  | **0.029**| **0.002**| 0.002| **0.030** | **0.004**| 0.001| 0.019| 0.001| 0.011|\n| DeepLIFT | 1.012| 0.291| 0.270| 0.193 | 0.089| 0.050| 0.143| 0.040| 0.302|\n| DeepLIFT+Curvature | **0.992**| **0.260**| 0.270| **0.192** | 0.089| **0.049**| **0.126**| 0.040| **0.301**|\n| GNNLRP | 0.531| 0.633| 0.193| **0.091** | **0.022**| **0.011**| **0.162**| 0.034| 0.231|\n| GNNLRP+Curvature  | **0.470**| **0.529**| **0.176**| 0.104 | 0.023| 0.013| 0.173| 0.034| **0.229**|\n| FlowX  | 1.276| 1.164| 0.466| 0.190 | 0.081| 0.054| 0.161| 0.039| 0.253|\n| FlowX+Curvature  | **1.267**| 1.164| 0.466| 0.190 | 0.081| 0.054| **0.160**| **0.038**| **0.249**|\n\n**Table 3:Relative Error (↑) with exact Ricci Curvature (%). The ⊕ in GNN is sum operation. (Best values in bold)**\n\n|Method|Cora|Citeeseer|Pubmed|BC-OTC|BC-Alpha|UCI|MUTAG|PROTEINS|IMDB-BINARY|\n|---|---|---|---|---|---|---|---|---|---|\n|GNNExplainer|4.5|-6.0|0.5|54.4|70.1|73.3|2.2|4.0|-0.2|\n|GNNExplainer+Curvature|**16.0**|**2.4**|**2.6**|**55.2**|**70.5**|**74.7**|**5.1**|**5.1**|**1.5**|\n|PGExplainer|2.7|6.1|2.6|3.8|-0.3|-0.8|-2.8|0.2|-1.2|\n|PGExplainer+Curvature|**4.5**|**8.4**|**5.6**|**6.4**|**1.9**|**1.9**|**3.9**|**2.6**|**1.3**|\n|Convex|8.4|41.8|32.5|35.3|60.2|47.1|0.4|-4.3|8.1|\n|Convex+Curvature|**15.9**|**60.6**|**37.7**|**56.3**|**71.2**|**58.2**|**5.6**|**1.1**|**12.3**|\n|DeepLIFT|11.6|15.8|13.4|0.3|-2.1|11.1|-2.5|0.5|-0.5|\n|DeepLIFT+Curvature|**16.7**|**23.9**|**14.6**|**1.3**|**1.9**|**11.5**|**1.4**|**2.1**|**1.9**|\n|GNNLRP|16.4|-2.0|19.4|7.5|5.1|19.8|-1.1|0.0|-2.7|\n|GNNLRP+Curvature|**22.7**|**6.2**|**23.3**|**9.1**|**5.7**|**24.4**|**2.5**|**1.5**|**2.7**|\n|FlowX|2.7|30.6|8.6|18.3|3.5|7.0|-1.8|-3.1|0.5|\n|FlowX+Curvature|**7.8**|**38.3**|**17.5**|**19.3**|**4.3**|**11.9**|**1.3**|**1.6**|**1.8**|\n\n\n**Table 4: $Fidelity_{KL}$ (↓) with exact Ricci Curvature. The ⊕ in GNN is sum operation. (Best values in bold)**\n\n| Method               | Cora | Citeeseer | Pubmed | BC-OTC | BC-Alpha  | UCI  | MUTAG  | PROTEINS | IMDB-BINARY |\n|-|-|-|-|-|-|-|-|-|-|\n| GNNExplainer | 0.961 | 0.578 | **0.549** | 0.184 | 0.245 | 0.093 | 0.177 | 0.068  | 0.284 |\n| GNNExplainer+Curvature | **0.924** | **0.518** | 0.566 | **0.175** | **0.242** | 0.093 | **0.170** | **0.067**  | **0.282** |\n| PGExplainer  | 1.762| 1.341 | **0.715** | 0.537 | 0.447 | 0.400 | 0.227 |0.048| 0.351 |\n| PGExplainer+Curvature | **1.754** | **1.339** | 0.726 |0.537| 0.447 | **0.399** | **0.225** |0.048| **0.344**|\n| Convex  | 0.353 | 0.277 | 0.075 | 0.132 | 0.009 | 0.011 | 0.010 | 0.001  | **0.017** |\n| Convex+Curvature | **0.340** | **0.273** | **0.073** | **0.053** | **0.008** | **0.010** | **0.008** |0.001| 0.021 |\n| DeepLIFT  | 1.020 | 0.481 | **0.277** | 0.178 | 0.134 | 0.202 | 0.206 | 0.054  | **0.315** |\n| DeepLIFT+Curvature  | **0.956** | **0.473** | 0.304 | **0.177** | **0.133** |0.202| **0.180** |0.054 | 0.321 |\n| GNNLRP | 0.472 | 0.589 | 0.293 | 0.174 | 0.202 | 0.234 | **0.116** | **0.033**  | **0.303** |\n| GNNLRP+Curvature   | **0.442** | **0.572** | **0.276** | **0.173** | **0.201** | 0.234 | 0.153 | 0.036  | 0.321 |\n| FlowX | 0.495 | 0.499 | 0.370 | 0.184 | 0.086 | 0.244 | **0.174** | 0.051| **0.298** |\n| FlowX+Curvature  | **0.489** | **0.470** | **0.365** | **0.182** | **0.085** |0.244 | 0.171 | 0.051| 0.301 |\n\n[1] Why the 1-Wasserstein distance is the area between the two marginal CDFs.\n\n>**Q2:The theoretical proof is not tightly related to the robustness definition.**\n\n**A2:** Thank you for your valuable feedback. We agree that the original theoretical formulation may loosely connected to the formal definition of robustness. To address this, we have revised the theoretical analysis to ensure a tight alignment with the robustness definition. Let $r=1-\\frac{|\\mathcal{E}_s|}{|\\mathcal{E}|}$.\n\nFor theorem 4.5, \n\n**If ⊕ is the sum operation**,  \n$|\\Pr_u(\\mathcal{G}) - \\Pr_u(\\mathcal{G}')| \\leq \\eta \\left( (1 + \\beta_{s})^{T-1} \\left(2 \\beta_{s}(1 - \\alpha_{s}) + 4 \\beta_{s}+2 \\right) + \\sum_{i=2}^{T} 2r (\\beta_{s})^i (1 + \\beta_{s})^{T-i} \\right), s.t, \\mathcal{E}_s \\subseteq \\mathcal{E}'.$\n\n**If ⊕ is the mean operation**,  \n$|\\Pr_u(\\mathcal{G}) - \\Pr_u(\\mathcal{G}')| \\leq \\eta \\left( 2^{T-1} \\left(4 \\beta_{s} + \\frac{3(1 - \\alpha_{s})}{\\alpha_{s}} +2\\right) + 2(T - 1)\\beta_{s}+2r(T - 1)\\beta_{s} \\right), s.t, \\mathcal{E}_s \\subseteq \\mathcal{E}'.$\n\nFor theorem 4.9, \n\n$|\\Pr_u(\\mathcal{G}) - \\Pr_u(\\mathcal{G}')| \\leq \\eta \\left( 2+2 \\beta_{s}^{T} (1-r^T)+\\bar{R}_s +\\frac{2}{dmin(1-\\gamma)}\\right), s.t, \\mathcal{E}_s \\subseteq \\mathcal{E}'.$\n\n>**Q3: The author could add some adversarial attack methods and test the performance after attack.**\n\n**A3:** Our work focuses on the robustness of graph model explanations, not on defending against graph attacks. So we do not compare with defense-related baselines, and to our knowledge, no such defense baselines exist for robust explanations.\n\nHowever, we do evaluate the robustness of explanations under graph perturbations (attack). For a given explanation subgraph, we randomly add and remove 10% of the edges that are not part of the explanation to create a perturbed graph G′. We repeat this process 300 times and compute the robustness score as defined in Definition 4.1. To provide a baseline, we also randomly sample some edges as explanation subgraphs, and apply the same process to get their robustness scores. We then compare the relative error between explanation method and the random baseline. The results show that our method clearly improves explanation robustness.\n\n>**Q4: How the explanation subgraphs are chosen in original base explainers?**\n\n**A4:** To ensure a fair and consistent comparison across methods, we fixed the fraction of selected edges for all explainers. Specifically, both the original base explainers and our proposed method select 10% of the total edges as the explanation subgraph. For the original explainers, the top 10% of edges are chosen based on their importance scores F(u,v) in descending order."}, {"review_id": "1tpe6Wg4dA", "reviewer": "Reviewer_MPaM", "summary": "This paper addresses the enhancement of robustness in explainability for Graph Neural Networks (GNNs). Prior approaches to improving robustness, such as model retraining and ensemble-based explanation, have significant limitations: retraining alters the model itself. At the same time, ensemble methods often yield overly divergent explanations, leading to high uncertainty. In contrast, this paper proposes a novel explanation method that emphasizes robust structural properties by leveraging the graph's geometric characteristics. Specifically, it quantifies the robustness of node connectivity using Ollivier-Ricci curvature and effective resistance, and theoretically proves that these quantities upper-bound the robustness of model predictions and explanations. The authors incorporate these measures as robustness scores into the objective functions of various existing explanation methods. Through experiments on nine benchmark datasets, the results demonstrate that incorporating the proposed method improves fidelity in over 80% of cases and achieves up to a 10% gain in explanation robustness.", "questions": "1. Could you discuss the possibility of determining the trade-off parameter lambda between robustness and fidelity not globally for the entire graph, but locally for each explanatory target?\n\n2. Could you elaborate on the potential situations where the approximation used in the Ricci curvature calculation, based solely on node degrees, may violate the underlying assumptions of the approximation?", "limitations": "The paper does not appear to consider the possibility that the trade-off between robustness and fidelity may vary locally across different explanatory targets. Additionally, there seems to be no discussion on scenarios where the assumptions underlying the approximation of Ricci curvature, based solely on node degrees, may not hold.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\nThe paper provides a theoretical proof that Ollivier-Ricci curvature and effective resistance can upper-bound the robustness of both model predictions and explanations. Building upon this, the authors incorporate these measures as robustness scores into the objective functions of various explanation methods, enabling explanations with guaranteed robustness for GNNs. This constitutes a significant contribution to the field of GNN explainability. Furthermore, the paper empirically validates the effectiveness of the proposed method across nine diverse datasets and multiple explanation techniques.\n\nWeaknesses:\nWhile the trade-off between robustness and fidelity is managed via a tunable parameter lambda, an elegant and theoretically sound idea, the method does not account for the inherent diversity of graph data. For example, even within node classification tasks, there may be cases where robust edges serve as the primary rationale for predictions, and others where fragile edges are more predictive. Therefore, using a single global lambda to represent this trade-off across an entire graph may be overly simplistic. A more nuanced discussion on this point could greatly enhance the persuasiveness and practical applicability of the proposed method.\n\nAdditionally, the Ricci curvature approximation relies solely on node degrees, which could be problematic. For instance, it is entirely plausible that two high-degree nodes are connected via a single bridge-like edge, and in such cases, the approximation may fail. This simplification appears to generalize the known result that, in one-dimensional real-valued distributions, the Wasserstein distance can be represented as the area between CDFs. However, whether this approximation holds in high-dimensional and non-Euclidean graph spaces warrants more careful examination.", "was_revised": true, "final_justification": "My concern regarding the potential oversimplification of using a single global lambda for the entire graph, and the possibility of determining it locally for each explanatory target, is largely consistent with the authors’ perspective. The additional experiments addressing whether the proposed approximation holds in high-dimensional and non-Euclidean graph spaces have improved the reliability of the approach. Based on these considerations, I remain inclined toward recommending acceptance; however, I will keep my rating unchanged.", "rebuttal": "Thank you for taking the time to review our paper!\n>**Q1:While the trade-off between robustness and fidelity is managed via a tunable parameter lambda, an elegant and theoretically sound idea, the method does not account for the inherent diversity of graph data. For example, even within node classification tasks, there may be cases where robust edges serve as the primary rationale for predictions, and others where fragile edges are more predictive. Therefore, using a single global lambda to represent this trade-off across an entire graph may be overly simplistic.**\n\n**A1:** Thank you very much for raising this important question. We agree that the trade-off between fidelity and robustness can vary depending on the explanatory target. As the reviewer pointed out, in some cases, fragile edges may be essential for the model’s prediction, while robust edges may contribute less. In such cases, increasing lambda to favor robustness can reduce fidelity. However, even without using lambda, it may still be difficult to find a subgraph that is both robust and faithful using other methods. This may be due to the structure of the graph itself. \n\nOn the other hand, in cases where many candidate edges have similar importance scores, using curvature-based robustness can help choose more stable edges and improve robustness without lowering fidelity. \n\nOur current experiments focus on global performance across all the explanation targets. Our results show that, in most cases, robustness can be improved without a significant drop in fidelity. We acknowledge that a more detailed analysis of target-specific trade-offs would be valuable.\n\n>**Q2: Could you discuss the possibility of determining the trade-off parameter lambda between robustness and fidelity not globally for the entire graph, but locally for each explanatory target?**\n\n**A2:** If it is a single target node, I think a more effective method may be to use grid search to obtain lambda. Using the optuan package in Python can speed up the search process.\n\n>**Q3: Additionally, the Ricci curvature approximation relies solely on node degrees, which could be problematic. For instance, it is entirely plausible that two high-degree nodes are connected via a single bridge-like edge, and in such cases, the approximation may fail. However, whether this approximation holds in high-dimensional and non-Euclidean graph spaces warrants more careful examination.**\n\n**A3:** We thank the reviewer for this insightful comment. Indeed, as pointed out, our original approximation relies solely on node degrees, neglecting explicit distance information within the graph. This simplification can indeed fail in specific scenarios, such as the one raised by the reviewer, where two high-degree nodes are connected via a single bridge-like edge. However, our experimental results demonstrate the effectiveness of this approximation method.\n\nTo address this limitation and strengthen our results, we have further derived the theory using the exact Wasserstein distance, explicitly incorporating graph distances.  The theorem 4.4 become as \n\n**If ⊕ is the sum operation**,  \n$|\\Pr_u(\\mathcal{G}) - \\Pr_u(\\mathcal{G}')| \\leq \\eta \\left( (1 + \\beta')^{T-1} \\left(2 \\beta'(1 - \\alpha') + 4 \\beta' \\right) + \\sum_{i=2}^{T} 2 (\\beta')^i (1 + \\beta')^{T-i} \\right)$\n\n**If ⊕ is the mean operation**,  \n$|\\Pr_u(\\mathcal{G}) - \\Pr_u(\\mathcal{G}')| \\leq \\eta \\left( 2^{T-1} \\left(4 \\beta' + \\frac{3(1 - \\alpha')}{\\alpha'} \\right) + 4(T - 1)\\beta' \\right)$\n\nWhen using exact Ricci curvature, Theorem 4.4 changes only in a **constant** term, and the conclusion remains **unchanged**. Additionally, we have conducted new experiments based on this precise Ricci curvature. Our updated experiments also support our theoretical claims. Considering Ricci curvature can improve the robustness of the explanation\n\n**Table 1: Relative Error (↑) with exact Ricci Curvature (%). The ⊕ in GNN is mean operation. (Best values in bold)**\n\n| Method| Cora | Citeeseer | Pubmed | BC-OTC | BC-Alpha  | UCI  | MUTAG  | PROTEINS | IMDB-BINARY |\n|-|-|-|-|-|-|-|-|-|-|\n| GNNExplainer|1.1 | 1.1  | 2.0  | 24.9  | 12.2 | 2.1  | 2.5  | 0.3  | 0.2  |\n| GNNExplainer+Curvature   |**4.8** | **6.8** | **5.5** | **27.9** | **13.5** | **5.3** | **1.3** | **0.5**  | **0.6**  |\n| PGExplainer    | -0.6  | -0.2 |-0.7 |-0.5 |-1.7 |-1. |-0.3|-0.2| -0.9|\n| PGExplainer+Curvature              | **2.4**  | **2.4**  | **1.2** | **1.8** | **0.5** | **0.9**   | **0.9**   | **0.5**   | **0.3**  |\n| Convex                 | 23.4 | -3.6  | 4.6  | 14.0  | 2.3  | 9.5  |0.5| -4.3  | -0.5  |\n| Convex+Curvature             | **37.9** | **10.8** | **11.3** | **16.8** | **3.5** | **16.7** | **3.2** | **1.1** | **0.2** |\n| DeepLIFT               | -0.1  | 55.3 | 14.9 | -0.9   | -2.3  | 0.8  | -0.2  | 0.0  | -1.8  |\n| DeepLIFT+Curvature           | **1.7** | **67.9** | **19.8** | **0.4** | **0.5** | **1.7** | **0.3** | **2.2** | **1.0** |\n| GNNLRP                 | 20.2 | 8.9  | 10.3 | 22.9  | 22.8 | -2.2  | -0.5  | -0.7  | 0.3  |\n| GNNLRP+Curvature     | **22.4** | **13.7** | **14.2** | **23.5** | **26.6** | **3.6** | **1.7** | **0.2** | **0.8** |\n| FlowX                  | 7.0  | 6.8  | 6.8  | 0.4   | -0.6  | -1.6  | 0.6  | -0.2  | 0.1  |\n| FlowX+Curvature  | **7.6** | **7.9** | **8.1** | **1.7** | **0.2** | **0.7** | **1.1** | **0.3** | **1.3** |\n\n**Table 2: $Fidelity_{KL}$ (↓) with exact Ricci Curvature. The ⊕ in GNN is mean operation. (Best values in bold)**\n\n| Method | Cora | Citeeseer | Pubmed | BC-OTC | BC-Alpha  | UCI  | MUTAG  | PROTEINS | IMDB-BINARY |\n|-|-|-|-|-|-|-|-|-|-|\n| GNNExplainer    | 1.458| 1.361| 0.162| 0.176 | 0.099| 0.063| 0.153| 0.034| 0.241|\n| GNNExplainer+Curvature      | **1.393**| **1.372**| **0.149**| **0.175** |**0.098**| 0.063| **0.150**| **0.032**| **0.235**|\n| PGExplainer                  | **1.647**| 1.534| **0.581**| 0.209 | 0.110| 0.054| 0.156| 0.025| 0.249|\n| PGExplainer+Curvature             |1.664| **1.483**| 0.582| **0.206** | **0.109**| **0.053**| **0.146**| **0.024**| **0.243**|\n| Convex                 | 0.086| 0.005| 0.002| 0.033 | 0.007| 0.001| **0.015**| 0.001| 0.011|\n| Convex+Curvature            | **0.029**| **0.002**| 0.002| **0.030** | **0.004**| 0.001| 0.019| 0.001| 0.011|\n| DeepLIFT               | 1.012| 0.291| 0.270| 0.193 | 0.089| 0.050| 0.143| 0.040| 0.302|\n| DeepLIFT+Curvature          | **0.992**| **0.260**| 0.270| **0.192** | 0.089| **0.049**| **0.126**| 0.040| **0.301**|\n| GNNLRP                 | 0.531| 0.633| 0.193| **0.091** | **0.022**| **0.011**| **0.162**| 0.034| 0.231|\n| GNNLRP+Curvature            | **0.470**| **0.529**| **0.176**| 0.104 | 0.023| 0.013| 0.173| 0.034| **0.229**|\n| FlowX                  | 1.276| 1.164| 0.466| 0.190 | 0.081| 0.054| 0.161| 0.039| 0.253|\n| FlowX+Curvature            | **1.267**| 1.164| 0.466| 0.190 | 0.081| 0.054| **0.160**| **0.038**| **0.249**|\n\n**Table 3:Relative Error (↑) with exact Ricci Curvature (%). The ⊕ in GNN is sum operation. (Best values in bold)**\n\n| Method               | Cora | Citeeseer | Pubmed | BC-OTC | BC-Alpha  | UCI  | MUTAG  | PROTEINS | IMDB-BINARY |\n|-|-|-|-|-|-|-|-|-|-|\n| GNNExplainer         | 4.5   | -6.0  | 0.5    | 54.4   | 70.1  | 73.3  | 2.2    | 4.0  | -0.2 |\n| GNNExplainer+Curvature    | **16.0** | **2.4** | **2.6**  | **55.2**  | **70.5** | **74.7** | **5.1**   | **5.1** | **1.5** |\n| PGExplainer          | 2.7   | 6.1   | 2.6    | 3.8    | -0.3  | -0.8  | -2.8   | 0.2  | -1.2 |\n| PGExplainer+Curvature     | **4.5**  | **8.4**  | **5.6**   | **6.4**   | **1.9**  | **1.9**  | **3.9**   | **2.6** | **1.3** |\n| Convex               | 8.4   | 41.8  | 32.5   | 35.3   | 60.2  | 47.1  | 0.4    | -4.3 | 8.1  |\n| Convex+Curvature          | **15.9** | **60.6** | **37.7**  | **56.3**  | **71.2** | **58.2** | **5.6**   | **1.1** | **12.3** |\n| DeepLIFT             | 11.6  | 15.8  | 13.4   | 0.3    | -2.1  | 11.1  | -2.5   | 0.5  | -0.5 |\n| DeepLIFT+Curvature        | **16.7** | **23.9** | **14.6**  | **1.3**   | **1.9**  | **11.5** | **1.4**   | **2.1** | **1.9** |\n| GNNLRP               | 16.4  | -2.0  | 19.4   | 7.5    | 5.1   | 19.8  | -1.1   | 0.0  | -2.7 |\n| GNNLRP+Curvature          | **22.7** | **6.2**  | **23.3**  | **9.1**   | **5.7**  | **24.4** | **2.5**   | **1.5** | **2.7** |\n| FlowX    | 2.7   | 30.6  | 8.6    | 18.3   | 3.5   | 7.0   | -1.8   | -3.1 | 0.5  |\n| FlowX+Curvature | **7.8**  | **38.3** | **17.5**  | **19.3**  | **4.3**  | **11.9** | **1.3**   | **1.6** | **1.8** |\n\n**Table 4: $Fidelity_{KL}$ (↓) with exact Ricci Curvature. The ⊕ in GNN is sum operation. (Best values in bold)**\n\n| Method               | Cora | Citeeseer | Pubmed | BC-OTC | BC-Alpha  | UCI  | MUTAG  | PROTEINS | IMDB-BINARY |\n|-|-|-|-|-|-|-|-|-|-|\n| GNNExplainer | 0.961 | 0.578 | **0.549** | 0.184 | 0.245 | 0.093 | 0.177 | 0.068  | 0.284 |\n| GNNExplainer+Curvature | **0.924** | **0.518** | 0.566 | **0.175** | **0.242** | 0.093 | **0.170** | **0.067**  | **0.282** |\n| PGExplainer  | 1.762| 1.341 | **0.715** | 0.537 | 0.447 | 0.400 | 0.227 |0.048| 0.351 |\n| PGExplainer+Curvature | **1.754** | **1.339** | 0.726 |0.537| 0.447 | **0.399** | **0.225** |0.048| **0.344** |\n| Convex  | 0.353 | 0.277 | 0.075 | 0.132 | 0.009 | 0.011 | 0.010 | 0.001  | **0.017** |\n| Convex+Curvature | **0.340** | **0.273** | **0.073** | **0.053** | **0.008** | **0.010** | **0.008** |0.001| 0.021 |\n| DeepLIFT  | 1.020 | 0.481 | **0.277** | 0.178 | 0.134 | 0.202 | 0.206 | 0.054  | **0.315** |\n| DeepLIFT+Curvature  | **0.956** | **0.473** | 0.304 | **0.177** | **0.133** |0.202| **0.180** |0.054 | 0.321 |\n| GNNLRP | 0.472 | 0.589 | 0.293 | 0.174 | 0.202 | 0.234 | **0.116** | **0.033**  | **0.303** |\n| GNNLRP+Curvature   | **0.442** | **0.572** | **0.276** | **0.173** | **0.201** | 0.234 | 0.153 | 0.036  | 0.321 |\n| FlowX | 0.495 | 0.499 | 0.370 | 0.184 | 0.086 | 0.244 | **0.174** | 0.051| **0.298** |\n| FlowX+Curvature  | **0.489** | **0.470** | **0.365** | **0.182** | **0.085** |0.244 | 0.171 | 0.051| 0.301 |"}, {"review_id": "6SqdrKl2Up", "reviewer": "Reviewer_Eh2L", "summary": "This paper adds an auxiliary loss on the classic GNN explanation objective, enhancing the robustness of basic explanations. \nSpecifically, the authors adopt the two concepts of Ricci curvature and effective resistance in graph theory to derive the auxiliary loss respectively, and theoretically draw a connection to explanation robustness.\nExperiments show that the proposed loss function consistently improves the robustness of GNN explanations while sacrificing less in terms of fidelity.", "questions": "Please refer to the weaknesses and limitations for suggestions.", "limitations": "Though the theory is well-established, the application value of this work may be limited: If our ultimate goal is robustness, why don't we transform the robustness objective function into a tractable loss ?  For example, using reparameterization or directly measuring the distance of the multivariate distribution for Eq. (2).", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n- The paper provides a fresh perspective on robust GNN explanation, with sufficient theoretical analysis.\n- The method shows significant robustness gains across multiple tasks, improving various base explainers.\n- The paper is well-organized. The authors have done a lot of work.\n\n**Weaknesses:**\n- The experimental results lack significance analysis: How many times was the algorithm run? Is the improvement significant?", "was_revised": true, "final_justification": "Although the authors' empirical results are not strong enough, my main concern has been addressed. Considering that the authors' responses to other reviewers are comprehensive with additional results, I believe this paper meets the acceptance criteria.", "rebuttal": "Thank you for taking the time to review our paper!\n>**Q1: How many times was the algorithm run?**\n\n**A1**: We run the base explanation algorithm only once to get the edge importance scores. Then, we generate the final explanation subgraph using the importance scores, Ricci curvature, and effective resistance. For robustness evaluation, we sample 300 random perturbation subgraphs and compute the robustness metric using Equation (2). For other metrics like fidelity, the algorithm also runs only once.\n\n>**Q2: Is the improvement significant?**\n\n**A2**: We did not perform formal statistical significance testing in this version of the paper. In our experiments, while some improvements over baselines are consistent, they are not always statistically significant across all datasets. We will consider adding significance analysis in future work.\n\n>**Q3: why don't we transform the robustness objective function into a tractable loss ? For example, using reparameterization or directly measuring the distance of the multivariate distribution for Eq. (2).**\n\n**A3**: We agree that transforming Eq. (2) into an optimizable loss is an interesting idea. However, it is not straightforward to implement for several reasons. First, it is difficult to define how to generate and update perturbation subgraphs based on the explanation subgraph during training. Second, most existing explanation methods rely on continuous edge/node masks, while the final explanation is discrete. This mismatch can lead to information loss and suboptimal performance. Instead, our method directly selects robust and important edges based on geometric properties of the graph, improving the robustness of the explanation without requiring a new loss function."}]

Author rebuttal:

Meta-review: The paper introduces a novel approach to enhance GNN explanation robustness using graph geometric properties, specifically Ricci curvature and effective resistance.  All reviewers acknowledged the novelty of applying Ricci curvature and effective resistance to GNN explanation robustness, with theoretical proofs establishing upper bounds on robustness. The method was validated across 9 datasets spanning node classification, link prediction, and graph classification tasks.  The approach can be applied to various base explanation methods, demonstrating broad applicability.

Initially, two reviewers have concerns on Approximation in Lemma 4.2, Statistical significance , Global vs. local lambda parameter  and Tight connection to robustness definition. All reviewers who engaged post-rebuttal were satisfied with the responses, with nZXN explicitly raising their score and VzwU addressing all concerns.

While this is a solid technical contribution with novel theoretical insights, it represents an incremental advance in GNN explainability rather than a breakthrough result.  I tis good to present in poster.
Decision: Accept (poster)