Global Graph Counterfactual Explanation: A Subgraph Mapping Approach
Authors: Yinhan He
Wendy Zheng
Yaochen Zhu
Jing Ma
Saumitra Mishra
Natraj Raman
Ninghao Liu
Jundong Li
Venue: Accepted by TMLR ()

Abstract: Graph Neural Networks (GNNs) have been widely deployed in various real-world applications. However, most GNNs are black-box models that lack explanations. One strategy to explain GNNs is through counterfactual explanation, which aims to find minimum perturbations on input graphs that change the GNN predictions. Existing works on GNN counterfactual explanations primarily concentrate on the local-level perspective (i.e., generating counterfactuals for each individual graph), which suffers from information overload and lacks insights into the broader cross-graph relationships. To address such issues, we propose GlobalGCE, a novel global-level graph counterfactual explanation method. GlobalGCE aims to identify a collection of subgraph mapping rules as counterfactual explanations for the target GNN. According to these rules, substituting certain significant subgraphs with their counterfactual subgraphs will change the GNN prediction to the desired class for most graphs (i.e., maximum coverage). Methodologically, we design a significant subgraph generator and a counterfactual subgraph autoencoder in our GlobalGCE, where the subgraphs and the rules can be effectively generated. Extensive experiments demonstrate the superiority of our GlobalGCE compared to existing baselines. Our code can be found at 
\url{https://github.com/YinhanHe123/GlobalGCE}.

Published in Transactions on Machine Learning Research (03/2025)
Global Graph Counterfactual Explanation: A Subgraph Map- ping Approach
Yinhan He
nee Deer7ne@vinginiaeee
Deportment of Electrical and Computer Engineering University of Virginia
Wendy Zheng
ned9ef@@virginia. edu
Department Computer Science University of Virginia
Yaochen Zhu
uqp4 @virginna edu
Department of Electrical and Computer Engineering University of Virginia
Jing Ma
jing.ma maa.gaa5@@cc.ccoommm edu
Department of Computer and Data Sciences Case Western Reserve University
Saumitra Mishra JP Morgon Chase Co.
saumitra. sawmitr. wwwmmmmmmmiiamm@@@pporggo.omm
Natraj Raman JP Morgon Chase Co.
natraj. www.n@immonnaaaamnnmmmmmmmmmaaaa com
Ninghao Liu
ninghao. liu &ruga. edu
School Computing University of Georgia
Jundong Li
j16gk @virgimia edu
Department of Electrical and Computer Engineering University of Virginia
Reviewed OpenReview: https: openreview. net/ formf id= Kdx
Abstract
Graph Neural Networks (GNNs) have been widely deployed in various real-world applica- tions. However, most GNNs are black-box models that lack explanations. One strategy to explain GNNs through counterfactual explanation, which aims to find minimum per- turbations on input graphs that change the GNN predictions. Existing works on GNN counterfactual explanations primarily concentrate on the local-level perspective (i.e., gen- erating counterfactuals for each individual graph), which suffers from information overload and lacks insights into the broader cross-grayy graph relationships. To address such issues, propose GlobalGCE, novel global-level graph counterfactual explanation method. Global- GCE aims to identify a collection of subgraph mapping rules as counterfactual explanations for the target GNN. According to these rules, substituting certain significant subgraphs with their counterfactual subgraphs will change the GNN prediction to the desired class for most graphs (i.e., maximum coverage). Mothodologically, we design a significant subgraph generator and counterfactual subgraph autoencoder in our GlobalGCE, where the sub- graphs and the rules can be effectively generated. Extensive experiments demonstrate the

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identified subgraph, explain y
added edge, Y+ y
Factual explanation
Counterfactual explanation
(a)
(b)
An illustration of GFE and GCE. (a) GFE. The graph is classified in the desired class because it contains a house motif, highlighted in blue. (b) GCE Initially, the graph does not have a house motif and is classified as undesired. To minimally perturb the graph into the desired class, an edge (in yellow) added, creating a house motif. The modified graph with the added edge is the counterfactual of the original graph.
Figure 1:
superiority of our GlobalGCE compared to existing baselines. Our code can be found at pttps://ggthub..cm//mmmaa 1230oloaa10100000 GlobalGCE
1 Introduction
Graph neural networks (GNNs) have achieved great success in many real-world applications, such as drug discovery (Jiang ct al. 2021). financial analysis (Wang et al. 2021), and epidemic forecasting (Panagopoulos et al. 2021). For example, GNN is applied in graph classification tasks to predict whether a drug would possess the desired therapeutic efficacy (Jiang et al. 2021). The wide deployment of GNNs in high-stake applications demands the explanations of their predictions (Prado-Romero et al.) 2022). Typically, GNN explainers aim to answer the following two questions: (1) What are the components (c.g., subset of graph nodes, edges, node features, and edge feal that contribute the most to the GNN predictions? and (2) What is the smallest perturbation that can be applied to for the input graph to change its prediction result from an undesired class to the desired class by the GNN?
GNN explanations that focus on the first question are called graph factual explanation (GFE) Tan
ct al. 2022), while those that focus on the second question are called graph counterfactual explanations (GCE) (Abrate and Bonchi 2021 Bajaj et al.) 2021 Huang et al.) 2023 Lucic et al. 2022
jet al. 2022). In Fig. [1] we give an example of GFE and GCE for GNNs. For simplicity, we assume that the GNN to be explained classifies input graphs into the desired and undesired classes, In this example, a graph is classified as desired if it contains a house motif and undesired if it does not. In Fig. [ (a), the smallest graph component that contributes the most to the GNN prediction is the highlighted house motif, so it serves as the factual explanation of the GNN prediction results for the input graph. In Fig. ! (b), the original graph does not have a house motif, the graph is classified as undesired. To minimally perturb it into the desired class, edge (in yellow) is added to create house motif. Here, the modified graph (the original graph with the added yellow edge) is defined as the counterfactual explonation of the original graph (a.k.a. 'counterfactual graph' or 'counterfactual" in the existing literature (Prado-Romero et al. 2022
Currently, most GNN explainers only function at the local-level GCE, generating individual counterfactuals for each input graph. Despite their effectiveness, local-level GCEs face severe limitations. The most signif- icant one is their inability to produce representative and concise global counterfactual rules. Here, global counterfactual rules refer to high-level graph perturbation strategies that apply to the largest proportion of graphs in an input graph domain. For example, global counterfactual rule might suggest: " "Modifying the non-drug molecules in the TUD-AIDS [Riesen and Bunke, 2008)) dataset by converting ketones and ethers into aldehydes will result in the majority of these molecules being classified as AIDS drugs." Such high-level explanations could effectively summarize model decision-makkin patterns and guide potential in- tervention directions. However, local CEs generate counterfactuals for each graph, resulting in a considcrable volume of highly individualized explanations for real sal-world datasets (ranging from thousands to millions of graphs). Therefore, the explanations impractical for humans to identify the global counterfactual rules.
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In this paper, we aim to address the problem of generating global GCEs. Recently, Huang al. (2023) propose GCFFsplainer, the only known global GNN counterfactual explainer, which aims to find a small set of counterfuctual graphs, typically no more than 100 graphs, as global GCE for GNNs. However, finding a small set of counterfactual graphs as GCE has two notable limitations: (1) Lack of straightforward global perturbation guidance. The counterfactual graph set given by GCFExplainer is still not straightforward for humans to discover general perturbation rules (e.g., where to perturb, and how to perturb) to achieve desired predictions. These perturbation rules, however, are exactly the key answers for the pivotal "what changes can be made' question of counterfactual explanation. As shown in Fig. 2 GCFExplainer still needs additional human effort to derive such answers by mapping and comparing the original graphs and counterfactual graphs, while our method can directly extract global rules as perturbation guidance. (2) Redundant information for explanations. Although GCFExplainer defines global GNN explanations as a small set of counterfactuals, each counterfactual a whole graph that may contain redundant information irrelevant to classification. In real-world graphs, GNN predictions are often only strongly influenced by cerlain "significant subgraphs", such as functional groups in molecule graphs, so editing these significant subgraphs greatly impact the classification outcomes. Therefore, the perturbation rules should focus on modifying these subgraphs for global GCE to generate counterfactuals effectively.
To address the above issues, we propose GlobalGCE (Global Graph Counterfactual Explanation by subgraph mapping). Given a set of input graphs, we aim to counterfactually explain a trained GNN model with humaau-understanddbbl rules, each involving the map of factual significant subgraph to a counterfactual subgraph. Specifically, the rules are learned such that by replacing the identified significant subgraph with the counterfactual subgraph according to the mapping, the explainee GNN will change the predicted class of as many of the graphs (i.c., maximum coverage constraint) in the representative dataset that contains the subgraph to another class (i.e., counterfactual constraint) as possible. However, the optimization is intractable due to the combinatorial complexity of the possible significant subgraphs and their combinations. To solve it practically, we show that the optimization target function is monotonic submodular and relaxes the problem by a greedy approximation. Specifically, we propose a novel significant subgraph generator and a counterfactual subgraph variational autoencoder, where the subgraphs and the rules can be effectively generated. This type of explanation our method provides is notably more concise and intuitive than existing efforts. We summarize our contributions as follows:
Problem Formulation: We formulate new global GNN counterfactual explanation problem based on subgraph mappings. Compared to existing works, ours enables compact global insights.
Novel Evaluation Metric:
We design a novel evaluation metric termed comprehensibility to measure the compactness of the graph edits produced by a GCE method, thus informing the viability of human understanding the counterfactual gencration process.
Novel Framework: We introduce GlobalGCE, novel framework for global GNN counterfactual explanations providing GCE in form of CSMs. The framework is based carefully designed significant subgraph generator and a subgraph counterfactual autoencoder.
Extensive Experiments: We perform extensive experiments on five real-world graphs to evaluate our proposed method. The results show the effectiveness of our GlobalGCE compared to other state-of-the--rt baselines across multiple evaluation metries.
2 Problem Definition and Evaluation Metrics
In this section, we first define the coverage of a CSM set C= (9: + - ) for G. Then, we formulate the global-level GCE problem using the coverage of the CSMs. We also apply another metric termed proximity to measure the average graph distance between an input graph and its countorfactual acquired by C. Finally, we propose novel evaluation metric called comprehensiblityy which aims to quantify how casily human can understand the counterfactual generation process for a given original graph.
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Ground Truth
Global Counterfactual Explanations imput graphs
original graphs
counterfactuals
counterfactual patterns
GCFExplainer
Subgraph Mapping
our method
(b)
Figure 2: Comparison of global GCE provided by GCFExplainer (Huang et al. 2023 and our GlobalGCE. (a) displays ground-truth local GCEs for input graphs. In (b), the upper right shows a subset of counterfactuals (in this case, only one), while the lower right depicts subgraph mapping rule as the global GCE generated by our approach. The global counterfactusl rule involves changing subgraphs from the left box area the right. This rule is not observable from the SxFExilainer output but is evident in our approach that generates subgraph mapping.
's output
2.1 Global Counterfactual Explanation for GNNs
Definition 1. (Coverage of CSM Sel) Given a inpul graph dalasel G and CSM set C= {g: - gf oi y (L.))==1) are non-isomorphic), graph G E G is 'covered' by C if valid counterfactual Gef can be produced by applying C. The 'coverage' of C, denoted coverage(C), the proportion of those covered graphs G.
Here, a valid counterfactual G represents a graph that is classified in the desired class by explainee GNN. Besides, applying a CSM (g. + - on graph G means replacing gi in with 9," at least once. Applying CSM set C G be combination of applications of any CSMs in C G, including the same CSM applied to mulliple posilions of G and multiple CSMs applied simullaneously to G. However, if two significant subgraphs in two CSMs overlap their positions in G, then the two CSMs cannot be applied simultaneously in the same modification process.
Problem 1. (Global Counterfactual Explanation for GNNs) Given a GNN that clussifies inpul graphs to the undesired class 0, CSM budget 0 <, k < n, the goal is to find the CSM set C marimizing the coverage:
max coverage(C)
(1)
s.t. |C
We can observe that this problem is NP-hard (see proof in Appendix Therefore, solving the exact optimization problem can be challenging. However, it is straightforward to see that the coverage function coverage(C) is monotonic submodular (proof omitted for brevity), we can optimize coverage greedily while maintaining a (1 - 2) performance lower bound Nemhaaner et al. 1978).
2.2 Evaluation Metrics for Global-level GCE
As suggested by the Section 2.1 we may evaluate the quality of the CSM mapping set generated by global-level GCE method by its "coverage." Besides, to evaluate global counterfactual explainer, also introduce proximily
prox.(() Doeg. min d(G, G
(2)
where Gc is the subset of G that is covered by C and the d(G,G'f) is the Le norm of the two graphs' adjacency matrices and node/edge feature matrices. A smaller prox.(C) indicates fewer adjustments to produce counterfactuals, thus making C more desirable global-level GCE. Finally, we design comprehensibility
comp. [EEeee CC(GAG Gel - 0.9
(3)
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Both GCEs have the same proximity of 0.25
Graph Nodes
Graph Edges
Removed Nodes
Removed Edges
Original Input Graph
GCE with comp. 10.7 (b)
GCE with comp. 440.0
(a)
(c)
Figure 3: Illustration of the usefulness of the proposed "comprehensibiltyy" metric. (a) shows that original input graph. (b) and (c) are two of its valid counterfactuals, they have the same proximity 0.25, but different comprehensibility 10.7 and 440.0 respectively. (b) is apparantly more human--comprehensivve, which we can interpret as "removing house motif may alther the GNN prediction label for graph." However, the graph edits for (c) is too scatter that human cannot conclude any GCE rule from the comparision of original input graph (a) and its counterfactual (c).
where CC(.) represents the number of connected components, and GAG is the symmetric difference graph (SDG) between graphs G and Gef, The SDG consists of edges that appear in either G or G but not both. The metric first calculates the average number of connected components across all SDGs between each graph G in our set Ge and its corresponding Gof Since every SDG must have at least one connected component, we subtract 0.9 (rather than 1, to avoid division by zero) to account for this baseline. Finally, we take the inverse of this value so that higher metric values indicate better comprehenssiiility.
2.3 Utility of the Proposed Comprehensibility Metric
As the proposed "comprehensibiltty" metric is claimed one of the main contributions of our work, now specifically elaborate the intuition and usefulness of our proposed comprehensibility metric. This metric aims to mcasure the compactness of the graph edits during the GCE process. In fact, with a certain amount of graph edits (node/edge addition/ addition/ddeltion feature changes) allowed, one intuilively expects to achieve a valid counterfactual with those edits being as compact (i.e., graph edits being near-in- -eastin-distance as possible, because concentrated edits can be more casily packaged into a unifiod, human-compprhhensille GCE rulc. For example, in the Fig. 3 the graph (b), (c) are two valid counterfactuals (also called GCEs) of the original input graph (a) with the same proximity 0.25. However, graph (b) has much higher comprehensibility (440.0) than that of the graph (b), which is 10.7. We can observe that the computed comprehensibility values align with how human-coo comprehensiee the generated GCE is: for GCE (b), we can straightforwardly conclude that "removing a house motif from the original input graph may achieve its counterfactual: However, it rather challenging to observe any coman-comprchensible GCE rule from the comparison of input graph (a) and the counterfactual (c) since the graph edits are too scattered. A higher comprehensibility score indicates that graph edits in the counterfactual explanations are more continuous, resulting in more comprehensible GCE,
2.4 Adapting Metrics to Locel-level GCE for Fair Comparisons
Local-level GCE Yuuan et al. 2022) methods yield counterfactual graph for each individual input graph. Although they cannot provide CSM set as the counterfactual explanation, we argue that our adopted three metrics are also applicable to local-GCE methods and compare the efficacy of both local-level and global-level GCE methods fairly.
Here, we first introduce how the metrics are applied in evaluating performance of local-level GCE, then we justify the fair comparison between global and local GCE methods under the metrics. The "coverage" of
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Graph
Significatt Subgraph Gencrntor
Saburaph Counteffccuall Ausoencoder
Applicatioon Subgraph Mapping
@ Added Nodt @ Deleted Nole
Added Edge
Deleted Edge
Subgraphi Comntrer Cactiala
Subgraph
Prequent
Groody Subgruph Selection by MAR
:
Encuder
Mis-
Sampling
Decoder
Generation
:
O
Latent Reproentation
Sinilariy Lis
Proliction
Figure 4: A overview of our GlobalGCE framework. Initially, we identify significant subgraphs from g based on their diversity and frequency. Then, with counterfactual subgraph autoencoder, we generate CSMs allowing all types of graph editions. In the final step, the model selects the most representative CSMs according to their coverage.
local-level GCE methods can be computed as the proportion of the input graphs that their corresponding counterfactual generated by the GCE method is valid (i.e., the counterfactual being calssified as in the desired class by the GNN). For the "proximity" and "comprehensibiltty" defined in Eq. 2 and Eq. 3 we simply change the Ge to the subset of the input graph set whose corresponding counterfactual provided by the certain GCE method is valid. Specifically, the "proximity" of the local-level GCE is calculated as the average graph distance between the input graphs and their counterfactual graphs. Here, the average is taken only for the input graphs whose counterfactual is valid. The "comprehensibility measures how scaltered is the graph edits from the original imput graph and its corresponding counterfactual when it is valid.
The global and local GCE methods' performance can be fairly mcasured via the three metrics since the metrics do not directly involve the CSMs but only requires the input graphs and their corresponding coun- terfactual graphs. The only difference between the global-level and local-level GCE methods is that, the global-level counterfactual graphs are acquired by applying CSMs on the input graph, while the local-level graph counterfactuals are produced directly by the explainer model (typically a deep learning model) by feeding in the original imput graph. The difference affects the acquisition process of graph counterfactuals, but this process is prior to and thus independent from the computation of the metrics. Therefore, the three metrics can be applied to fairly compare the performance between global and local GCE.
3 The Proposed Framework: GlobalGCE
Given the number of required CSMs k global GNN counterfactual explainer aim to find (output) k- I-CS set such that applying those CSMs to the input graph datasel achieves the highest coverage. Achieving this goal requires that (1) the subgraphs in the CSMs set be frequent and diverse enough so that each appears in large number of different imput graphs; (2) When substituting a subgraph in an input graph with its corresponding counterfactual subgraph, there should be a high probability of altering the GNN's prediction for that graph. Based on these requirements, we design our GlobalGCE, with three modules (an overview of GlobalGCE is shown in Fig.
Generation of Significant Subgraphs. To address the first requirement, we identify frequent subgraphs in G. We select. K(K > k) of them greedily based on their appearance rate (AR), i.e., the proportion of the input graphs that contain the subgraph, ensuring its diversity.
Counterfactual Subgraph Autoencoder (CSA). Toward the second requirement, GlobalGCF) generates a counterfactual subgraph with our designed Conditional Graph Variational Autoencoder (GVAE). Our CSA is a lightweight framework allowing all forms of graph edits.
Greedy Summary of Subgraph Mappings. After the CSA generates candidate CSMs, we greedily select & CSMs from the CSM candidate set by their marginal coverage in the input graph dataset. They serve the global-level GCE of the GNN w.r.t. the input graph dataset.

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3.1 Generation of Significant Subgraphs
Based on Problem 1 we expect to identify a significant subgraph set S that is frequent, diverse, and highly influential in GNN predictions. 'Frequent' means each subgraph in S appears in most input graphs in g. 'Diverse' indicates that the subgraphs in S collectively cover as many different input graphs as possible. Highly influential' refers to the likelihood that modifying a significant subgraph will change the graph's classification outcome in the GNN. To satisfy the frequency requirement, we apply gSpan (Yan and Han 2002), a classic algorithm for frequent subgraph generation, to the input graph dataset 9 with a given minimum appearance rate T and generale a set of frequent subgraphs S. We greedily select set S from the frequent subgraphs by their appearance rate. The greedy selection guarantees the diversity of the significant subgraphs since, in each iteration, we pick the new subgraph that maximizes the number of input graphs containing it but does not contain any subgraph already selected in S.
The last requirement is to ensure that the significant subgraphs are highly influential to the GNN prediction. However, we claim that the requirement unnecessary in this module. This is because, during the summary stage, the subgraphs that are not influential to GNN prediction will have low coverage (see Definition [] in the input dataset. Thus, they will be filtered out during the greedy summarization process. In the next subscction, introduce the counterfactual subgraph autoencoder that provides the counterfactual subgraphs of those significant subgraphs.
3.2 Counterfactual Subgraph Autoencoder
With the identified significant subgraphs, we proceed to train the counterfactual subgraphs corresponding to those significant subgraphs utilizing a generative framework. Inspired by the application of variational autoencoder (VAE) on graphs (Simonovsky and Komodakis 2018), we design counterfactual subgraph autoencoder, composed of encoder component and decoder component, that enables fast counterfac- tual subgraph optimization and generalization in the input graph domain. The encoder function maps each significant subgraph onto multi-dimensionn Gaussian distribution in the latent space, denoted as W(Hg T * I), where fgg and 0g are the mean and variance of g's latent distribution. Then, we sample a latent representation 2g from this Gaussian distribution. By utilizing Σy, the decoder generates g's counter- factual subgraph gef. We optimize f by maximizing the log- log-likelihood that the application of the CSM {g- + } successfully generates the counterfactuals of the input graphs containing g.
In (y = EEGDD In P(G Iy"" G),
(4)
where Dy is the distribution of graphs in the input graphs distribution D where each graph contains the subgraph g, Gef = [G') ( n] U of is acquired by editing in G to got and y" is the desired graph class. We estimate Equ. 4 utilizing the Monte Carlo approximation Gol
1 Neeg In P(G ly", ,g. G), where G3 = {G e GIG > g). G is the imput graph dataset sampled from D. However, In P(G "o", G) is intractable since the posterior In P(g|G*f G) is necessary to compute P(G ,,,,,"".... G), but In ( G) is not accessible. Therefore, we utilize viable substitute optimization target, which is a lower bound of Equ. based on the Evidence Lower Bound (ELBO) (Kingma and Welling 2013):
1
Daea In P(G ,g. .g.G) G.)
[gl
(5)
1
2 |0,1
Bo(x,lawr) In Iaes, P(G (2gg" G.g) =I(((<//.."''))/(((>>/>
where QZ2 g, )²) refers to the approximation of the posterior distribution P(zglg.8) modeled by the en- coder, and KL(-Il.) means Kullback-Lcibler (KL) divergence. In Equ. 5 the first term in the RHS represents the probability of the generated graph to be a counterfactual of G conditioned on the latent representation 2g and the desired label y", However, due to the lack of ground- truth labels, the maximization of this term substituted with the loss function that encourages the generation of valid counterfactual subgraphs: -Eqld(g. (f) + a I(O(G))) "),, where d(-.)) measures the distance between g and gef, and I(-) is the counterfactual prediction loss which is the negative log-likelihood between the GNN prediction d(Gef) and
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the desired label y". The parameter controls the weight of the counterfactual prediction loss. Overall, we can write the loss function of our counterfactual subgraph autoencoder as:
L = g + a ((6(()))))))) + (L(Q(=g19.)))) x=:(((=))1
(6)
Encoder. In the encoder component, the input comprises of the node attribute matrix X, (one-hot representation of node types), edge attribute matrix Eg (one-hot representation of edge types), well as the graph adjacency matrix Ag of a subgraph the subgraph 9- The encoder, modeled as a GNN, is trained to model the conditional distribution Q(2119. (l)..vv) = (g)( (9)g We minimize the KL divergence between Q(zgl9. y*) and the adopted prior P(z y²) = N( .) to ensure the proximity of the prior and its approximation. The latent variable 2g sampled from the learned distribution Q(2glg. y")) with the reparameterization Kingma and Welling (013
The output is the latent representation 2g of
Decoder. In the decoder, the imput variables are the latent representation Zg of the subgraph g, all graphs in Dg, and target counterfactual label y" The output is formulated as the counterfactual subgraph -Ager, X get, Eger). To make our model optimizable, we adopt a probabilistic adjacency matrix, node attributes and edge attribute matrix, whose elements are continuously from the interval [0, 1]. We employ a similarity loss function defined as the graph distance
d(g. g =Ag Ages) + 3 xx(xx:xyx1) + bdy.dd(BB dg( Eget).
Here, da,ds dx and dE are 12 norm the two matrices' subtraction. After training our designed autoencoder, we discretize the adjacency, node attribute, and edge attribute matrices to generate the counterfactual subgraphs. We calculate the binary adjacency matrix Ayet by thresholding these probabilistic values, setting the corresponding value exceeds 0.5 and 0 otherwise. Analogously, one-hot node attribute matrix X ges and edge attribute matrix E yee are generated by assigning to the entry with the largest value and 0 to all other entries in each row.
entries
3.3 Greedy Summary of Subgraph Mappings
J Now, we have generated set of K CSMs candidates {g: gi Ji=i Given user-defined CSM budget k, we aim to select k (k < K) CSMs from those candidates to serve GCE for the largest proportion of G. Specifically, we greedily select k CSMs by the coverage function coverage(C). Since the coverage function is monotonic submodular the selection provides a (1 - 1/e) approximation. Here, we highlight three advantages of our GlobalGCE:
Providing Global Explanations. We offer global-level GCE for GNNs, which differs from most existing GNN explainers focusing on generating local explanations. We are also the first model generating counterfactual subgraph mappings as GCEs.
Identifying Significant Subgraphs. Compared with GCFExplainer, our model offers significant subgraphs and perturbation rules in the global-level graph counterfactual explanation, which is more informative and human-comppeeenssblle for global explanation.
Faciliting Domain Knowledge Integration. Tt is easy incorporate domain constraints to GlobalGCE. For example, domain experts may rule out the counterfactual subgraphs that are in- feasible to increase the model accuracy.
4 Experiments
In this section, evaluate GlobalGCE with extensive experiments five real-world datasets. Our ex- periments aim to answer the following research questions: Quantative Analysis: How does GlobalGCF perform w.r.t. the evaluation metrics compared with the state-of-tho-art baselines; How do different com- ponents in GlobalGCE contribule to the performance? Qualitative Analysis: How does GlobalGCE
see proof Theorem Appendix
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Table The performance of different methods of CFEs on graphs (mean-sstandard deviation over five repcated exccutions), where "cove.", "prox." and "comp." represent coverage, proximity and comprchensibil- ity, respectively. We also abbreviate the methods names by substituting "Explainer" with "Exp.". The best results are in bold, and the runner-up up results underlined.
GNNEXp. CF- GF-GNN CLEAR
RegExp. GCFExp.
GlobalGCE
cove.
0.00 n/a
0.00+0.00
0.35+t018
0.00 n/a 10.03+1.17
0.34 93.34+0.00
AIDS
prox.
n/a 15.67+2.34
16.10+0.03
comp.
n/a
n/a
1.40+0.23
n/a
0.18+0.01
1.64+t.00
cove. 14.32:00.61
23.47#1.61
0.17+0.08
8.6811.09
0.91
93.07=0.00
NCII
prox.
7.09++0.05
8.68++0114
14.28+00.00
11.54+0.82
4.21+0.33
15.59+0.03
comp. 0.07+0000
0.09+0.00
10.00:0.00
0.49:10.11
0.08
5.19:0.12
cove. 18.15=10.57
36.52:10.25
1.66:10.86
1.66-22.32
0.25
97.6810.00
Mutagenicity prox.
18.25+0.25
15.80+0.099
27.28+4.78
7.66+4.97
5.22+0.36
25.22++0.08
comp.
0.12
0.09:00.00
0.29
0.05:10.01
0.05+0000
1.11
0.00+10.00
0.00+10.00
20.47
0.00+10.00
22.7910.93
72.09-0.00
PROTEINS prox. n/a
n/a
15.6442.78
n/a
2.35:1.225
18.67
comp. n/a
n/a
2.62++0.48
n/a
0.50+0.04
10.00+0.00
cove. 26.51=0.000 51.81+0.00
14.21+11.00
18.55:1..05
9.88
98.8010.00
ENZYMES prox.
12.54=0.18
11.64:0.13 18.01
9.33::..2
3.1620.54
20.87+0.00
comp. 0.15++001
0.15+0.01
2.17+0027
0.27+L0.11
0.05++0.00
9.09+0.00
provide global insights from its global counterfactual explanation results? Please refer to the Appendix for supplementary experiment results.
4.1 Experiments Setup
Datasets. Our experiments utilize five real-world datascts (NCI1, Mutagenicity, AIDS, ENZYMES, PRO- TEINS) from TUDatasel (Morris et al. 2020),, where graphs represent chemical compounds with nodes as atoms and edges as bonds. They are labeled according to relevance to lung cancer, mutagenicity, HIV activity, and certain properties of proteins, respectively. See Appendix for details.
Baselines. We compare our GlobalGCE with the following stato-of-tho-art basclines. (1) GNNEx- plainer (Ying et al. 2019 is GFE model; we apply it for GCE with some modifications on the loss function, see Appendix for details. (2) CF-GNNExplainer (Lucic et al.) 2022) generates counterfactual ego graphs in node classification tasks. We adapt it for graph classification by taking the whole graph as input and optimizing the model using the graph label instead of the node labels. (3) CLEAR (Ma et al. 2022 a generative model that produces counterfactuals of all input graphs simultaneously while preserving causality. We adapt by removing the causality component for fair comparison. (4) RegExplainer (Zhang) ct al. 2023 generate subgraphs of input graphs as counterfactuals by graph mix-up approach and contrastive learning. (5) GCFExplainer (Huang et al. 2023) is the only global-Level GCE for GNNs. This model finds counterfactuals by perturbing the original graph based on vertex-reinforeed random walk. As justified in the Section 2.4 we may apply our introduced evaluation metrics directly on those local-level GCE baselines.
GNN Classifier. We train a two-layer graph attention network (GAT) (Velicković et al. 2017) for graphs with edge attributes (AIDS and Mutagenicity) as a ground-truth GNN classifier. For datasets without edge attributes, we substitute the GAT with a two-layer graph convolutional network (GCN) (Kipf and Welling 2016). Both GAT and GCN have an embedding dimension of 32, max pooling layer, and a fully connected layer for graph classification. The model is trained with SGD with learning rate of 1c-3 for 500 epochs. The train/validale test split is 50% 20% 25%. GNN accuracies are in Appendix D.1.1
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AIDS
ENZY. 150
PROT.
150
GlobalGCE-NA
GlobalGCE-NS
125
GlobalGCE
125
GlobalGCE
100-15490:9% +3..999%%%440%% +38.0%
+17.1%
Coverage (%)
100
100
93.3%
75
+17.4%
75
+55.0%
50
50
25
25
0
AIDS
ENZY.
PROT.
AIDS
ENZY.
PROT.
(a) GlobalGCE v.s. GlobalGCE-NA
(b) GlobalGCE v.s. GlobalGCE-NS
Figure 5: Effectiveness of the GlobalGCE's two components (shadowed bars represent the original Global- GCE framework). Here, ENZY. means ENZYMES, and PROT. mcans PROTEINS. We conduct experiments for GlobalGCE-NA and GlobalGCE-NS with counterfactual rules budget, k = 30, k = 10, and k = 20 for each dataset respectively. (a) presents the coverage comparison of GlobalGCE and GlobalGCE-NA; (b) presents the coverage comparison of GlobalGCE and GlobelGCE-NS.
Exploiner Settings. In the frequent subgraph generation, set the minimum and maximum number of nodes to three and twenty respectively. The minimum appearance rate for different datasets of the frequent subgraphs is shown in the Appendix. For the counterfactual subgraph autoencoder, we allow at most two CSMs to be applied on the same input graphs to avoid combinatorial complexity (if m CSMs are applicable to one input graph, (*) combinations need to be evaluated). We set the latent space dimension as 64 and the dropout rate for the autoencoder be 0.5. Please refer more hyperparameters in Appendix D.1.2
4.2 Quantitative Analysis
We compared GlobalGCE's coverage, proximity, and comprehensibility against state-of-the-art baselines. coverage
Results in Table show: (1) Coverage and Proximity. GlobalGCE achieves across all datasets, with significant leads in AIDS (93%) and NCI (76%). Our method achieves suboptimal results in proximity since generating large number of counterfactuals requires more deviation from the original graph. GlobalGCE is able to give higher coverage while maintaining relatively low proximity (2) Comprchensibility. GlobalGCE outperforms all basclines by editing only one or two self-connected signifi- cant subgraphs, resulting in concentrated edits. In contrast, Explainr,,, the only other global graph-level GCE method, uses a random restart mechanism (i.e., randomly reselect node/edge edit) that produces scattered edits, loading to lower comprchensibility.
Additionally, we design two variants of GlobalGCE to examine the effectiveness of its each component. GlobalGCE-NS and GlobalGCE-NA. For GlobalGCE-NS modifies the significant subgraph generator to obtain the least significant subgraphs. GlobalGCE-NA removes the counterfactual subgraph autoencoder, and instead randomly perturbs the input graph. Fig. 3 shows that removal of any of the two components lead to significant performance degradation with the effects being the most obvious in AIDS. Next, we conduct parameter study to test our GlobalGCE's performance under different settings of hyperparameters (i.e., CSM budgets h and learning rate 1). Specifically, vary & from k {5. 10, 15, 20, 25, 30) and the learning rate from 1 E {0.0001, 0.001, 0.01, 0.015, 0.02, 0.025]. According to Fig, 6 observe from (a) that higher CSM budgets load to better coverage and comprchensibility since they enable more varied graph edits and increase the likelihood of valid counterfactuals: and our GlobalGCE demonstrrates robust. performance in terms of both coverage and comprehensibility across different learning rates. Lastly, we compare the average experiment time, displayed in Table 2 At first glance, GlobalGCI does not have the best performance across all datasets. However, GNNExplainer, ---NNxlpaarerrr,, and CLEAR are local explanation methods, which means that they only need to perform individual graphs. Thus, is more reasonable to compare our GlobalGCF.
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Coverage
Comprehensibility
1.0
2.0
1.0
2.0
0.9
1.8
0.9
1.8
Comprehensiiilly
Comprehemsibility
0.8
Coverage
1.6
0.8
Coverage
1.6
0.7
1.4
0.7
1.4
0.6
1.2
0.6
1.2
0.5
1.0
000.00
1.0
10
20
30
0.01
0.02
CSM budget (k)
Learning Rate (b)
(a)
Figure 6: Parameter Sensitivity Analysis. We conduct experiments on GlobalGCE test its coverage
(in green) and comprehensibility (in purple) performance under different parameter settings the AIDS dataset. (a) presents the GlobalGCE's performance with different numbers of counterfactual rules k. (b)
presents GlobalGCE's performance with different / under k = 10.
(a) ground truth
(b) GCE by GlobalGCE
(c) GCE by GCFExplainer
Figure Case Study on A Synthetic Dataset. (a) shows the ground-truth GCE rule. (b) the GCE in form of CSM generated by GlobalGCE. (c) displays the GCE in form of high-coverage counterfactual graphs produced by Explaae,,,,r, the only known gloabl-level GCE. We can observe that GlobalGCE successfully recovers the ground-truth GCE rule with minor noise (the diagonal lines). However, one cannot uncover the ground-truth GCE from GCFExplainer due to its high noise and structual complexity.
Table 2: Average wall-clock time in seconds experiment (i.e. generate counterfactual explanations) for each method-dataset combination.
GNNExp. CF-GNNEXp.
CLEAR
RegExp.
GCFFxp.
GlobalGCE
AIDS
248.38
529.39
10.15
1751.03
497.68
243.20
NCT1
53.64
91.69
3.92
331.54
160.79
181.77
Mutagenicity 114.54
205.53
300.84
839.33
4964.66
1386.20
PROTEINS
58.93
51.12
82.80
988.73
108.42
297.83
ENZYMES
9.23
20.55
3.93
57.65
254.21
130.11
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R
R
H
H
Br
O N
c
C
=
S
C
C
O
H
Na
C
R
R
(a)
(b)
(c)
(d)
Figure 8: Case Study on Real-world Dataset. We perform GlobalGCE on AIDS and Mutagenicity with the number of rules budget k = 10. We show two generated CSMs: {(a) + (b)} for AIDS and {(c) + (d)} for Mutgenicity. Both CSMs reveal some chemical insights. "R" represents the rest of the compound.
to the only other global baseline GCFExplainer. As seen in Table 2 the running time for our method is comparable to that of GCFExplainer, achicving better results on AIDS, Mutagenicity, and ENZYMES.
4.3 Qualitative Analysis
In this subsection, show how our GlobalGCE provides easily understandable global-level GCE both synthetic dataset and real-world dataset First, we craft a synthetic dataset with one thousand graphs where each graph possesses nodes randomly from four to fifteen. In the graph dataset, one half of the graphs are labeled undesired (i.c., the graph does not contain any cycle), the other half are labeled desired (i.c., the graph holds at least one rectangle). The graph construction design indicates that the ground-truth GCE rule adding one edge to enclose four-node line achieve a rectangle, shown in Fig. (a). We conduct our GlobalGCE on the crafted dataset and found that it successfully recovers the ground-truth GCE rule (displayed in Fig. 7 (b)) with minor noise (the two diagnal lines). Meanwhile, perform GCFExplainer, the only known global-level GCE method, on the dataset and demonstrate two top-coverage counterfactuals found by it in Fig. 7 (c). It is extremely challenging for one to extract the ground-trute GCE rule from Fig. 7 (c), proving that our GlobalGCE provides more complete global insight than GCFExplainer.
Taking AIDS and Mutagenicity (Morris et al. 2020 datasets as examples, our GlobalGCE identifies global counterfactual rules that increase the likelihood of non-AIDS drugs/non-mutagen becoming AIDS drugs/mutagens, respectively. Fig. [ (demonstratee two representative cases (a) (b) and (c) + (d). The effectiveness of these transformations stems from several key chemical modifications: (1) Introduction of Reactive Elements: The addition of sulfur (=S) and bromine enhances molecular interactions, since sul- fur provides antiviral properties (De Clercq and Li 2016 and bromine forms reactive intermediates with DNA (āgg et al. 1994). (2) Electronic Configuration Changes: Both cases involve modifications that alter electron distributions, the sodium alkoxide group (-ONa) improves bioavailability (Srajuudiin (2007) while triple bonds create regions of high electron density for nucleophilic interactions (Tornde
2002
(3) Enhanced Molecular Interactions: Adding nitrogen (=NH) enables hydrogen bonding with target
proteins Meanwell 2011 similar to the modified electrons interact with genetic material (DeMarini
2004
5 Related Work
5.1
GNN Counterfactual Explanation
There are few studies related to GNN counterfactual explanations (Prado-Roment el al. 2023 Ying et al. 2019 Bajaj et al. 2021 Lucic et al. 2022. Ma et al. 2022 Tan et al.) 2022 Huang et al. 2023). GNNExplainer (Ying al. 2019) aims to find the counterfactual by maximizing the mutual information between the GNN's prediction and distribution of possible subgraph structures. However, it is not robust to input noise. To address this problem, RGCExplainer (Bajaj et al. 2021) generates robust counterfactuals
Hyperparameters adjusted for best visualization
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by removing edges such that the remaining graph is just out of the decision boundary. The explanation is robust. because the decision boundary is in GNN's last-layer feature space, where the features are naturally robust under perturbations. Similarly, CF-GNNExplainer (Lucic et al. 2022) and CF2 (Tan et al. 2022) also generate counterfactuals by removing edges. In those methods, some generated counterfactuals may violate causality, Ma et al. (2022) propose a generative model, named CLEAR, to generate causally feasible counterfactuals. Besides, there are also some methods particularly designed for a certain domain, such as biomedical and chemistry (Abrate and Bonchi, 2021 Numeroso and Bacciu 2021 Wu et al., 2021p. However, understand. Recently, Huang et al. (2023) propose the first global-level GCE model, GCFFxplainer, which formulates global GCE as finding a small set of representative graph counterfactuals. Nonetheless, GCF ExpExaaaieer is lack of straightforward global perturbation guidance and redundant information for explanations.
those models are all local-level GCE models, which fail to provide GCEs that humans
5.2
Global Counterfactual Explainer on i.i.d. Datasets
Global counterfactual explanation is attracting more attention (Rawal and Lakkaraju 2020 Ley et al. 2022b a
Becker et al., 2021). Among them, AReS formulates the global counterfactual explanation problem as finding a set of triples = ((g (, (.))) Here, each (gi, c)) represents a counterfactual explanation rule: under the condition q:, we can change an undesired sample to be classified as desired by changing the sample feature C to 4. They find the counterfactual explanation rules by simultinuously optimizing their defined rules' correctness, coverage, and cost. Despite being insightful, ARcS is computationally expensive because AReS selects counterfactual explanation rules from the set of all combinations of conditions (æs) and sample features (cs). To tackle this problem, Ley et al. (2022b) propose a fast rule evaluator as a preprocessor of ARes to filter out those less-likely counterfactual rulcs. Since using triples to represent global counterfactuals does not fit contimuous sample features, GLOBE-CE (Ley et al. 2023) formulate the global counterfactual explanation as translation of the sample features with fixed direction and varied magnitude. Specifically, for all the samples {x GLOBE-CE learns a normalized vector 8. The counterfactual of the samples are fx; + 6k; 1 where {k are scalars computed by GLOBE-CE that varios among the samples. However, those models are tailored for i.i.d. datasets and non-trivial to adapt to graph dalasets.
6 Conclusion
We study an important problem of global-level counterfactual explanations for GNNs. Specifically, we aim to find set of CSMs global-level GCE to achieve better global insights and to find the significant subgraphs that contribute the most to the GNN prediction. To address this problem, propose GlobalGCE, which is composed of a significant subgraph generator, counterfactual subgraph autoencoder, and a greedy summarization module for CSMs, To evaluate the generated counterfactuals, we design novel and reliable comprchensibility metric. Extensive experiments validato the superior performance of our GlobalGCE.
Acknowledgments
This work is supported part by the National Science Foundation under grants (IIS-2006844, ITS- ITS-2223769, CNS-2154962, BCS-2228534, and CMMI-2411248) and the JP Morgan Chase Faculty Research Award.
References
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A Proofs
Theorem 1 The Problem 1 is NP-hard.
Proof. We aim to transform the problem as maximum coverage problem, which is NP hard, thus proving that the Problem is NP-hard. We assume that the node number of input graphs and their possible counterfactuals are upper limited by L. Then the set of all possible CSMs can be constructed as follows. For the sigmificant subgraph set {oi of CSMs, we build it with all subgraphs with all variations subfeatures of the nodes and edges. Meanwhile, the counterfactual subgraph set {0, }a is composed of all possible graphs with sizes smaller than or equal to L. Therefore, the CSM set is the set of all graph pairs where one graph is from {oi) and the other from {9, } for every CSM. As graph sets {gi}; and {g ) have finite sizes, the CSM set is also finite, denoted {CSM:) |_- Furthermore, we may write the set of input graphs that is covered by CSM, Sg. Now, the Problem L is transformed to find subset
{S such that [S] s k and the total number of the covered graphs is maximized. This is exactly the definition of the maximum coverage problem.
The log-likelihood of applying {o + g successfully generate the counterfactual of original input graph G can be lower estimated with
Theorem
Eoes, In P( 1g", .g.G) G)
[gl
(8)
E ((( In Ilaee,, P(G [zg."" o G, -KI0@k///...""///pt/////..
where the notations are consistent with those in Section
Proof. We derive the evidence lower bound of the optimization target follows. For certain frequent subgraph 9.
Eesg Pl G)
-Inllasg u 1Y,,, G).
= -Inllos9 P(G U 2g 2g .Y',, G)dz,
= in / Q(Eglg, Y')l tv
P(G \. U Ugef 2 (,GG)))) dāg
Q(zgl9) Y*)
(9)
P([G \g) U Jef Zg (y,,, dzg
Q(=glg, V*)In
Q(2glg, Y*)
P[G s) U 2g Q( Y+)
Y+ ",G.g) dāg
Q12glg,Y' Y") Y')Egog In
Q(zy1g, Y Eg P(GG U 9ot Zgg Y Yr,Ggg) Y G.9) dzg.
Q(zgg, Y.)
For any input graph G satisfying G 3 g, we have P(zy Y "Y""G,g) = (Y g) due to the assumption that G does not effect the choice gef- Hence, the Equ. can be expressed as
/ Q2 ("Es In P U G..g)- g In Q(2g|Y*, ,9" P(z, Y*, )dz, Eqq In P Ug kz.. Y G. - KL Y Y')l Y
(10)
Since we do not have accurate ground-truth subgraph counterfactual gef, the first term can be estimated
End(g, 9es) a [[[(G)) \ g) U ges), Y*).
(11)
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Therefore, the loss function is concluded as
= ge1) + a = KL((()))))
(12)
Theorem The coverage(C) is monotonic submodular.
Proof. We will prove that coverage C is both monotonic and submodular. By definiton, a set function f 2 + R is monotonic if AA C B S f(A( s f(B), submodular if VA c B c V and e e V. f(AUe) - f(A) 2 (Bue) (BB).
Next, we prove each properly for coverage C: We first. prove for Monotonicity. Let AC B be two sets of CSMs. For any graph G E G that is covered by A, G must also be covered by B since B contains all CSMs in A. Therefore
coverage(A) KGG G|G is covered by AH HG G|G is covered by B}] = coverage(B).
(13)
9
[
Then we prove for Submodularity. Let A S B be two sets of CSMs and == (g - ged be a new CSM not in B. We need to prove coverage(A U {ee) coverage(A) 2 coverage( U (et) coverage(B). We define GA as the set of graphs covered by A, GB as the set of graphs covered by B; and G. as the set of graphs that can be covered by applying CSM e. Then, the marginal gain when adding e to A is coverage(A A U {e}) coverage(A) G.MGal Similarly for B, coverage(B U (e}) coverage(B) G.AGGl 19
191
Since A B, we have GA C GB, which implies Ge \ AB S Ge GA. Therefore |G. GBl S |G, \ GAl Dividing both sides by [9], we have GGAGn
[GAAAAl This proves that coverage( (BunugeeBBBee) (B) coverage S [9] S [9]
coverage (AU fet) - coverage(A), Thus, coverage(C) is submodular. As we have proven both monotonicity and submodularity, we can conclude that coverage(C)) is monotonic submodular.
B Limitation
While our proposed GlobalGCE framework demonstrates significant improvements in generating global coun- terfactual explanations for GNNs, there are several limitations that warrant further investigation. First, the computational complexity of our approach increases with the size and number of input graphs, which may limit scalability to extremely large graph datasets. Additionally, our method currently focuses on binary classification tasks, and extending it to multi-class or regression problems requires careful consideration. The subgraph mapping rules generated by GlobalGCE, while interpretable, may not. capture all nuances of the GNN's decision-malee making process, especially for highly complex graph structures. Furthermore, the effective- ness of approach relies the quality of the identified significant subgraphs, which may not always align perfectly with the most influential substructures for the GNN's predictions.
C Time Complexity Analysis
Subgraph search. We utilized gSpan Yan and Han (2002). a mature algorithm for frequent subgraph generation. The time complexity of gSpan is xl v!), where & is the average size of subgraphs explored (which is significantly limited by the minimum appearance rate r), n is the mumber of graphs in the dataset, and is the average number of vertices in each graph. This complexity arises from the algorithm's exploration of the DFS Code Tree, where cach node requires subgraph isomorphism testing (contributing the v! factor), and the depth of exploration is controlled by both the graph size and the minimum appearance rale т. The n factor comes from needing to check each candidate subgraph against all graphs in the dataset. A higher 7 roduces io by pruning more branches of the search space, as only frequent subgraphs meeting the threshold are explored. Therefore, although the time complexity can be theoretically high, we may still conduct the gSpan with adjustable when the graph size scale to be very large. Specifically, if gSpan does not return
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result after a determined time period, we increase the minimum appearance rate, the running time will decrease syyyoomilllyysuper-ppyymmmmm
(refer to Fig 5 of Yan and Han (2002)).
Counterfactual generation. The time complexity for training the subgraph counterfactual VAE can be analyzed per training iteration. For each significant subgraph 9, we process all graphs containing it (denoted as [Ggl) in batches of size For each graph a batch, we perform VAE operations on the subgraph (O(nš) for n-node subgraphs), subgraph matching to find insertion locations (O(Nn) for N-node original graphs), insert the counterfactual subgraph (O(n2)), and run forward/ /aaaar//bbwwwarddd backward passes through the GNN (O(N3)). With training iterations, the total time complexity is Oii* g /bb (nă +N NN)). In general, shown in Table 2, our method consumes significantly less time than the only existing global-level graph counterfactual explanation work GCFExplainer.
D Reproducibility
In this soction, we provide more details of model implementation and experiment setup of our evaluation results.
D.1 Details of the Model Implementation
D.1.1 GNN the Prediction Model
We train a two-layer graph attention network (GAT) for graphs with edge attributes (AIDS and Mutagenic- ity) as a ground-truth GNN classifier. For those datasets without edge attributes, we substitute the GAT with a two-layer graph convolutional network (GCN). Both the GAT and GCN have embodding dimension 32, a max pooling layer, and a fully connected layer for graph classification. The model is trained with the PyTorch Adam optimizer using dataset-specfic hyperparameters shown in Table
Table 3: GNN Prediction Model hyperparameter values for cach dataset.
NCI1
AIDS
Mutagenicity PROTEINS ENZYMES
train epochs
100
300
100
400
400
learning rate
0.1
0.01
0.01
0.0001
0.001
weight decay 0.001
0.01
0.001
0.0001
0.01
The Beinn/validate/test split is 50%/25%/25%
The accuracy of the GNNs is shown in Table.
Table 4: Graph classification accuracy (%) of the GNN classifiers.
NCI1
Mutagenicity
AIDS
PROTEINS
ENZYMES
Training
91.32
97.62
99.89
91.00
99.00
Validation
95.92
67.24
99.78
64.00
72.00
Testing 68.40
72.84
99.57
62.00
64.00
D.1.2 Details of GlobalGCE
Our GlobalGCE comprises three stages: generation of significant subgraphs, counterfactual subgraph au- toencoder, and greedy summary of CSMs.
Table 5: Minimum appearance rate of the frequent subrgaphs in the input graph dataset. Here, min. AR means minimum appearance rate.
NC11
AIDS
Mutagenicity
PROTEINS ENZYMES
Min. AR 64.10%
20.44%
92.59%
28.57%
10.20%
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Published in Transactions on Machine Learning Research (03/2025)
Generation of Significant Subgraphs. We first find frequent subgraphs, where set the minimum and maximum number of nodes of frequent subgraphs to three and twenty, respectively. In addition, the minimum appearance rate for different datasets of frequent subgraphs (i... a subgraph is defined as frequent subrgaph if it appears in more than T proportion of the input graphs) shown Table 5 which is determined by trials to ensure that the generated number of frequent subgraphs can achieve the predefined budget K.
Counterfactual Subgraph Autoencoder. We allow at most two CSMs to be applied on the same imput graphs. We set the latent space dimension as 64 and the dropout rate to 0.5. We train the model with dataset-specific hyperparameters, which is listed in Table 0
Omission of Greedy Sclection by Appcarance Rate. According to Section 3 the significant subgraphs are greedily selected from the frequent subgraphs with the appearance rate function; however, extensive empirical evaluations suggest that the frequent subgraphs themselves are diverse enough, and the greedy selection does not gain higher performance for our GlobalGCE. Therefore, in our experiments, we omit the greedy selection and identify significant subgraphs as those subgraphs with k highest appearance rates.
Graph Distance Calculation. We approximate graph edit distance (GED) Gao et al. 2010 with
d(g. (( d.. Agr) + 3 (Xg: ggr) + +y de dE(Eg. Eger).
(14)
Here, dA = 0 dx and dg are calculated with (2 pairwise distance. As the computation of d is time-consuming, we simplify this value as cross-entropy with logits in the training of the counterfactual autoencoder. In all other situations, such as the evaluation of GlobalGCE and all
baselines, adopt the definition with the dot product. We set the 10, = 7 = 1 to emphasize the counterfactual's structural change.
Table 6: GlobalGCE hyperparameter values for each dataset.
NC11
AIDS
Mutagenicity PROTEINS
ENZYMES
train epochs
100
300
100
100
100
topk
20
20
learning rate
0.01
0.01
0.1
0.01
0.01
D.2 Details of Experiment Setup
D.2.1 Baseline Settings
For each graph, GNNExplainer (Ying et al., 2019 outputs an edge mask that estimates the importance of different edges in model prediction. We adapt this model to counter- factual generation taking the prediction loss term as the negative likelihood of the masked graph a being classified as desired by GNN rather than undesired. We remove edges with edge mask weights smaller than the threshold 0.5. The perturbation on node features cannot be designed as straight- forwardly as the perturbation on the graph structure; thus, we do not perturb graph node features in GNNExplainer
GNNExplainer:
CF-GNNExplainer: CF-GNNExplainer (Lucic et al. 2022) is originally proposed for node classi- fication lasks, and it only focuses on the perturbations on the graph structure. Originally, for each explainee node, it takes its neighborhood subgraph as input. To apply on graph classification tasks, the whole graph the neighborhood subgraph and assign the graph label the label for all nodes in the graph. We set the number of iterations to generate counterfactuals as 500.
CLEAR: CLEAR (Ma et al. 2022) is a generative model that produces counterfactuals of all the input graphs simultancously while preserving causality. This model generates both a graph adjacency matrix and a graph node feature matrix perturbation, which allows all forms of graph edition
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Published in Transactions on Machine Learning Research (03/2025)
such node/edge additionn/delltionnffaauu perturbation. We adapt it by removing the causality component for fair comparison. We train the model with 500 epochs, other hyperparameters are in default according to Ma et al. (2022).
Explaiii:::: RegExplainer (Zhagg et al. 2023 explains graph regression models with informa- tion bottleneck theory to help solve the distribution shift problem. Although the original model is tailored for graph regression models, it can be directly applied to the graph classification tasks. We implement the RegExplainer with GNNFxplainer as the base explainer model in Algorithm of (Zhang et al. 2023) and train the model for 500 epochs. Other hyperparameters are tuned for the best perlormance in each dataset.
GCFExplainer (Huang et al. 2023 is the only global-level GCE for GNN utilizing vertex reinforcement random walk. "Vertex-reinforced" means that the transition probability at each step of the walk is inversely proportional the frequency of visits neighboring nodes. We apply this model for each dataset with the maximum steps as 1000. Other hyperparameters are in default according to Huang al. 20233
GCFExplainer:
D.3 Experiment Settings
All the experiments are conducted in the following environment:
Python==3.9
Pytorch==1.11.0
torch-goometri--221100
torch=scatter==200999
forch-sparse==0.66111
Scipy=11.9.3
Networkx--3.0
Numpy=11.23.4
Cuda==11.6
gSpan-mining=:022
D.3.1 Datasets
We utilize five real-world datasets from TUDataset Morris el al. 2020 for the meta-data, see Table
Table 7: Metadata of the adopted real-world datasets. Here, Muta. means Mutagenicity. Av. Nodes and Av. Edges are the average number of graph nodes and edges, N.C. means the number of node class
Dataset.
Graphs
Av. Nodes
Av. Edges
N.C.
NCII
4,110
29.87
32.30
10
Mutagenicity
4,308
30.32
30.77
10
AIDS
1,837
15.69
16.20
9
PROTEINS
1,113
39.06
72.82
ENZYMES 600
32.63
62.14
2
The NCI1 dataset is a collection chemical compounds screened for their ability to inhibit the growth of panel of human tumor cell lines. Each compound represented as graph, where vertices are atoms, and edges represent chemical bonds. This dataset widely used in graph kernel methods for machine learning, especially for graph classification tasks.
NCI1:
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Published in Transactions on Machine Learning Research (03/2025)
Mutagenicity: The Mutagenicity dataset comprises chemical compounds labeled according to their mutagenic effect on bacterium. Similar to NCH1 each compound is graphically represented, focusing on the atom and bond structure. It serves as key resource for studying graph machine- learning methods.
This dataset contains chemical compounds that are potential anti-HIV agents. Each graph represents compound's molecular structure, capluring the connectivity and properties of atoms. The AIDS dataset is often used in graph-based machine learning for tasks such as compound clas- sification and pattern recognition.
AIDS:
ENZYMES: ENZYMES is set of protein tertiary structures. Each protein structure is represented as graph, where nodes are amino acids, and edges connect neighboring amino acids. This dataset is particularly useful for research in graph-bases protein function prediction and structural analysis.
PROTEINS:
The PROTEINS dataset consists of protein structures, represented graphs. Nodes in these graphs represent secondary structure elements, connected by edges if they are neighbors in the amino acid sequence 3D space. This dataset facilitates the study of graph-based methods in understanding protein structure-functt relationships.
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Reviews:
[{"review_id": "RelkXvsaGY", "reviewer": "Reviewer_aPzD", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\nThe paper's motivation is clear and the subgraph mapping-based counterfactual generation method is a novel contribution to GNN interpretability.\n\nThe use of a variational autoencoder for counterfactual subgraph generation is well-justified and efficiently optimized. And the comprehensibility metric is a valuable addition that enhances the interpretability of counterfactual explanations.\n\n**Weaknesses**\n\n- While the proposed method is scalable compared to local GCEs, the subgraph search and counterfactual generation remain computationally expensive. I would like to see more insights here. \n\n- The method assumes a fixed set of counterfactual subgraph mappings, which may not generalize well to highly heterogeneous graphs.\n\n- The method is evaluated on binary graph classification tasks and may not generalize to multi-class classification or regression problems.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper presents GlobalGCE, a novel method for global counterfactual explanation (GCE) of GNNs. The main contribution is the identification of subgraph mapping rules that provide high-coverage, interpretable counterfactual explanations, distinguishing it from existing local-level GCE methods. Also, the authors propose comprehensibility, a metric that quantifies the human interpretability of counterfactual graph edits.\n\nOverall, I find the work interesting and valuable, with contributions to GNN interpretability.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Overall, I find the work interesting. Here are some suggested revisions:\n\n- Clarify the introduction, especially the discussion on GCFExplainer (\"In this paper, we aim to...\"), to make the motivation and limitations clearer upfront. I find it hard to follow without reading the following sections. \n- Provide a detailed computational complexity analysis of each component (subgraph generation, autoencoder training, greedy summarization) to assess scalability.\n- Discuss the generalization of the method to multi-class classification and regression tasks.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "1m5nf8Y7sr", "reviewer": "Reviewer_TSmf", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The paper addresses a significant and timely problem within the domain of GNN explainability.\n- The newly proposed comprehensibility metric is intuitive and effectively complements standard interpretability metrics such as proximity.\n- The experiment results seem reasonable.\n\nWeakness:\n- Stating that “existing works on GNN counterfactual explanations primarily concentrate on the local-level perspective” is not entirely precise. Several model-level (i.e., global) explanation approaches already exist—e.g., XGNN [1], GNNBoundary [2], GDM [3], D4-Explainer [4], and Graphon-Explainer [5]. Direct comparison with these methods would help validate the effectiveness of the proposed approach.\n- Additional datasets and baselines from these aforementioned works would strengthen the empirical evaluation and bolster claims about method generalizability.\n- The idea of identifying common, influential motifs is not entirely new. Relevant efforts such as [6-7] also aim to discover representative motifs to explain model behavior. A comparison or discussion regarding these works would contextualize the contributions more convincingly.\n- Although Figure 3 is illustrative, the assertion that comprehensibility is unequivocally “better” than other metrics (e.g., proximity) demands more rigorous justification, such as theoretical analysis or more systematic empirical motivations. In many real-world scenarios, there is no readily available “ground-truth motif” to benchmark comprehensibility. The quality of the set $\\mathcal{G}$ can strongly influence comprehensibility scores, and the approach may perform differently on larger, more complex graphs.\n\n[1] XGNN: Towards Model-Level Explanations of Graph Neural Network. KDD'20\n\n[2] GNNBoundary: Towards explaining graph neural networks through the lens of decision boundaries. ICLR'24 \n\n[3] Globally interpretable graph learning via distribution matching. WWW'24\n\n[4] D4Explainer: In-Distribution GNN Explanations via Discrete Denoising Diffusion. NeurIPS'23\n\n[5] Graphon-Explainer: Generating Model-Level Explanations for Graph Neural Networks using Graphons. TMLR'24\n\n[6] MAGE: Model-Level Graph Neural Networks Explanations via Motif-based Graph Generation. ICLR'25\n\n[7] Interpretable Sparsification of Brain Graphs: Better Practices and Effective Designs for Graph Neural Networks. KDD'23", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper tackles the challenge of generating global-level counterfactual explanations for Graph Neural Networks (GNNs). Rather than producing separate edits for each individual graph, the authors propose a small set of subgraph-mapping rules that, when applied, flip many graphs’ predictions to a desired label. The framework—called GlobalGCE—operates in three stages: (1) it identifies significant subgraphs that appear frequently across the dataset, (2) uses a lightweight autoencoder to learn how to replace each significant subgraph with a counterfactual subgraph, and (3) greedily selects the best subgraph-mapping rules to maximize coverage. The paper also introduces a new comprehensibility metric. Experiments on multiple datasets demonstrate that GlobalGCE can flip a larger proportion of graphs’ labels than prior methods, often with more compact edits and clearer global insights.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Aforementioned in weakness", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "0PkHWkQJsA", "reviewer": "Reviewer_yrGS", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength\n\nThe author proposes the novel task of global graph counterfactual explanation, evaluation metrics for the proposed task, and effective method to tackle the proposed global graph counterfactual explanation task. The paper has good novelty.\n\nThe proposed evaluation metrics and proposed prediction methods are well-discussed and seem suitable for the proposed task.\n\nThe proposed method demonstrates good performance with respect to the proposed metric.\n\nThe author conducts several ablation studies and visualizations.\n\n\nWeakness\n\nAs for the proposed metric, proximity and comprehensibility indirectly depend on the coverage since they are calculated only on the covered samples. Therefore, it may be difficult to compare the methods with different coverage scores using these scores.\n\nIn the bottom of page 3, the author discusses the case where we apply the subgraph transition multiple times. I find this case interesting. However, it seems the proposed model training method and evaluation does not care this case. In fact, how many test case exist where multiple application of the transition is required.\n\nAccording to 2.4, is the coverage of existing local method same as a sort of success rate ignoring k? If so, I would like to see the reason of much worse performance in Table 1.\nFurther, can we compare the methods with existing metric?\n\nIn the equation (2), the definition of the distance d is not explained. Since there is an ambiguity, e.g. graph edit distance, number of transition rule in C applied, distance proposed in the equation (7). It would be clearer to add explanation just before or after the equation (2).\n\nIn section 3.2, the author says the ‘lack of ground-truth labels’. Is that indicate y*? I would like the clarification of the problem setting i.e. what sort of information can we access during the training of subgraph extraction rule. Further, the more discussion about justification of the replace of this term with equation (6) would be helpful for the clarification.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a novel problem of global graph counterfactual explanation that is to extract a set of subgraph transformation rules that can make the most of the input graph to be classified to the target class.\n\nFor this task, the author proposes the novel three metrics: coverage calculates haw many input graphs can be transformed to the desired class by the rules. Proximity calculates the magnitude of the change of the graph. Comprehensibility calculates the number of connected subgraphs required to add or remove in the graph transition process.\n\nAlso, the author proposes a novel method for the global counterfactual extraction consists of three modules. First extracts the candidate subgraphs from input graphs considering frequency, diversity, and influence. Then trains the encoder-decoder model to transform the input subgraph to the different subgraph such that the transformed graph is classified to the desired category. Finally filters the candidate subgraph transformation rules with respect to the coverage.\n\nThe author experimentally demonstrates that the proposed method demonstrates better performance than the existing local counterfactual explanation methods in the proposed three metrics. Furthermore, the author conducts ablation studies and visualizations to demonstrates the effectiveness of the proposed method.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Overall, my evaluation is positive at this stage even without the suggestion below.\n\nAdditional discussion about the proposed metric would clarify the importance of each metric.\n\nDiscussion about multiple transition rules case are just from my interest.\n\nDiscussion about evaluation of existing methods would help clearing the fairness of the comparison.\n\nThe explanation about distance in equation (2), problem setting and about ground-truth labels in section 3.2. would help improve the clarification.", "broader_impact_concerns": "The paper does not have broader impact section. \n\nThe paper has algorithmic limitation section in the appendix.\n\nI think the potential negative impact would be small, since the contribution is the extension of the existing task.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "WmbHBRLa7s", "reviewer": "Reviewer_u3YN", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n1. The paper is well-organized in general. The authors define a new problem, introduce a new evaluation metric, and then propose their own approach to solve the problem.\n2. It’s good that the proposed measures are not only applicable for global GCE, but also for local GCE for a fair comparison with existing methods.\n3. The authors performed diverse experiments, and the proposed approach shows good empirical performance compared with recent models.\n\nWeaknesses\n1. I can understand the meaning of comprehensibility thanks to Section 2.3, but the mathematical intuition behind Eq. (3) is hard to capture. I suggest improving the representation and explanation of Eq. (3) in Section 2.2.\n2. Eq. (7) seems a little too naive. Since we know that X and E are one-hot vectors, there should be a better way to measure the difference, e.g., KL divergence, or the likelihood of X_g (or E_g) given their counterfactuals.\n3. It would be nice to include a proof on why coverage(C) is monotonic submodular in Appendix, since it is a core part of this work.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "1. This work introduces a new problem, global GNN counterfactual explanation based on subgraph mappings.\n2. The authors design a new evaluation metric termed comprehensibility to measure the compactness of graph edits produced by a GCE method.\n3. The authors introduce GlobalGCE, a new framework for global GNN counterfactual explanations providing GCE in form of CSMs.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Please refer to the weaknesses above, and address them as much as you can.\n2. In Eq. (2), it seems wrong that $C$ is in the subscript of the min function.\n3. The comprehensibility of graphs in Fig 3 is presented differently in text and in the caption.", "broader_impact_concerns": "I have no concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes GlobalGCE, a novel framework for global counterfactual explanations in Graph Neural Networks (GNNs). Instead of editing each graph individually, GlobalGCE identifies a small set of subgraph-mapping rules that can flip predictions of many graphs to a desired class. The method involves three steps: identifying frequently occurring subgraphs, learning subgraph replacements via an autoencoder, and selecting rules to maximize coverage. The authors also introduce a new comprehensibility metric to evaluate interpretability. Experiments show that GlobalGCE achieves higher label-flipping rates and provides more concise and interpretable explanations than existing local-level approaches.

Overall, the approach is interesting, and the reviewers are satisfied with both the paper and the rebuttal. I therefore recommend acceptance.
Decision: Accept (poster)