RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
Authors: Jingxiang Qu
Wenhan Gao
Jiaxing Zhang
Xufeng Liu
Hua Wei
Haibin Ling
Yi Liu
Venue: ICML 2025 poster (2025)
Keywords: 3D Molecular Graphs
3D Graph Explanation
Radius of Influence
Primary area: applications->chemistry_physics_and_earth_sciences
TL;DR: We introduce a novel pipeline to explain 3D GNNs by identifying interpretable subgraphs derived from node-wise radii of influence.

Abstract: 3D Geometric Graph Neural Networks (GNNs) have emerged as transformative tools for modeling molecular data. Despite their predictive power, these models often suffer from limited interpretability, raising concerns for scientific applications that require reliable and transparent insights. While existing methods have primarily focused on explaining molecular substructures in 2D GNNs, the transition to 3D GNNs introduces unique challenges, such as handling the implicit dense edge structures created by a cutoff radius. To tackle this, we introduce a novel explanation method specifically designed for 3D GNNs, which localizes the explanation to the immediate neighborhood of each node within the 3D space. Each node is assigned an radius of influence, defining the localized region within which message passing captures spatial and structural interactions crucial for the model's predictions. This method leverages the spatial and geometric characteristics inherent in 3D graphs. By constraining the subgraph to a localized radius of influence, the approach not only enhances interpretability but also aligns with the physical and structural dependencies typical of 3D graph applications, such as molecular learning.

RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
Jingxiang Qu Q
Wenhan Gao 2
Jiaxing Zhang²
Xufeng Liu Hua Wei
Haibin Ling' Yi Liu
Abstract
progress in drug discovery, protein engineering, and ma- terials science (Zhang et al., 2025; Bronstein et al., 2021). Traditionally, molecules have commonly been modeled as 2D planar graphs (Gao et al., 2021; Liu et al., 2021; 2020), with atoms represented as nodes and chemical bonds as edges, disregarding their geometric configurations. How- ever, it is widely believed that chemical behaviors and bi- ological functions of molecules are largely determined by their 3D geometric structures (Tusnády & Simon, 1998; Hansch & Fujita, 1964). The limitations of 2D represen- tations in capturing geometric information have led to a growing focus on 3D graph learning (Gainza et al., 2020; Townshend et al., 2021). Consequently. 3D graph neural net- works (GNNs) with geometric features have demonstrated superior performance and thus become dominant across var- ious tasks (Schütt al., 2017: Gasteiger al., 2020; Liu et al., 2022; Wang et al., 2022b; Brandstetter al., 2022: Subedì et al., 2024; Yan et al., 2022; Wang et al., 2023).
3D Geometric Graph Neural Networks (GNNs) have emerged as transformative tools for mod- eling molecular data. Despite their predictive power, these models often suffer from limited interpretability, raising concerns for scientific ap- plications that require reliable and transparent in- sights. While existing methods have primarily focused on explaining molecular substructures in 2D GNNs, the transition to 3D GNNs intro- duces unique challenges, such as handling the implicit dense edge structures created by a cut-off' radius. To tackle this, we introduce a novel expla- nation method specifically designed for 3D GNNs, which localizes the explanation to the immediate neighborhood of each node within the 3D space. Each node is assigned defining the localized region within which mes- sage passing captures spatial and structural inter- actions crucial for the model's predictions. This method leverages the spatial and geometric char- acteristics inherent in 3D graphs. By constraining the subgraph localized radius of influence, the approach not only enhances interpretability but also aligns with the physical and structural dependencies typical of 3D graph applications. such as molecular learning. The code is available at https: /ttp//gtttttt github....... com/ com/QuJX/
radius of influence.
Although 3D GNNs have gained appealing performance in molecular graph learning, they are largely regarded as black-box models. GNN explanation methods seek to find crucial information from a molecular graph that significantly impacts the model's predictions. The main goal is to iden- tify a compact subset of edges (a subgraph) that accurately captures the behavior of the original graph (Ying al., 2019; Yuan et al., 2020; Luo et al.. 2020; Yuan et al., 2021; Huang et al., 2023; Liu et al., 2025). Chemists can verify whether these explanatory subgraphs align with actual chem- ical substructures, such as functional groups and motifs, that contribute the most to the chemical properties. While these explanation methods have demonstrated effectiveness for 2D GNNs, they overlook the fundamental distinctions of 3D GNNs, leading to explanations with inferior fidelity.
1. Introduction
With recent advances in deep learning, molecular learning has become pivotal field of research, driving significant
"Equal contribution Department of Computer Science, Stony Brook University, Stony Brook, NY, United States Department of Applicd Mathematics & Statistics, Stony Brook University, Stony Brook, NY, United States 'Department of Informatics, New Jer- sey Institute of Technology, Newark. NJ, United States "School of Computing and Augmented Intelligence, Arizona State Uni- versity, Tempe, AZ, United States. Correspondence to: Yi Liu <yi.liu.4C 4e stonybrook.edut
There are two major differences between 2D and 3D GNNs that should be taken into account when designing explana- tion methods for 3D) molecular learning. (1) Difference in Representation: In 3D molecular graphs, the edges are no longer actual chemical bonds but are instead constructed based on cut-off distances (Schütt et al., 2017; Gasteiger et al., 2020; Liu et al., 2022), resulting in rapid increase of edges; (2) Difference in Learning: 3D GNNs aim to model both bonded and non- non-bonded interactions (e.g., elec-
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
trostatics, hydrogen bonds) through distances and spatial modeling. These differences lead to a broader search space and significant difficulty in optimization for existing explanation methods, which will be discussed further in Sec. 3.1.I. More importantly, these characteristics render the explanatory results from existing methods chemically uninterpretable. In 2D molecular graph learning, edges directly represent chemical bonds, which naturally define explanatory substructures such as functional groups (Wu et al., 2018; Diao et al., 2022; Ma et al., 2020). However, this direct correspondence no longer holds in 3D GNNs and the optimized substructures often consist of a set of "random edges" that are chemically uninterpretable. As a result, there a pressing need for a principled explanation framework that considers the differences and closes this gap.
variable Y. The target Y can be discrete labels for graph classification tasks continuous values for regression tasks. Without loss of generality, we focus regression tasks.
There is a notable absence of well-founded explanation methods specifically designed for 3D GNNs, as most ex- isting approaches work primarily for 2D GNNs. A 2D molecular graph G is represented as G = (V. A.A, X), where
Fp denotes a set of n nodes, A denotes n X n adjacency matrix where each entry aij {00,1 1} indicates the presence or absence of an edge connecting nodes i and j, and X = X2,....... Xnl E R=xdd is the node feature matrix, with each x, E I& denoting the dy dy-dimensional node feature associated with node U,.
V {1L, u2,...., .....
Following Ying et al. (2019), instance-level graph explana- tion aims to find a subgraph Gs S G that is important to the target Y. This is formally expressed as:
In order to design principled 3D GNN explanation methods considering these differences, natural question arises:
(2) In atomic systems, interactions between atoms or nodes often weaken with distance. Does the 3D GNN inherently adhere to this physical principle in its learning process? If so, can we design explanation methods by referring to distances?
G arg min ((): s.t. |Gsl S B, GG
(1)
where L denotes the task-dependent loss function, and B represents size constraint on the subgraph to avoid trivial solutions. Eq. (1) can be rewritten as:
In partieular, find that the proximity (distances between atoms) has substantial impacts on the message-passing scheme that messages between closer nodes are more im- portant. Building on these insights, we propose a principled approach that finds the radius of influence for each atom to extract the explanatory subgraphs. These radii of influence define the localized regions that capture the most important interactions critical for the prediction.
G = arg min :( @ A) s.t. || S B, (2) M
where M E {0, 1]mxn are binary masks, indicating whether to retain or remove an edge, applied to extract subgraph.
2.2. Existing Works
GNN explanation methods (Huang et al., 2023; Luo et al., 2020; Ying et al., 2019; Zhang et al., 2024; Yuan et al., 2021; Miao et al., 2023; Xie et al., 2022) aim to understand and interpret the decision
Overall, we summarize our contribution as follows: ① We are the first to identify and introduce two crucial yet previ- ously overlooked differences between 2D and 3D GNNs. @ Building on this, we propose a principled approach, known as RISE, that leverages radii of influence to extract the explanatory subgraphs for 3D GNNs. @ Our proposed ap- proach does not require the relaxation from binary masks to continuous masks as in existing methods, allowing exact op- timization. + Radii of influence naturally account for both differences, making RISE the only explanation pipeline that can produce chemically interpretable explanatory subgraphs as discussed in Sec. 3.3. 5 Experiments demonstrate the superiority of our proposed framework, not only in quanti- tative evaluation metrics but also in producing chemically interpretable results.
processes of GNNs. We can classify GNN explanation into two categories instance- level and model-level explanations. Instance-level expla- nation aims to identify important graph components that drive the model's decisions. As a pioneering work, GN- NExplainer (Ying et al., 2019) finds the most important subgraph in transductive setting by masking the edges. PGExplainer (Luo et al., 2020) extends GNNExplainer to the inductive setting by parametrizing masks with neural networks. Various works have been proposed to further extend and refine these methods. For example, MixupEx- plainer (Zhang et al., 2023) consider the subgraph distri- bution to further improve explanatory results. In contrast, model-level methods aim to provide global explanations of model's behavior, independent of specific input graphs. For instance, XGNN (Yuan et al., 2020) utilizes a graph generator to interpret the model's decision- while CGE (Fang et al., 2023) generates explanatory sub- graphs along with their corresponding subnetworks. Despite the advancements in GNN explanation methods and their
2. Background and Related Work
2.1. GNN Explanation
process,
The notations used throughout the paper are summarized in Table 4 in Appendix A. A GNN & mapping from graph G a prediction Y corresponding to the target
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RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
Learning Soft Edge Masks
Edge Mask:
Threshold
D.T
0.3
Existing Explainers
Select Top-k Edges based on Threshold
Binary Edge Masks
Optimized Edge Masks
0.3
[1.0, 1.0, 1.0|0, o....
Binarization
po... 0.8, 0.7 0.760.6, 0.5, 0.4,..
Discrepancies Between Optimized and Applied Masks
O. Edge Mask
Selected Edges-
Unselected Edges
543
Learning Radii of Influence
Atomic Influence:
Select Edges within the Influence Radius
RISE
Binary Edge Masks
Radius of Influence
Edge
[1.0, 1.0, 1.0, 0,
Radiust.12 171
Soloction
Learned Edge Masks Are Naturally Binary
Badiaa:1 532
Mass
Distance Between
Figure 1. Comparison with existing approaches. Existing approaches require relaxation from binary masks to soft continuous masks, leading to inconsistencies between the optimized masks and the explanatory binary masks. These inconsistencies not only compromise explanation fidelity quantitatively but also produce chemically uninterpretable results, making the model explanation itself black-box. On the other hand, RISE introduces a novel framework for extracting explanatory substructures based on atomic radii of influence that aim find localized regions that capture the most important interactions decisive the prediction under some budgets. Given an appropriate small budget, RISE can precisely extract chemical bonds and chemical bonds only. For instance, for Ethane (CH,CHa) in the figure, the radii of influence from our experiments assign the of interest with radius of 1.532 and the H of interest with a radius of 1.171. Under similar radii of influence for C atoms and H atoms, respectively, RISE extracts the precise chemical bonds and chemical bonds only: C-H (1.171 > 1.095); C-C (1.532 1.530); all other edges have distance greater than 1.532 and will be masked out by RISE. Note the unit of distance is A, which is used throughout the paper unless otherwise specified.
proven effectiveness, existing approaches struggle to effec- tively explain 3D GNNs (Miao et al., 2023), as they often overlook the fundamental differences between 2D and 3D representation learning discussed later in Sec. 3.1. In this work, we focus addressing the unique challenges of 3D graph representation and learning at the instance level.
3. RISE: Radius of Influence based Subgraph Extraction
In this section, we begin by analyzing the two geometric characteristics unique in 3D GNNs in Sec. 3.1: First, we discuss the difference in representation and the challenges it brings to existing explanation methods in Sec. 3.1.1; @We then discuss the differences in learning, highlighting the crucial role of proximity in the peessag--passing scheme in Sec. 3.1.2. Motivated from these geometric characteris- tics, we introduce our proposed Radius of Influence based Subgraph Extraction (RISE) framework, specifically refor- mulated for 3D GNNs, in Sec. 3.2. Finally, in Sec. 3.3, we discuss the benefits that RISE offers, especially its consis- tency in optimization and, most importantly, its capability to provide interpretable explanatory subgraphs. An overview of RISE is given in Fig. 1.
Relations with Prior Works. There is a significant lack of work in 3D GNN explanation. To our best knowledge, there only one published work specifically concerned with 3D GNN explanation, known as LRI (Miao et al., 2023). LRI introduces a probabilistic masking framework based on Bernoulli and Gaussian noise to evaluate the importance of atom existence and spatial location, then masking the edges based on the importance of its vertices. However, it only considers the difference in representation between 2D and 3D molecular graphs; there is an important piece that missing: The difference in learning mechanism for 2D and 3D GNNs resulting from different representations. The message- passing framework operates differently with 3D molecular representations. Building on this, we propose principled approach based on the radius of influence, which accounts for both representational differences and variations in the learning mechanism. This design offers several the- oretical advantages and leads to significant performance improvements, demonstrated in Sec. 3.3 and Sec. 4.2.
3.1. The Missing Piece: Geometric Characteristics
The primary difference between 2D and 3D molecular leam- ing stems from the inherent structural distinctions between the molecular graph representations. In 2D graphs, nodes are presented in planar layout and the edges are the chem- ical bonds. In 3D graphs, nodes are presented in 3D Eu- clidean space, offering a more general depiction of geo- metric relationships, and the edges are usually constructed
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RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
based cut-off distance (Schütt et al., 2017; Gasteiger et al., 2020; Liu al., 2022; Brandstetter et al., 2022), re- sulting in very dense graphs (difference in representation) as illustrated in Fig. 2. This construction is motivated by the underlying physical and chemical interactions (Wang et al., 2022a; Leach, 2001), aiming to model both bonded and non-bonded interactions (difference learning). This distinction between 2D and 3D molecular learning leads to two geometric characteristies that should be considered but are missing in existing GNN explanation methods.
GNNExplainer (Ying et al., 2019) and follow-up works (Luo et al., 2020; Yuan et al., 2021) attempt to lessen this issue by including additional losses penalizing the extent to which the masks deviate from 0 or 1, such as the entropy loss. Moreover, the budget is enforced through sparsity regular- ization loss, such as the L1 regularization. Therefore, the final optimization objective becomes
G = arg mY ((Mon @ AA + all alIMMollla Man
(4)
+ 3- (Man):
where and 33 are loss balancing terms. However, adding additional loss terms introduces a trade-off between the optimization objective and the regularization. This trade-off' often requires careful tuning of hyperparameters to balance the sparsity or discreteness of the masks against the primary objective of accurately explaining the model's predictions. This issue exists in 2D GNN explanation well but worsens in 3D GNNs due to rapid increase of edges, where small inconsistencies for each mask compound into significant deficits in explanation fidelit
(a)
(b)
1. DIFFERENCE IN LEARNING
Figure 2. (a): 2D representation of CgH1s-Nodes are atoms, and edges are chemical bonds, No geometric information; typically small number of edges. (b): 3D representation of CyHis-Nodes are atoms with spatial locations. Edges are constructed with a specified cut-off distance, resulting in dense graph. (c): 3D representation of CaHls with all non-bonding edges removed.
The dense edges in 3D GNNs aim to capture interatomic interactions, including both bonded and non-bonded inter- actions. These interactions and forces usually decay signifi- cantly with distance. Interactions between nodes separated by large distances are typically negligible due to the rapid decay of force magnitudes, e.g., London dispersion poten- tial decays as 1 and strong interactions usually have a short-distance depen- dency (1 2A for covalent bonds and 2 3. for hydrogen bonding).
ze or faster, Pauli repulsion decays as e aad
3.1.1. DIFFERENCE IN REPRESENTATION
We find that the difference between 2D and 3D molecu- lar representations poses significant challenges in adapt- ing 2D GNN explanation methods for 3D GNNs. Specifi- cally, to allow gradient- based optimization, the binary hard- masks in Eq. (2) are relaxed to continuous soft-masks, Maan E [0, 1|axn_ During testing, an explanatory subgraph is generated from the soft-mask based on the budget; the masks are now made strictly 0-s and 1-s, even though some values, such as 0.70, might be thresholded to 1, while oth- ers, such as 0.68, are assigned to O. Clearly, a significant drawback of relaxing discrete masks to soft masks lies in the difference between optimized continuous masks and the final binary masks representing the explanatory subgraph (inconsistency in optimization). Mathematically, we give the following bound to illustrate such an inconsistency:
We find that 3D GNNs can fully leverage the underlying physical principles and truly learn to capture the proximity- dependent nature of interactions. In Sec. 4.1, we conduct extensive experiments to demonstrate that 3D GNNs learn the proximity -eepnitedddeeeencyyy in message passing: Messages between closer nodes in general carry greater importance. This learning difference between 2D and 3D molecular graphs underscores the need for tailored explanation frame- work that effectively captures the unique ability of 3D GNNs to learn range-dependent relationships.
These two geometric characteristics make 2D GNNs poorly suited for 3D GNNs, which can be seen from their poor per- formance in Sec. 4.2. We are the first to propose a principled explanation method based on our findings on the geomet- ric charucteristics of 3D molecular graphs. The geometric influence on the -aassagg scheme suggests reformu- lating 3D graph explanations in terms of atomic radii of inftuence, explicitly accounting for the varying importance of interactions based on distance.
G arg min (G)))) < C(Y; Maon @ A)) GGCG + L(O(XX Masn @ A); +(X, M @ A)),
optimization objective
difference that ignored during optimization
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RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
3.2. RISE Reformulation
explanation in Eq. (1) can be reformulated as:
Due to the difference in representation, where 3D molecu- lar graphs constructed based on cut-off distance, 3D graphs can be treated as proximity graphs, in which edges are not explicitly defined.
G3 arg min L(Y; M" @ R, X)) s.t. [r B, M
(5)
where M E [0, are continuous masks defining the radius of influence of each node as shown in Fig. 3. B
Definition 3.1. A directed proximity graph (DPG) is a ge- ometric graph constructed from a set of points in a metric space, where directed edges are formed based on a prox- imity rule. Specifically, each node ii has an associated radius 0 S ri. and directed edge Gi-ij exists if and only if dij < ri where dip is the distance between U; and V/- Remark 3.2. Any 3D graph constructed based on node radii can be viewed as a directed proximity graph. 3D graph constructed based on cut-off distance (the same radius for all nodes) a proximity graph under uniform radi;; the graph is undirected as all the radii are the same.
the budget to avoid trivial solutions and can be set as a ratio B - p [|]]||
Clearly, the resulting explanatory graph is DPG and is subgraph of the original graph. The main objective of 3D GNN explanation is to provide interpretable insights into the predictions. The concept of the radius of influence inher- ently contributes to chemical explainability, as it represents the spatial extent over which each atom interacts with others, depending on the property of consideration. Different chem- ical properties are influenced by different types of atomic interactions, which may only be significant within specific spatial ranges.
In 2D GNNs, edges are explicitly defined, leading to the formulation of 2D graphs described in Sec. 2.1, where subgraphs are considered subsets of these edges. In contrast, 3D GNNs do not have explicitly defined edges; instead, edges are determined based on specific cut-off radius. Building on this concept, we generalize graph construc- tion using the notion of DPGs; consequently, explanatory subgraphs are also identified as DPGs as presented below.
Let A E {0, 1]mX* be the adjacency matrix induced by the radii, R, of the original graph. Eq. (5) can be optimized by masking of the computation graph of A" through:
G arg min C M X))).
(6)
M
where M E {0, 1 indicates the removal or retention of edge after reducing the radii of influence, M". As strict discrete values are not differentiable, we reparametrize each element Mij in M with bounded differentiable function f such that Mjj = f (MF; dej) that satisfies Mij + 1 when dıj < M and Mi - O when dij > M where di is the distance between nodes v; and t;y, and Mr denotes the radius of influence of the node u4 Generally, any bounded differentiable function that satisfies this requirement should suffice. However, smooth monotonic function is better for the purpose of gradient- based optimization. Therefore, we use the following function in particular:
We define 3D graph as G (V; R, R, X), where V repre- sents the set of nodes and X denotes their features, similar to 2D GNNs. The matrix P E R= contains the Euclidean coordinates of the nodes, and R = [r1, r2,-.., rn]] E IR" specifies their radii for edge constructions.
Mij 1 e K(M d,,)'
where is typically large. With a sufficiently large k; obvious that the requirement above satisfied.
(a)
(b)
3.3. Merits of RISE
Figure 3. (a): Original 3D graphs constructed based on com- mon cut-off distance; this is the approach taken in most 3D GNNs (Schütt et al., 2017; Gasteiger et al., 2020; Liu et al., 2022; Brandstetter et al., 2022). The edges are bidirectional and dense.
Consistency in Optimization. RISE is consistent between the optimization objective and the explanation process, directly optimizes the radius of influence, which is naturally continuous; there is relaxation from discrete masks to continuous masks. The optimized radius mask M naturally represent subgraph G, = (V, P, AM'.. R, X) whereas in existing methods, the optimized continuous soft edge or node masks do not represent a subgraph G, # (V, Mgott O A", X). Additionally, we can enforce the budget exactly to
(b): Explanatory substructure identified by finding the radii of influence. The radii of influence are optimized (the circles shrink dynamically; see an illustration in Fig. 6 in Appendix B) such that the critical messages that are most relevant to the prediction task are preserved. The edges are directed and more sparse.
With this new definition of 3D graphs, instance-level graph
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RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
avoid having sparsity regularization loss:
With appropriate radii of influence under small budget, RISE is able to extract chemical bonds and chemical bonds only as illustrated in Fig. 1, while all other methods may fail due to the aforementioned issues. In Fig. 4, we show the explanatory substructure from real experiments; is obvious that all baseline methods extract chemically uninterpretable results while RISE extracts exactly the chemical bonds. We provide more results in Fig. 8 in Appendix D. Therefore, designing interpretable explanation methods like RISE for 3D GNNs is critical.
MI- B exp (0+) EJ=1 exp (0,) (S2.),
(8)
where 0, E Rn are both learnable parameters and (-) is the sigmoid function. It is ensured by construction that [M S B. Therefore, RISE is consistent in optimization, as it does not require converting discrete values into contin- uous ones for optimization and does not include penalty or regularization terms to promote discreteness or enforce the budget, different from existing works where the inconsis- tency is induced as seen in Eq. (4). Such inconsistency not only compromises explanation fidelity but also presents a far more severe issue: The explanatory substructures become chemically uninterpretable, making model explanation itself' a black-box. We demonstrate this below.
4. Experiments
In this section, we begin by addressing the critical question of how the distances between nodes influence the impor- tance of message passing; as discussed in Sec. 4.1, it is clear that the distances indicate the importance of message pass- ing. Next, we evaluate our RISE method, developed based on insights from the aforementioned findings, and com- pare its performance against several state-of-the-art (SOTA) baselines in Sec. 4.2. Several representative 3D GNNs, in- cluding invariant models SchNet (Schütt et al., 2017) and DimeNet (Gasteiger et al., 2020) and an equivariant model SEGNN (Brandstetter et al., 2022), are used backbone models. An introduction to these 3D GNNs is given in Appendix E. All explanation methods are evaluated on the widely used datasets QM9 (molecules in the equilibrium state) (Ramakrishman et al., 2014) and GEOM (geometric ensemble of molecules) (Axelrod & Gomez-Bombarelli 2022). The results highlight the superiority of RISE as a principled approach specifically designed for 3D GNNs.
Interpretable Subgraph Extraction. In 2D GNNs, edges correspond directly to chemical bonds, naturally forming the basis for chemically interpretable substructures. In 3D GNNs, due to rapid increase of edges, different continu- ous edge masks can all yield very high explanation fidelity, leading to ambiguity in explanation. On the other hand, enforcing discreteness regularization loss can compromise explanation fidelity, ultimately reducing the reliability of the extracted substructures. LRI shifts edge masks node masks by defining each edge mask the product of the node masks of its end nodes. However, this approach still fails to provide chemically interpretable substructures: two nodes with high mask values will indicate a high impor- tance for their connecting edge, even if they are distant (see more details in Appendix C). This misrepresentation con- tradicts both the learning dynamics of 3D GNNs and actual chemistry.
4.1. Proximity Decides Messages To Pass or Not To Pass
To study where proximity (distance) decides the importance of message passing, we conduct experiments centered on two key questions: 1 Does excluding short-distance edges from message passing impact the model's predictions more significantly than excluding long-distance edges? @ Do edges with similar distances have comparable importance in influencing the model's predictions?
GNNixplainer
PGlxxplainer
L.II-Bemoulli
Setup. We group edges based on their distances di into 5 annular bins: {0 I de S dij < dx-1 S deu k=1" where all dæ-values are determined such that, on average over the entire testing dataset, each annulus contains ~ 20% of the total edges. O To answer the first question, we remove all the edges in each annulus and measure the resulting decline in predictive fidelity. If the importance of message-passing edges correlates with distance, the removal of edges from inner annuli (shorter distances) should lead to more pronounced decline in performance. @ To answer the second question, we randomly remove 10% of the edges within each annulus and evaluate the model performance after the removal of edges; if the intru-anulls variance is
RISE
Actual Chemical Bonds
Figure 4. Explanatory substructure produced from experiments by different explanation methods on the Ethane molecule (CH; CH1s) the QM9 dataset (Ramakrishnan et al., 2014). The same budget (number of edges) is used for different explainers. It is obvious that only RISE yields chemically interpretable results that conform interpretable chemical structures. It should be noted that, der larger budgets, baseline methods yield a more "chaotic" set of edges that uninterpretable at all, whereas RISE identifies atomic regions of influence, enabling chemical interpretation. This underscores the need for interpretable 3D methods like RISE.
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RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
Table 1. Comparison of masking edges in different radius ranges on a and ELUMO properties of the QM9 dataset with SchNet and SEGNN as the representative of invariant and equivariant models, respectively. Values in most cells are strictly greater than the previous ones in the row; clearly, proximity decides the importance of message passing.
MAE After Annulus of Edges Removed
Property
Backbone
Original MAE
80-100%
60-80%
40-60%
20-40%
0-20%
SchNet SEGNN
0,118
0.248
0.253
0.373
0.373
0.739
0.110
0.196
0.342
0.418
0.826
0.895
SchNet
0.046
0,051
0.051
0.058
0.070
0.128
CHOMO
SEGNN 0.032
0.034
0.035
0.036
0.048
0.088
consistently small, it suggests that the importance of edges in each annulus (similar distance) is approximately uniform.
polarizability (a), the highest occupied molecular orbital energy (<HoMo), and the lowest unoccupied molecular or- bital energy (eLUMO). For the GEOM dataset, the learning task focuses on predicting the energy of conformers at their low-energy states.
Results. The quantitative results for setting are presented in Table 1. It is evident that removing closer annuli leads to more significant drop in MAF, values in most cells strictly decrease compared to the previous row. A sample of the qualitative results for setting @ is shown in Fig. 5, with the full results, all sharing the same trend, available in Appendix F. From these results, we observe that edges at similar distances have comparable importance, as indicated by the relatively flat trend lines with minimal fluctuation. Additionally, shorter edges play a more significant role in message passing. In summary. proximity determines the importance of messages. This pattern holds consistently across molecular graphs in the dataset, reinforcing the gen- eral trend. For each molecular graph, we identify the atomic radii of influence to retain the most critical interactions within the given budget.
Setup. Experiments are conducted on all three represen- tative 3D GNNs as the backbone models. Specifically, in- variant models are tested with the original QM9 test dataset and 1, 000 randomly extracted molecules from the GEOM dataset, while the equivariant model is tested using the first 1, 000 molecules from the QM9 test dataset and the same 1, 000 molecules from the GEOM dataset due to the compu- tational hurdle of tensor decompositions in SEGNN. RISE is compared with the two representative explanation methods, GNNExplainer and PGExplainer, and the only existing 3D explanation method -LenddL-R---e LRI-e--uoullllll Note there is another method in LRI, LRI- RGGasssii; however, it is perturbation- based and cannot be directly compared with RISE. More de- tails about baseline explanation methods are in Appendix G. Following existing works (Yuan et al., 2022), report the mean absolute error (MAF, the lower the better) of the predictions made using the explanatory subgraphs.
-00
SEGNNN
Mean: 0.885. Varianca: 4,80e
Moan: 0.826.
Variance: 1.70e
To thoroughly evaluate performance across varying levels of edge sparsity, we set the budget parameter p to different values: 0.3 to 0.6 for the QM9 dataset, and 0.4 0.6 for the GEOM dataset. Since different ex explanation methods have different budget constraints, we compare them in a unified way based on the number of preserved edges. The baseline models' budgets are set such that they strictly preserve more edges than RISE, making the comparison even more advantageous for the baselines. More experimental details including hyperparameter settings and training details are attached in Appendix H.
Mean: 0,418 Variance: 7.35e*
Mean: 0.342, Variance: 5.04e"
20405
Mean: 16, Variance: 3.89e"
13.04
Mean: 0.1 Variance
MAE
10
14
Times
Figure 5. The visualization of the quantitative results of of the QM9 dataset on SEGNN when randomly masking 10% edges in different annuli. It shows that the influenec of edge masking has significant correlation with the distances, i.c., masking short edges will cause larger perturbation than long edges. Moreover, the small variance of each annulus demonstrates the comparable importance among edges with similar distances.
Results. We present the results the QM9 dataset using SchNet as a representative of invariant backbone models in Table 2. When SchNet is used as the backbone model, RISE consistently outperforms all baselines across various budgets, except for the <hомo property, where it performs on par with GNNExplainer. We present the results on the QM9 dataset using SEGNN as a representative equivariant backbone in Table 3. Similarly, when SEGNN is used as the backbone model, RISE consistently outperforms all basclines across various budgets, except for the a property
4.2. The Power of Proximity: RISE Performance
We evaluate the performance of RISE on two widely used molecular datasets- QM9 and GEOM. For the QM9 dataset, the learning objective is a regression task to predict molecu- lar properties for stable molecules; we perform experiments on important properties: the dipole moment (n), isotropic
7

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RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
Table 2. Resulting MAE (the lower the better) with SchNet on the QM9 dataset. The original SchNet employs cut-off of 10A, leading an average of 316 edges per graph. The average number of edges preserved is indicated in parentheses. The best and second best results respectively highlighted in red and blue, both in red if they very close and the lower one has more edges preserved.
Explainer
f
0.3
0.4
0.5
0.3
0.4
0.5
GNNExplainer
2.972(50.0%)
1.162(69.9%
0.859(79.9%)
2.836(50.0%
2.2.6(69,.9)
2.054(79.9
PGExplainer
1.884(45.6
1.756(65.19) 1%)
1.684(82.8%)
10.420(45.995 .9%)
9.520(63.9%%
9.221(81. 9%%
LRI-Bernoulli
6.087(44,1 1%)
5.325(70.0%)
4.127)00.0%)
14.459(47
1..877667.%%
10.144(80.35) 38))%%
RISE (ours)
0.596(44.11% %))
0.518(64.8%)
0.422(79.22%
2.670(44.6%)
2.127(63.7%
1,430(77.0%
Explainer
CHOMO
CLUMO
0.3
0.4
0.5
0.3
0.4
0.5
GNNExplainer
0.334(49.9% %
0.239(69.9%)
0.154(80.0%)
1.181(50.0%
0.766(69.9%
0.209(89.955
PGExplainer
0.567(50.0%)
0.580(70.0%)
0.598(8000%)
1.322(50.2%)
1.355(71.6%)
1.418(85.99) 9%)
LRI-Bernoulli
0.622(67.4%) 0.601(47.7%)
0.623(80.4%)
1.355(50.0%)
1.379(80.3%)
1.401(90.0%)
RISE (ours)
0.333(44.3%) %))
0.248(63.5%% %)
0.172(76.8%)
0.544(45.2%)
0.370(66.0%)
0.207
Table 3. Resulting MAE (the lower the better) with SEGNN on the QM9 dataset. The original work employs a cut-off of 5.4, leading to an average of 280 edges per graph. The average number of edges preserved indicated in parentheses. The best and second best results are respectively highlighted in red and blue, or both in red if they are very close and the lower one has more edges preserved.
Explainer
/
0.4
0.5
0.6
0.4
0.5
0.6
GNNExplainer
1.183(9.88%)
1.056(00.0%)
0.939(59.8%)
27.590(29.899 8%)
23.026(44.958
16.388(59.88)
PGExplainer
1.197(30.1
1.140(40. 1))
0.974(60.2%)
27.191 (3)5%%)
23.320(41.599
17.673(52.2%)
LRI-Bemoulli
1.051(42.1%) 1.040(27.2%)
1.005(50.0%
0.895(60.0%)
26.247((0044))
15.201(60.2%) 22.625(43.2%)
RISE (ours)
0.872(40.0%)
0.674(51.8%)
26.261(27.9%)
20.274(40.2%)
15.557(51.6%)
Explainer
6HOMO
GLUMO
0.4
0.5
0.6
0.4
0.5
0.6
GNNExplainer
1849(29.8%
0.725(39.8%)
0.457(59.8%
1.671(29.8%)
1.45((3.8%)
1.014(5988%)
PGExplainer
0.848(29.8%)
0.766(40.0%)
0.461(60.0%)
1.778(30.6%)
1.582(40.6%)
1.129(60.5%)
LRI-Bernouli
3.677(30.0%)
4.646(40.0%)
3.363(60.0%)
1.006(30.0))
0.724(40.0%
0.578 %)
RISE (ours)
0.838(26.8%)
0.551(39.4%)
0.339)50.%%)
0.988(26.8%
0.679(39.1% 1
0.438(50.5%)
under budget of 0.4, where it performs on par with LRI- Bernoulli. When using SEGNN as the backbone model, higher MAEs are observed across the board.
bonds represent the strongest interactions, and RISE effec- tively captures this crucial aspect of molecular structures.
Additionally, in Appendix I, we present results on the QM9 dataset using DimeNet (invariant) and on the GEOM dataset. RISE consistently achieves the best performance across all configurations on the GEOM dataset and outperforms other methods in most configurations on QM9 with DimeNet. These results further emphasize RISE'S ability to capture meaningful interactions essential for molecular prediction while generating interpretable explanatory substructures that generalize across different datasets and tasks.
5. Conclusions, Limitations, and Future Work
In this work, we introduce a principled explanation method for 3D GNNs by representing 3D graphs as directed prox- imity graphs and determining the radius of influence for each atom. Our approach is the first to explicitly address the fundamental distinctions between 2D and 3D GNNs in both representation and learning dynamics. Consequently, it surpasses existing SOTA explainers in quantitative perfor- mance on 3D GNNs. More importantly, existing methods often yield chemically uninterpretable explanations for 3D GNNs, as they do not specifically account for 3D character- istics, rendering the explanation method itself a black box. In contrast, our method consistently produces interpretable and chemically meaningful results.
While achieving exceptional quantitative results is impor- tant, another crucial aspect is the interpretability of the explanation results. As discussed in Sec. 3.1, due to the differences between 2D and 3D GNNs in molecular rep- resentation and learning dynamics, existing methods often produce chemically uninterpretable results when applied to 3D GNNs. We provide several visualizations of the ex- planatory substructures in Appendix D. Notably, under small budget, when explanation methods can preserve only limited number of edges, RISE is the only method that selec- tively retains edges corresponding exclusively to chemical bonds. This is particularly significant because, chemically,
Limitations and Future work. This work focuses on ex- plaining relatively small 3D molecular graphs. However, macromolecules, such as proteins, exhibit more complex interactions and intricate structural hierarchies that may in- troduce long-range dependencies. It would be interesting to see if our method can be generalized to macromolecules with long-range interactions.
8

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RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
Acknowledgments
Conference on Web Search and Data Mining, WSDM 23, pp. 616-624, New York, NY, USA, 2023. Associa- tion for Computing Machinery. ISBN 9781450394079, doi: 10. 10.1145/3339599733777788 URL ht /doii.
The work was partially supported by National Science Foun- dation award #2421839 and institutional funds of Stony Brook University.
org / 10. 1
Impact Statement
Fey, M. and Lenssen, E. Fast graph representation learning with PyTorch Geometric. In ICLR Workshop Repre- sentation Learning on Graphs and Manifolds, 2019.
This paper advances the field of 3D Graph Neural Networks by introducing a novel framework that enhances 3D GNN explanation through localized influence radii, capturing spa- tial and structural interactions critical for predictions. Our method aligns with the geometric and physical properties of 3D data, supporting trustworthy AI applications in scientific domains like molecular discovery. There are limited poten- tial societal consequences of our work, and none require specific highlighting at this time.
Fuchs, F. B., Worrall, D. E., Fischer, V., and Welling, M. tran)srranfoormess:: 3d roto-translation equivariant at- tention networks. In Advances in Neural Information Processing Systems 34 (NeurIPS), 2020.
Gainza, P., Sverrisson, F., Monti, F., Rodola, E., Boscaini, D., Bronstein, M., and Correia, B. Deciphering interac- tion fingerprints from protein molecular surfaces using geometric deep learning. Nature Methods, 2020.
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RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
A. Table of Notations
Table 4. A summary of used notations.
Symbol
Description
A GNN mapping graph G' to prediction Y.
G
A molecular graph, defined as G (V, A, X) for 2D and G (V. for 3D.
V
Set of nodes in the graph, V = {U1, ........ vn
A
Adjacency matrix of size X n, where dij E {0, 1 indicates edge presence.
X
Node feature matrix, X E Raxd, where X; E Rd is the feature of node t,.
P
Euclidean coordinates of nodes in 3D space, P E Ruxs.
R
Initial radii for edge construction in 3D graphs, Rt [11, 22. ................................ rnl R"
8
Loss balancing terms.
C
Task-dependent loss function.
dej
Euclidean distance between nodes o; and V..
G
A subgraph of G.
M
Binary (hard) edge mask, M E {0, 1jxxn, used for subgraph extraction.
M Moan
Continuous (soft) edge mask, Mon [0, 1 for approximation of M for gradient-
based optimization
M
Radius of influence masks for nodes in 3D graphs, M" [0.
A"
Adjacency matrix induced by radii R in 3D graphs
Learnable parameters for determining radii of influence in RISE.
a.))
Sigmoid activation function.
B
Budget constraint on subgraph size, B = p. |R]| with the ratio
H
Entropy function used in soft mask regularization.
B. An Animation of Optimizing the Radii of Influence
As we optimize the masks for the radii of influence, we are essentially shrinking the atomic influence circles. An illustration given in Fig. From the top-left to the bothom-right, the sequence represents the entire evolution.
Figure 6. An animation of optimizing the radii of influence.
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RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
C. Issues with Masking Nodes in 3D GNNs
The structural differences between 2D and 3D molecular graphs not only alter the representation but also impact the learning mechamism. The importance of message passing should be based on distances. A.RI-Bemoulli accounts for the difference in representation by offsetting the edge masks node masks. Namely, let Mnode represent the set of masks on nodes, with Mode indicating the mask node U; LRI-Bernoulli conditions the message passing by converting node masks edge masks through:
M soft = MMode M node
This node masking approach, which disregards the differences in learning dynamics between and 3D GNNs, may result in significant issues. We now assume fully connected graphs for simplicity: in fact, with a cut-off distance of 5 many 3D graphs for small molecules are fully connected (Satorras et al., 2021). Taking CH,CH3 as an example, both C-C and same-side C-H bonds are critical, we aim to assign high mask values to the corresponding C and H atoms. However, since all C and H atoms participate in C-C and same-side C-H bonds, assigning high mask values to all C and H atoms fails to differentiate the relative importance of specific bonds. This approach also inadvertently assigns high mask values to less significant interactions, such as H-H edges across different sides. On the other hand, by accounting for the differences in learning dynamics, the RISE explanation, with appropriate radii of influence, precisely preserves the chemical bonds and chemical bonds only. This is demonstrated in Fig. 7.
0.9
H
C
2. 543
0.9
0.97
0.9
0.9
0.9
1.0
1.0
1-530
765
0.9
1.0
0.9
0.9
0.9
Radius:1.1
0.9
Center of Mass
Distance Between
Radius:1.
Atoms
(b)
Figure 7. Comparison between node- malking methods (a) and RISE (b). (a): Node-masking methods fail to accurately capture important interactions because they not consider spatial proximity. Two nodes with high mask values will always result in high importance for their connecting edge, even if they are far apart. This contradicts how 3D GNNs learn and the principles of actual chemistry. (b): RISE can correctly capture the most important interactions by considering proximity. Given an appropriate small budget, this example, B = 0.15, RISE can precisely extract chemical bonds and chemical bonds only, i.e., for Ethane (CH; (HH) given in the figure, the radii of influence from our experiments assign the C of interest with radius of 1.532 and the interest with a radius of 1.171. Under similar radii of influence for C atoms and H atoms, respectively, RISE extracts precisely chemical bonds and chemical bonds only: C-H (1.171 > 1.095); C-C (1.532 > 1.530); all other edges have a distance greater than 1.532 and will be masked out by RISE.
D. Visualization of Explanatory Substructures
The qualitative results of explanations are depicted in Fig. 8. These results highlight the unique advantage of RISE not only delivering the highest explanation fidelity (quantitative metric) also in extracting interpretable subgraphs that are chemically meaningful. Existing explanation methods, including GNNExplainer, PGExplainer, and LRI, often produce fragmented, overly dense, and chemically inconsistent explanations, and this issue gets worse in larger graphs (for visualization purposes, we provide relatively small molecules here).
For instance, in the case of methylene oxalate (C)H20,), other methods fail to isolate all C-O bonds, instead erroneously highlighting distant or non-adjacent nodes due to their reliance on continuous or indirect edge attributions. In contrast, RISE successfully reconstructs the precise chemical structures, as confirmed by structural comparisons with the ground-truth molecular bonds.
13

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RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
The failure of baseline methods stems from their inability resolve the ambiguity of 3D explanations-eitthr by producing overly distributed importance scores (e.g., LRT) by failing enforce discrete structural preservation (e.g.. GNNExplainer) as discussed in Sec. 3.3. These limitations underscore the necessity of RISE, principled approach that resolves the aforementioned issues by considering the two major differences between 2D and 3D graphs.
IUPAC Name
GNNExplainer
PGExplainer
LRI
RISE
Chemical Bonds Cutoff-based Graph
Ethane (CH,CH))
Methylene oxalate
(CgH,Ou)
N-Methylformmmide (C2H,NO)
Cyclopropane (CsHa)
Figure 8. The visualized samples of molecules within the QM9 dataset. The explanatory results of GNNExplainer, PGExplainer, LRI- Bernoulli, and RISE are inferred based on SchNet. The budgets are the same across different explanation methods; other words, the same number of edges retained in the explanation results. All edges are directed due to masking; however, RISE consistently identifies both directions, as interactions should be bidirectional in nature, whereas other methods often capture only one direction, leading to a less faithful representation of the underlying interactions. As result, other explanation methods appear to retain bidirectional and unidirectional edges are visualized as a single edge). Apparently, it can be seen that only our method preserves the chemical structures well, while other methods produce ambiguous explanations without interpretable chemical structures.
edges visually (both
E. 3D Geometric GNNs: Invariant and Equivariant GNNs
In 3D GNNs, molecular or spatial structures are typically represented as 3D graphs, where nodes correspond to atoms spatial points, and edges indicate interactions proximity relationships. The most common approach to constructing a 3D graph is to connect nodes based on a distance cut-off, where an edge is established between nodes whose Euclidean distance falls below a specified threshold, i.c., the cut-off. Based on different geometric feature representations the constructed graph, the corresponding 3D GNNs can be divided into invariant and equivariant GNNs.
Invariant GNNs, also referred to scalarization GNNs (Duval al., 2024), 3D geometric graph neural networks that rely exclusively invariant features, such distances, angles, and bond angles. These features remain unchanged under Euclidean transformations, including translations and rotations. As result, these GNNs are inherently invariant to Euclidean transformations, making them particularly suitable for invariant tasks, such as predicting energies, GHOMO-ELUMO gaps, and other scalar molecular properties. Notable, but far from complete, examples of invariant GNNs include SchNet (Schütt et al., 2017), DimeNet (Gasteiger et al., 2020), and SphereNet (Liu et al., 2022).
Equivariant GNNs, on the other hand, are designed to handle vectorial or tensorial properties that transform predictably under Euclidean transformations. These networks incorporate both invariant and equivariant features to ensure that the outputs transform in consistent manner when the inputs are subjected to translations rotations. For example, when predicting forces, dipole moments, or gradients, the outputs of an equivariant GNN should rotate or translate in accordance with the same transformation applied to the input geometry. Equivariant GNNs leverage advanced mechanisms, such as spherical harmonics and tensor algebra, to ensure that all the operations preserve equivariance throughout the neural network. Notable examples of equivariant GNNs include SE(3)- Transformer (Fuchs et al., 2020), ((3ivvqqivarian GNNs (Batzner al., 2022), and SEGNN (Brandstetter et al., 2022).
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RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
F. Proximity Decides Messages To Pass or Not To Pass
Table 5. Corresponding cut-off distance range of annuli.
Cut-off Distance
0-20%
20-40%
40-60%
60-80%
80-100%
3A
[0,2.1]
2.1, 2.6
[2.6,3.
[322, 4.0)
[4.0,
10A
2.2
2.2, 2.8
[2.8, 3.5)
3.5.4.4
[4.4,
To investigate the correlation between edge distance and importance, we conducted extensive experiments on the QM9 dataset using both invariant and equivariant backbones. First, categorize edges into distinct annuli, we analyzed the edge distance distribution within QM9, as presented in Table 5. The visual and quantitative results are illustrated in Fig. 9.
Our findings reveal that as the distance of masked edges increases, the perturbation induced by edge masking correspondingly decreases, indicating a reduction in edge importance. Additionally, the low variance observed across 20 trials suggests that edges within the same annulus exhibit similar importance. Collectively, these results establish a strong correlation between edge proximity and its role in message passing, reinforcing the significance of RISE, which explains 3D graphs based on spatially localized interactions.
G. Descriptions of Baseline Explanation Methods
Graph explanation methods aim to find a subgraph that best preserves the predictive signal of the original graph G while satisfying budget constraint B on the size of the subgraph (Ying et al., 2019). Mathematically,
arg min L(;; G
|Gsl S B
(10)
GGSG
where L denotes the task-dependent loss function, and B represents size constraint on the subgraph to avoid trivial solutions. Eq. (10) can be rewritten as:
G3 arg min min L(Y; @ X)) s.t. ||M]]a B,
(11)
M
where M E (0, 1)xm) are binary masks, indicating whether to retain or remove an edge, applied to extract a subgraph.
Existing explanation methods relax binary masks into continuous soft masks, Moott E [0, 1jmxa, to enable gradient- optimization. practice, GNNExplainer (Ying et al., 2019) initializes these masks as learnable parameters, applying a sigmoid function constrain their values to be between 0 and and optimizes them in transductive setting.
PGExplainer (Luo et al., 2020) extends this idea by introducing parameterized explainer that employs multi-layer perceptron (MLP) to generate edge masks from learned feature embeddings. This approach enables the collective explanation of multiple instances, improving generalization and making it applicable in the inductive setting.
Learnable Randomness Injection (LRI) (Miao et al., 2023) introduces two different methods -iiionds-------A L.RI-Berooulli and LRI- Gaussian. LRI-Bernonde Bernoulli injects Bernoulli noise into nodes to evaluate the importance of node-wise existence for the final prediction; in simple terms, applies binary masks nodes. LRI-Gaussian, on the other hand, a perturbation-barl method that adds Gaussian noise to the positions of nodes to assess the significance of their geometric features. Unlike methods that aim to identify substructures, LRI-Gaussian does not identify subgraph but rather focuses on identifying important geometric aspects of the input geometric graph. Therefore, it is not directly comparable with RISE or other extraction-based baselines, including GNNExplainer and PGExplainer.
H. Experimental Details
We conduct comparative experiments on molecular graph regression tasks using two widely used molecular datasets, QM9 and GEOM. For the QM9 dataset, four key molecular properties are taken for testing purposes: dipole moment (p), isotropic polarizability (a), highest occupied molecular orbital energy (CHoMo), and lowest unoccupied molecular orbital energy (cLUMO). In contrast, for the GEOM dataset, the regression task focuses on predicting the energy. The QM9 dataset was obtained from the PoToh Geometric (PyG) library and used its original form. The GEOM dataset was sourced from the
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RISE: Radius of Influenee based Subgraph Extraction for 3D Molecular Graph Explanation
Meen:
Mean: 0.739, Verience: 2.096
SchNeet
SchNet
Mean: 0.573, Varence 2,74
Mean: 0.374. Varianee 3.18e
000
Mean: 0:160, Variance
Variance:
0.118 Variance: 9.36e
Mean: 0.248, Varienoe: 1.016
Mean 0.119 Varianee
Mean: 0,130 Varinnee
0.118, Variance:
1922
SEGINN
Mean: 0.317 Verlance: 1.546"
SEGNN
Moam: O 0,895, Variance: 4.80c
Meen: 0.825,
Variance: 1.70e
Mean: D, 0,086, Vaniance: 4.020
100
Mean: 0,418 Variance: 7.350
Meen: 0,061, Veriance, 2,28
Mean: 0.342, Variance: 5.04e
Mean: 0.061. Variance: 1,32e
Mean: 0.046, Verience 1.786
Meen: 0.196, Variance: 3.86e
101101
Mean: 0.110, Vaniacce 0
Mean: 000, Variance!
Text
Teat Times
(a) μ
(b)
Meen: Variance: 6.540
SchNNet
SchNet
Meen: 0.393, Variance: 4.10a
0.070, Variance: B.67e
238
Moan: 0,058 Variance: 7,00
0.00
Mean: 0,051, Variance: 1.386
Meen: 0.089, Variance: 6 0.055*
Mean: 0.051, Varianoe: 3.896
Mean: O 055
Variance: 8.66
Mean: 0.045, Verjence 1,09e
0,046, Variance:
Mean:
08-005
Ong MAX
Mean: 0.038, Variance:
020
SEGNN
9.35e
SEGNN
Mean: D. 161, Vaniance: 1.02e
Mean: 0.0 0404, Variance: 1.39e
6.0
Mean: 0.008
Variance: 2.460
006
Mean: 0.044, Varlence: 1.706"
608h
Mean: 0:055,
Vallance: 7:26
Mean: 0.032 2 2,16e
Mean: 0.034, Varlanos: 1.426 Mean: 0:032, Vanianze
Mean: 033,, Variance: 1.25e
Meeni 0 030, Variance:
Mean: 0,038, Variance:
M92
Tos
(c) Ehomo
(d) Ejumo
Figure 9. The visualization of MAE prediction μ4, a, сHомO, and éLUMO with randomly mask 10% edges within the top 0 20%. 20 - 40%, 40 - 60%, 60 - 80%, and 80 100% distant range, respectively. The results on four properties jointly indicate the correlation between edge distance and corresponding importance, while edges within the same annulus have similar importance.
official GEOM paper (Axelrod & Gomez-Bombarelli, 2022), and we preprocessed it by removing all conformers except the lowest-energy one for each molecule.
As for the backbone models, on QM9 dataset, we parameterize SchNet and DimeNet models with the pre-trained weight
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RISE: Radius of Influence based Subgraph Extraction for 3D Molecular Graph Explanation
Table Hyperparameter Search Space: To ensure fairness and reproducibility, we present the parameter search space used in
experiments. All models were evaluated under range of different hyperparameter settings, and the best pest-performing results are reported. Explainer
Setting
Search Space
Loss Weights
Aprdd
1, Asize O, Aen =
RISE
Training Epochs
50, 100, 300
Loss Weights
Apred 1, Apred E {0.1. 0.5, L.0,, A {0.5,1)
GNNExplainer
Training Epochs
50, 100, 300
Loss Weights
Aprrat Aprod E (..1, 0.5, 2 E [1, 10 Amm E 00,,0,
PGExplainer
Training Epochs
50. 100, 300, 500
Loss Weights
Apecal I, Apred E [0.5, 10], 20), Aen E {0.01, 0.1,
LRI-Bernoulli
Training Epochs
50. 100, 300, 500
provided by the PyG (Fey & Lenssen, 2019) library with their original settings. The SEGNN model was trained on QM9 following the official settings described in the paper. For the GEOM dataset, model settings were maintained the same as the experiments on QM9. The cut-off for all models was set to 5.0 on GEOM. The backbone models are first trained on 50, 000 molecules from GEOM dataset, which then filtered and split into 10, 000 molecules for training explanation models, and 1, 000 for testing. All models are trained independently on a single NVIDIA RTX A6000 GPU.
We configure our RISE method described in Sec. For the baseline methods, we adjust the weights of the loss function, which given as
C >xrea + Msize|/Msotlll + [enn
(12)
where the Cped is the prediction loss. The search space of weights and training epochs shown in Table 6. After training, baseline methods select the top & edges based on their mask values. The original LRI method lacks the size loss, which makes it learn dense edge mask with all values close to 1. Therefore, we incorporate the size loss into its original loss function as well.
I. Additional Experimental Results
Table 7. Resulting MAE (the lower the better) using DimeNet as the backbone model the QM9 dataset. The setup from the original work employs a cut-off distance of 5', leading to an average of 279 edges per graph. The average number of edges preserved indicated in parentheses. The best and second best results are respectively highlighted in red and bluc.
Explainer
/
0.4
0.5
0.6
0.4
0.5
0.6
GNNExplainer
0.897(29.9%)
0.795(39.9%)
0.667(50.0%
25.873(29.995 9%%)
17.769(39.955
10.073
PGExplainer
1.202(31.3%)
1.410(48.4%)
1.423(57.15%
56.00(30.0%)
112.81(40.0%)
208. (5000%%)
I.I--ernoulli
7.932(33.3%)
6.087(44.
5.265(52.8%)
15.963(34.7% 7
15.142(41.2%)
13.740(54.3%)
RISE (ours)
0.697(25.2%)
0.570(37.8%%
0.458(49.8%)
5.004(25.2%)
1.485(37.8%)
1.087
Explainer
<нOMO
GLUMO
0.5
0.7
0.5
0.7
GNNExplainer
1.505(39.8%)
0.995(50.0%)
0.493(69.8%
1.055(39.8%
0.882(50.0%)
0.368 )
PGExplainer
3.712(42.6%)
3.596(49.799
3.268(71.33)
1.039(40.0%)
1.002(50.0%)
0.950(70.0%)
LRI-Bernoulli
3.646(40.0%)
3.547(50.0%)
3.220000,0%
0.944(40.0%
0.927(50.0%)
0.911(70.0%)
RISE (ours)
5.091(37.6%)
2.727(49.1%)
1.486(65.9%
0.749(37.8%)
0.731(47.5%)
0.197(66.4%)
Table 8. Resulting MAE (the lower the better) using SchNet and DimeNet as the backbone models on the GEOM dataset. The setup from the original work employs cut-off distance 5.1, leading to an average edges per graph. average number edges preserved is indicated in parentheses. The best and second best results are respectively highlighted in red and blue, both in red if they are very close and the lower one has more edges preserved.
SchNet 0.5
DimeNet 0.5
Explainer
0.4
0.6
0.4
0.6
GNNExplainer
0.105(00.0%)
0. 106(4000%))
0.102(50.0%)
10.397(30.0%)
4.224(40.0%
2.292(50.0%
PGExplainer
0.071(25.8%)
(3084(39.5%)
0.097(53.2%)
47.605(20.0%)
41.645(30.0%)
29.949(50.0%%
L Re--Benoulli
17.535(20.0%)
17.532(40.0%)
17.531650.0%
33.018(20.0%)
24.633(30.0%)
12.01 12.018(50.0%)
RISE (ours)
0.098(19.6%)
0.096(31.2%)
0.082(43.2%)
0.845(17.3%)
0.198(27.3%)
0.081(40.55%
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RISE: Radius of Influenee based Subgraph Extraction for 3D Molecular Graph Explanation
I.I. RISE Performance: DimeNet QM9 Dataset
On the QM9 dataset, we additionally provide the results using DimeNet as the backbone model in Table 7. The results indicate that RISE outperforms all baselines in most cases across different budgets. For instance, at B 0.6, RISE achieves the lowest MAE for μ., a, and eLUmO predictions, as well the second-lowest MAE for Chомо- while preserving fewer edges. These results demonstrate the superiority of RISE in capturing meaningful interactions.
1.2. RISE Performance: Comparison on GEOM Dataset
To demonstrate the generalizability of our method, we also conducted experiments on the GEOM dataset, which contains the molecules related to experimental data the biophysics, physiology, and physical chemistry domains. The quantitative results are shown Table 8. RISE consistently outperforms or is at least on par with existing explanation methods. These findings highlight the robustness of our approach in adapting diverse molecular structures and learning meaningful edge attributions.
18


Reviews:
[{"review_id": "aV6tet4tZp", "reviewer": "Reviewer_U9Dy", "summary": "The paper presents RISE, a method for explaining 3D molecular GNNs by identifying key substructures using a radius of influence for each atom. RISE formulates the explanation process as an optimization problem that finds a compact, chemically interpretable subgraph while maintaining prediction fidelity. Instead of soft edge masks, RISE optimizes the radius of influence for each atom, ensuring consistency between the explanation and the model’s learning process.\nThe authors evaluate RISE on QM9 and GEOM datasets, comparing it against GNNExplainer, PGExplainer, and LRI-Bernoulli. The results show that RISE produces more interpretable explanations, capturing actual chemical substructures like functional groups and bonds while maintaining high explanation fidelity.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for reviewing our work. We respond below.\n\n## Claims And Evidence\n- The baselines are given budgets **preserving more edges than RISE, favoring them in comparison**. \n- Despite this, RISE **consistently** shows strong performance. \n\n> Baselines old; existing methods ineffective on 3D graph\n- **The issues of applying existing methods to 3D GNNs have also been noted in Sec. 2 of the LRI paper [1].**\n- **There are not too many explanation methods that can be used.**\n    - The provided reference **MAGE is not comparable to instance-level methods (our work)**, as it is **model-level**. MAGE does not generate subgraphs but identifies key motifs **across the whole dataset** for a prediction class. \n    -  The other reference, SubgraphX, is older than some baselines like PGExplainer and LRI. **Nevertheless, we have included its results (see \"Methods and Evaluation Criteria\")**. \n\n\n\n## Methods And Evaluation Criteria\n> Not a fundamentally new method\n- Our method is **not just a reformulation** of existing methods. \n- **Eq. (1) defines a well-established general framework for subgraph extraction.** Many works [1-4] reformulate Eq. (1) to Eq. (4) for gradient-based optimization with budget and sparsity constraints. \n- We reformulate Eq. (1) to Eq. (5) using **directed proximity graphs (DPG) tailored for 3D graphs**. **This is the first and only reformulation to 3D GNNs that is directly optimizable using radii of influence. This is fundamentally different from Eq. (4), which requires additional constraints to address issues from soft mask relaxation.** \n\n> Comparison with subgraph-level explainability baselines\n- **The provided reference may not be directly comparable to our method.** \n    - **MAGE is a model-level explanation method** that identifies the most influential motifs **across the entire dataset** for a given class label. Our method is instance-level, identifying **a subgraph for each graph**. \n    - SubgraphX searches through various node combinations, and the search space for hard mask optimization scales exponentially with the number of nodes. **Our method and all baselines relax hard masks to enable efficient gradient-based optimization.** Consequently, **SubgraphX is 5× slower than our method on QM9, with an even larger gap for larger molecules**. \n\n- Nevertheless, SubgraphX is still instance-level; we compare it with RISE. **The results below clearly show that RISE outperforms SubgraphX** (metric is MAE, lower, better). We present results on the first 1,000 molecules from the QM9 test data due to time constraints. We will include the full results in the revision if you feel necessary.\n    - **Beyond time complexity, SubgraphX shares the same limitation as all other node-masking methods (see Appendix C).** Its exponential scaling prevents extension to edge masking.\n\n|Method|μ(0.3)|μ(0.4)|μ(0.5)|α(0.3)|α(0.4)|α(0.5)|\n|-|-|-|-|-|-|-|\n|SubgraphX|10.844|10.528|10.354|2.550|2.632|2.816|\n|RISE|0.482|0.320|0.134|1.983|1.311|0.525|\n\n|Method|HOMO(0.3)|HOMO(0.4)|HOMO(0.5)|LUMO(0.3)|LUMO(0.4)|LUMO(0.5)|\n|-|-|-|-|-|-|-|\n|SubgraphX|0.387|0.282|0.272|0.610|0.557|0.540|\n|RISE|0.375|0.190|0.095|0.474|0.251|0.114|\n\n> RISE extracts meaningful results\n- **We have already provided several qualitative results in Appendix D**, as discussed in the last paragraph of Sec. 4.2. If you believe it is necessary, we will include more in the next revision. \n\n\n## Experimental Designs Or Analyses\n> Different distance-based formulations\n- Our work is motivated by the finding that distance is a key factor in message passing for 3D geometric graphs. **No existing explanation method uses a distance-based formulation**.\n\n- That said, we are unsure if you suggest designing alternative distance-based formulations for comparison with RISE. **Our work is among the first to explore 3D GNN explanation. RISE is currently the best method we can propose based on our insights. Our findings can inspire the community to develop more geometry-oriented subgraph extraction methods.** \n\n\n\n## Essential References Not Discussed\n> Comparison with the provided reference\n- **The scope of the provided reference [5] is very different from our work.**\n    - **[5] does not explain 3D geometric graphs but rather 2D graphs.**\n        - \"*Molecular graphs were generated from the SMILES strings of the molecules*\" (p2, 1st paragraph under Methods, 2nd last sentence)\n    - While they also use QM9, their focus is on explaining models predicting the X-ray absorption spectrum (XAS). In contrast, we propose a general method for 3D GNN explanation and evaluate it across multiple quantum-level properties. \n\n\n[1]Interpretable Geometric Deep Learning via Learnable Randomness Injection\n\n[2]Parameterized explainer for graph neural network\n\n[3]GNES: Learning to explain graph neural networks\n\n[4]Stratified GNN Explanations through Sufficient Expansion\n\n[5]Integrating Explainability into Graph Neural Network Models for the Prediction of X-ray Absorption Spectra", "claims_and_evidence": "1. Claim: **Unlike traditional 2D molecular graphs, 3D molecular graphs introduce implicit dense edges based on spatial proximity, making existing explainability methods ineffective.** However, the experiments in this paper show that RISE doesn't significantly outperform GNNExplainer and PGExplainer. Since both baselines are very old, the motivation of this paper -- existing explainability methods ineffective on 3D graph, doesn't seem to be valid.", "methods_and_evaluation_criteria": "1. While the idea of using spatial relationships in 3D GNN explanations is valuable, the approach is mainly a reformulation of standard explainability techniques rather than a fundamentally new method. \n2. RISE is compared against GNNExplainer, PGExplainer, and LRI-Bernoulli. The comparisons are reasonable, but the lack of additional subgraph-level explainability baselines limits the scope of the evaluation since this is a subgraph-based method. (For example, SubgraphX, MAGE)\n3. Unlike other methods that produce random subgraphs, RISE extracts explanations that correspond to actual chemical bonds, making them useful for domain experts. But authors are encouraged to provide results showing that by comparing with the existing method.", "theoretical_claims": "No", "experimental_designs_or_analyses": "1. The paper demonstrates that distance is a key factor in message-passing but does not explore alternative explanation strategies or compare different distance-based formulations. \n2. Experiment is not thorough. See \"Claims And Evidence\".", "relation_to_broader_scientific_literature": "Recognizing Learning Differences in 3D GNNs: Rather than treating graphs as fixed node-edge structures, RISE treats them as node-radius systems, which better aligns with molecular interactions.", "essential_references_not_discussed": "[1] Integrating Explainability into Graph Neural Network Models for the Prediction of X-ray Absorption Spectra, 2023\n\nThis paper also evaluated on QM9. How do you compare your method with it?", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6KePlpJCxV", "reviewer": "Reviewer_4kQJ", "summary": "This paper proposes a novel explanation method that localizes interpretability within each node’s immediate 3D neighborhood. By defining a \"radius of influence,\" the approach constrains message passing to spatially and structurally relevant subgraphs. This enhances interpretability while maintaining alignment with the physical dependencies in molecular applications.\n\n## update after rebuttal\nThe author's response has resolved many of my questions, and I will increase the score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your valuable feedback! We provid our responses below.\n## Experimental Designs Or Analyses\n> Comparison with baselines on 2D GNNs\n- **First of all, we want to emphasize that our work is not just an extension of its 2D counterparts.** Our work reformulates 3D graphs as directed proximity graphs (DPG), and based on this formulation, we determine the radii of influence. This is a principled approach specifically designed for 3D GNNs based on their unique characteristics. \n- We are not sure whether you are suggesting that we compare our method with 2D explanation baselines, even though they are designed for 2D GNNs, or that we evaluate our method on 2D GNNs. \n- **If you are suggesting a comparison with baselines for 2D GNN explanation, we have already included them in our work.** GNNExplainer and PGExplainer are the two franchise works for 2D GNN explanation. We have also compared our method with LRI, the only available explanation method designed for 3D GNNs. \n- **If you are suggesting an evaluation on 2D GNNs, our method is not applicable to them, as it explicitly considers the properties of 3D geometric graphs and 3D GNNs.** 2D topological GNNs cannot be formulated as directed proximity graphs, and there is no concept of distance or radius in 2D topological graphs. Our method aims to find the radii of influence for 3D geometric GNNs.\n## Other Comments Or Suggestions\n> Missing bolded 'Results'\n- Thanks for pointing this out! We will revise it in the next version.", "claims_and_evidence": "This paper provides a detailed analysis of the differences between 2D and 3D models, proposes a graph explanation method for the 3D domain, and demonstrates its superiority through experiments.", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "I am so sorry that I am not an expert in this field. Based on the reasoning from 2D to 3D, these formulas might be correct, but I am not entirely sure.", "experimental_designs_or_analyses": "The experiments compare the proposed RISE with some 3D baselines. However, from the paper, it can be seen that the proposed 3D model is an extension of the 2D model. Shouldn't an experiment be designed to compare the differences between the 2D and 3D models and examine the advantages of the 3D model in terms of interpretability?", "relation_to_broader_scientific_literature": "Not found", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "None", "other_comments_or_suggestions": "Section 4.1 is missing the bolded Results.", "questions_for_authors": "None", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YODlE3m5wK", "reviewer": "Reviewer_JV7x", "summary": "This contribution introduces RISE (Radius of Influence based Subgraph Extraction), an innovative explanatory approach for 3D geometric Graph Neural Networks (GNNs) in molecular learning. \n\nRISE's principal contribution is the allocation of a \"radius of influence\" to each atom (node). This delineates the confined area where message passing encapsulates the most significant spatial and structural interactions for the model's predictions. RISE reconfigures 3D networks into directed proximity graphs (DPGs), whereby each node possesses a designated radius that governs the establishment of edges. By optimizing radii of impact, RISE extracts subgraphs that accurately reflect the model's predictions and are chemically interpretable.\n\nThe submission evaluates RISE across several 3D GNN architectures (SchNet, DimeNet, and SEGNN) and datasets (QM9 and GEOM), demonstrating its constant superiority over established explanation techniques, such as GNNExplainer, PGExplainer, and LRI-Bernoulli. Significantly, RISE generates chemically interpretable explanatory subgraphs that correspond with genuine chemical linkages, whereas current methodologies frequently give chemically uninterpretable outcomes.\n\n## Update after rebuttal\nThe authors provided detailed discussions to address my concerns. I appreciate their effort and would like to retain my original score: accept this submission.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you so much for your efforts reviewing our work. We respond to your questions below.\n## Questions For Authors:\n> Extension to larger systems with long-range interactions\n- Even in larger atomic systems, short-range interactions typically dominate chemical bonding and molecular stability. Covalent bonds, hydrogen bonds, and van der Waals forces are strongest at short distances, meaning **our method remains highly applicable and effective in most cases**. \n- However, there are scenarios where long-range interactions play a crucial role. For example, in proteins, tertiary and quaternary structures depend heavily on long-range interactions such as salt bridges, hydrophobic packing, and π-π stacking. **We have acknowledged this in the discussion of future work in Sec. 5.** \n- **Our approach also has the potential to be extended to such scenarios.** Instead of using a single value as the radius mask, we could allow multiple radii of influence, represented as annular regions that collectively adhere to a predefined budget. For example, for a given atom, the model could identify edges with distances in the ranges [0.0, 0.3] and [0.6, 0.8] as important. This extension aligns well with chemical principles, as different types of interactions—such as covalent bonding, van der Waals forces, and electrostatic interactions—tend to dominate at specific distance ranges. \n\n> Attention-based explanation methods\n- There are some attention-based methods in the literature, with Graph Attention Networks (GAT) being a representative. **Attention-based methods are ante-hoc**, e.g. [1]. Ante-hoc methods are built into the model itself, making it inherently interpretable (like based on evolving attention scores) while **our work's focus is on post-hoc explanation of any existing 3D GNNs** that use radius cut-off graphs. The scope of our work and that of attention-based explanation are different, so there is not a direct comparison.\n- In terms of incorporating insights from attention mechanisms, attention-based GNNs (e.g. GAT) provide explicit edge importance scores for each node with respect to all other nodes. This is different from subgraph extraction, where we try to find the important edges (substructures) with respect to the entire graph. Therefore, we think that attention mechanism might be incorporated into RISE as a joint optimization process. We jointly optimize attention weights and explanation masks, where the explanation seeks to refine or filter the attention. **Although this is a very interesting perspective, it is out of the scope of our current work.**\n\n> Small perturbations in molecular geometry\n- This is a good point! We did not test RISE under small perturbations in molecular geometry in the original version, because **Small perturbations will alter the ground truth values of chemical properties, making evaluation intractable.** Since the exact chemical property values after perturbation are unknown (intractable for us to compute each time a small perturbation is introduced), it is not possible to faithfully assess or compare different explanation methods in such cases. \n- Assuming minor perturbations in atomic positions do not alter chemical properties, we conducted a robustness test on the same 4 molecules as in the qualitative results in Appendix D (discussed in the last paragraph of Sec. 4.2). Results indicate that when adding normal Gaussian noise to a small fraction (1/5) of atoms, RISE maintains explanation stability. While perturbations affect radii of influence, the binary edge mask remains unchanged, which means subgraphs extracted by RISE remain the same. **This demonstrates the robustness of RISE under small perturbations** (and for sure, again, if we assume minor perturbations in atomic positions do not alter chemical properties.)\n\n[1] Interpretable and Generalizable Graph Learning via Stochastic Attention Mechanism, Siqi Miao et al., ICML 2022", "claims_and_evidence": "The claims made in the submission are well supported by empirical evidence.\n\nClaim 1: Proximity (distance) determines the importance of message passing in 3D GNNs. To support this claim, the author conducted two experiments: The first one is removing edges from different distance-based annular bins and measuring the model performance. Secondly, randomly removing edges within each annulus to assess the variance in importance. The results in Table 1 clearly show that \"removing closer annuli leads to a more significant drop in MAE, as values in most cells strictly decrease compared to the previous row.\" Figure 5 further demonstrates that \"edges at similar distances have comparable importance, as indicated by the relatively flat trend lines with minimal fluctuation.\"\n\nClaim 2: RISE outperforms existing explanation methods for 3D GNNs. This claim is supported by comprehensive experimental results across multiple backbone models (SchNet, DimeNet, and SEGNN) and datasets (QM9 and GEOM). Tables 2 and 3 show that \"RISE consistently outperforms all baselines across various budgets\" for most molecular properties and model configurations. The authors even note that they set \"the baseline models' budgets... such that they strictly preserve more edges than RISE, making the comparison even more advantageous for the baselines,\" which strengthens the credibility of their results.\n\nThe third major claim is that RISE produces chemically interpretable explanatory subgraphs, unlike existing methods. This claim is supported by visualizations in Figure 4, which clearly show that \"only RISE yields chemically interpretable results that conform to interpretable chemical structures.\" The authors provide a concrete example with the ethane molecule (CH₃CH₃), where \"the radii of influence from our experiments assign the C of interest with a radius of 1.532 and the H of interest with a radius of 1.171,\" allowing RISE to extract the chemical bonds precisely.", "methods_and_evaluation_criteria": "The methodological approaches and assessment criteria employed in this submission are well-suited and thoughtfully constructed.\n\nThe submission formulates the instance-level graph explanation, described as identifying \"a subgraph $G^S \\subseteq G$ that is important to the target $Y$.\" The authors effectively highlight the shortcomings of current techniques when implemented with 3D GNNs.\n\nThe RISE methodology demonstrates considerable theoretical rigor. By recasting 3D graphs as directed proximity graphs (DPGs), where \"each node $v_i$ has an associated radius $0 \\leq r_i$, and a directed edge $e_{i\\rightarrow j}$ exists if and only if $d_{ij} < r_i$, where $d_{ij}$ is the distance between $v_i$ and $v_j$,\" the authors create a framework that corresponds well with established physical principles of molecular interactions. This approach acknowledges that \"interactions between nodes separated by large distances are typically negligible due to the rapid decay of force magnitudes.\"\n\nThe evaluation strategy is appropriate and direct. The use of MAE as the primary performance indicator offers a straightforward measurement of how effectively the explanatory subgraph maintains the model's predictive accuracy.\n\nVisual representations of the explanatory subgraphs offer compelling qualitative evidence of RISE's capacity to generate chemically meaningful interpretations. The researchers demonstrate that \"under a small budget, when explanation methods can preserve only a limited number of edges, RISE is the only method that selectively retains edges corresponding exclusively to chemical bonds,\" representing a significant advancement for molecular science applications.", "theoretical_claims": "The theoretical foundations presented in this submission regarding RISE demonstrate logical consistency.\n\nThe introduction of directed proximity graphs (DPGs) as a mathematical extension of 3D graphs presents a rigorous framework. This formulation establishes a suitable mathematical structure for representing molecular graphs with variable influence radii.\n\nThe authors' assertion that \"any 3D graph constructed based on node radii can be viewed as a directed proximity graph\" and that \"a 3D graph constructed based on a cut-off distance (the same radius for all nodes) is a proximity graph under uniform radii\" (page 5) follows directly from the DPG definition and exhibits proper mathematical derivation.\n\nThe paper establishes that RISE maintains \"consistency in optimization, as it does not require converting discrete values into continuous ones for optimization and does not include penalty or regularization terms to promote discreteness or enforce the budget\" (page 5,6). This claim receives adequate support through the formulation of RISE, which optimizes continuous influence radii directly rather than relaxing binary masks to continuous values.\n\nThe bound provided to illustrate inconsistencies in existing methods, \"$L(Y; \\Phi(G^S)) \\leq L(Y; \\Phi(X, M_{soft} \\odot A)) + L(\\Phi(X, M_{soft} \\odot A); \\Phi(X, M \\odot A))$\" (page 4), correctly identifies the optimization objective discrepancy that remains unaddressed during the optimization phase in soft mask-dependent methods.", "experimental_designs_or_analyses": "The experimental methodology and analytical procedures presented in this submission are sufficient to demonstrate the effectiveness of RISE. \n\nTwo complementary approaches effectively examine the relationship between proximity and message passing significance in 3D. This experimental protocol appropriately isolates distance effects while maintaining control over confounding variables, thereby testing the proximity-importance hypothesis with methodological rigor.\n\nThe experimental outcomes, presented in Table 1 and Figure 5, demonstrate that \"removing closer annuli leads to a more significant drop in MAE\" (page 7) and that \"edges at similar distances have comparable importance\" (page 7). These findings provide substantial empirical support for the theoretical basis of RISE and its radius-of-influence conceptualization.\n\nMethodological fairness is ensured by configuring \"the baseline models' budgets... such that they strictly preserve more edges than RISE\" (page 7), thereby introducing a conservative bias against the proposed method. This experimental constraint strengthens the validity of the comparative analysis by demonstrating RISE's superior performance despite operational disadvantages in edge preservation capacity.\n\nThe visualization analysis of explanatory subgraphs in Figure 4 offers qualitative evidence supporting RISE's capacity to generate chemically meaningful interpretations. The authors observe that \"under a small budget, when explanation methods can preserve only a limited number of edges, RISE is the only method that selectively retains edges corresponding exclusively to chemical bonds\" (page 8). This observation underscores the practical relevance of RISE for molecular science applications.\n\nThe experimental methodology and analytical procedures presented in this manuscript provide compelling evidence for RISE's effectiveness in explaining 3D molecular GNNs through rigorously designed and systematically executed evaluations.", "relation_to_broader_scientific_literature": "This submission effectively positions its contributions within the domains of molecular graph learning, graph neural networks, and explainable artificial intelligence scholarship.\n\nBuilding on established literature concerning the limitations of molecular representations, the authors articulate that \"chemical behaviors and biological functions of molecules are largely determined by their 3D geometric structures\" (page 1) while simultaneously recognizing contemporary advancements in three-dimensional graph learning methodologies.\n\nThe work extends the foundations established by previous explanation frameworks, such as GNNExplainer and PGExplainer. The authors identify a substantial methodological gap in the current literature, noting that \"existing approaches struggle to effectively explain 3D GNNs\" (page 2). They appropriately acknowledge the previously published research specifically addressing 3D GNN explanation while critically assessing its methodological constraints and limitations.", "essential_references_not_discussed": "N.A.", "other_strengths_and_weaknesses": "N.A.", "other_comments_or_suggestions": "N.A.", "questions_for_authors": "1. The paper focuses on relatively small molecular graphs. How would RISE scale to larger molecular systems that would exhibit long-range interaction?\n2. How does RISE compare with attention-based explanation methods that might be adapted for 3D GNNs? Are there insights from attention mechanisms that could be incorporated into RISE?\n3. Have you investigated the robustness of RISE explanations to small perturbations in molecular geometry?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HEtepvJJXN", "reviewer": "Reviewer_FVPY", "summary": "The paper proposes a new instance-level method to “explain” the predictions of 3D molecular GNNs. Following earlier work, this is formulated as an optimization problem over subgraphs, where the objective is to minimize the loss of predictive power when removing edges, under a given budget of edges. The author's main observation is that the edges between close-by atoms generally should affect the performance of 3D GNNs more. They support this with experiments by deleting random edges of a certain distance and showing a stronger drop in performance the shorter the deleted edges. Based on this they formulate the optimization problem using a “radius of influence”, which is a radius around each atom beyond which the edges are being cut. This contrasts with previous literature, that optimizes soft masks without any distance-based inductive bias.\nThe authors highlight two major benefits: 1) The radius of influence is smooth and therefore optimizable with gradient-based methods, which is claimed to eliminate the need for thresholding soft masks into hard masks. 2) The budget can be enforced strictly without relying on penalty terms.\nThe authors test the method by measuring the drop in predictive power when a certain percent of edges are removed and show that their method performs best compared to previous methods. They also claim that their method is the only one that preserves known important features such as covalent bonds.\n\n\n### update after rebuttal\n\nThank you for the clarifications. I think the experiments are well conducted, but I am still unconvinced as to what the usefulness exactly is. The authors say, “ The main purpose is not to identify novel chemical interactions unknown to chemists, but rather to understand how ML models make decisions. This is particularly important for scientific applications, where explainability is crucial for domain scientists to trust ML models.” Unfortunately, at the current point in time, I do not trust the ML models more based on the explanations given by RISE. Showing that the model is most sensitive to covalent bonds is expected by the design of the networks, which have a strong near-sightedness bias built in.\n\nHowever, I can imagine that my opinion would change if the authors could present less trivial cases where the method can pick up known non-trivial interactions, which would make me believe that the model learns physical many-body interactions. Another convincing experiment would be if the method can explain a negative case, so a case where the ML model fails, and the explainer shows that the model pays too much attention to some unnecessary/unphysical edge. This way the absence of weird edges could make me trust the model more. It would also expand the usefulness of the method to let practitioners find failure points and correct them in a targeted fashion.\n\nThe idea of the annuli for long-range interactions makes sense but needs to be proven. Again, I think this has a lot of potential to be impactful, but at the current point I am not convinced enough and therefore keeping my score.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you so much for your detailed and constructive comments! We provide our responses here.\n\n## Claims And Evidence\n\n> Interpretability of explanation\n- The radii of influence can be intepreted as **the spatial extent within which an atom can significantly affect its surroundings**.\n\n> Additional edges by other approaches due to the bigger budget\n- There is a misunderstanding here. **The budget is exactly the same for all methods.** This is stated in the caption of Fig. 8. We will include clearer statements in the revision to further emphasize this. \n- In terms of interpretability, **the edges found by all explanation methods are directed as in their original works**. RISE successfully finds both directions of the covalent bonds, while other methods find one direction of the covalent bonds and some less interpretable edges. We will include this clarification in the revision.\n\n## Methods And Evaluation Criteria\n\n> Long-range interactions\n- The current form of our method is mainly for small molecules as we note in our discussion for future works in Sec. 5.  **Our approach also has the potential to be extended to scenarios with long-range interactions.** We provide some key insights below and will include such discussions in Appendix in the next revision.\n- We could allow multiple radii of influence for nodes, represented as annular regions that collectively adhere to a predefined budget. For example, for a given atom, the model could identify edges with distances in the ranges [0.0, 0.3] and [0.6, 0.8] as important. This extension aligns well with chemical principles, as different types of interactions—such as covalent bonding, van der Waals forces, and electrostatic interactions—tend to dominate at specific distance ranges. **As a result, to capture important long-range interactions, only relevant annuli will be included; it won't result in unnecessary edges.** \n\n## Theoretical Claims\n\n> Relaxation to Soft Masks\n- **There might be some misunderstanding here.** Indeed, both RISE and baseline methods relax the weights of edges from discrete values to continuous values. However, **there is a fundamental difference**. **The loss function of RISE is a function of radii; it can be extremely close to discrete values with the relaxation, e.g., Eq. (7). Similar approaches, e.g. applying Eq. (7) to edge/node masks, won't change the minima of the loss function of baseline methods, still resulting in mismatch.** \n- **RISE optimizes the radius masks, which are naturally continuous.** In Eq. (7), we reformulate the radius masks to be edge masks to allow gradient-based optimization. **As explained in sentences above Eq. (7), we can apply a function such that the edge masks will be extremely close to $0$ or $1$.** \n- Consider a node with three neighbors of distances $0.5$, $0.6$, and $0.7$, respectively. Suppose the optimized radius is $0.55$. **The resulting edge masks using Eq. (7) with $k=1,000$ are $1.00$, $1.93\\times 10^{-22}$, and $7.18\\times 10^{-66}$**, respectively . \n- Additionally, we present below the percentage of edge masks in different ranges resulting from real experiments on RISE and GNNExplainer, respectively. The closer the edge mask values are to $0$ or $1$, the smaller the mismatch between the mask and the binary optimization objective. **It can be seen clearly that the mismatch is almost negligible for RISE, but very severe for GNNExplainer.**\n|Edge Mask Value| GNNExplainer(%)|RISE(%)|\n|-|-|-|\n|[0, 0.1] and [0.9, 1.0]|0.003|99.314|\n|[0, 0.2] and [0.8, 1.0]|11.463|99.438|\n|[0, 0.3] and [0.7, 1.0]|54.508|99.741|\nExperiment constructed on the first 1000 molecules in QM9 test dataset for lumo using SchNet. On average, there are 314 edges per graph.\n## Essential References Not Discussed\n\n> Discussion of 3D GNNs\n- **We have provided a discussion of different 3D geometric GNNs, including the ones used in this paper, in Appendix E.** We will include more details, such as how messages are constructed, in the next revision.\n\n## Other Comments Or Suggestions:\n\n> The need of explanation\n- **In the introduction, we have briefly introduced the motivation and elaborated on the need of 3D GNN explanation methods.**\n-  We will include further discussion in the revision. **The main purpose is not to identify novel chemical interactions unknown to chemists, but rather to understand how ML models make decisions**. This is particularly important for scientific applications, where explainability is crucial for domain scientists to trust ML models.\n\n> The number of edges\n- We do not mean that the number of edges is $O(C^N)$ ($N$ is the number of nodes) but rather a term to describe the large number of edges 3D GNNs have. **We will change to the word \"rapidily\" in the next revision.**\n\n> Search spaces before introducing optimization\n- We will make necessary modifications.\n\n## Questions For Authors:\nThe questions have similar responses to the content of 'Other Comments Or Suggestions'. Please refer to the section above.", "claims_and_evidence": "The claim that atoms close to each other have a stronger influence is generally accepted and even enforced by 3D GNN architectures using a cutoff function that decreases a message's magnitude with distance. Since RISE is building on this observation it is generally a sensible idea, which is supported by the QM9 and GEOM experiments.  However, the central claim that the subgraphs found by RISE are more interpretable is less well supported, or even defined what is meant by interpretable. The biggest piece of evidence is given in figure 4 and 8, where RISE only leaves the covalent bonds, whereas the other methods also include other, less interpretable edges. However, all approaches include the covalent bonds, so the fact that there are additional edges found by the other approaches could also just be due to the bigger budget afforded to them.", "methods_and_evaluation_criteria": "The authors use the drop-off in QM9 and GEOM performance given a certain budget of edges as a figure of merit and show that their method retains better performance than previous methods. This shows that the subgraphs picked out by RISE are more predictive on average on the given datasets. However, it is not clear to me if this would still be the case for systems where long-range and non-local effects are important since RISE would be forced to generate very large radii of influence to find these interactions, thereby including many potentially unnecessary edges, whereas previous methods could find only a few important edges that mediate the effects.", "theoretical_claims": "The authors say that other methods have to threshold soft edges which induces a mismatch between optimization and resulting graphs, whereas RISE does not require this thresholding. As far as I understand RISE also includes a form of soft masks due to equation 7, such that there is still a mismatch in the mask and the optimization. \nThe other claim about exactly enforceable budgets seems solid though.", "experimental_designs_or_analyses": "Discussed above", "relation_to_broader_scientific_literature": "The paper is an adaptation of existing ideas to 3D GNNs", "essential_references_not_discussed": "The paper does a good job of discussing the most relevant explainability literature. I think a section explaining the key pieces of the used 3D GNNs, in particular how messages are constructed would be a good addition.", "other_strengths_and_weaknesses": "The paper is original and the presentation is good", "other_comments_or_suggestions": "In my eyes, the two most pressing suggestions is to contextualize the vision of the paper better. Why do we care about these explainers? Are there any unexpected explanations made by RISE that a chemist wouldn't have thought of easily? Show some example where RISE picks out a non trivial edge that correlates to interesting physics.\nThe paper mentions several times that the number of edges in a 3D GNN grows exponentially. I am not sure if I misunderstand what the authors are trying to say, however, the number of edges in a fully connected graph is quadratic in the number of nodes, not exponential.           \nAnother small suggestion to improve the flow of the paper is that the paper talks about search spaces (line 48) before explaining that these explanation methods are formulized as optimization problems. Adding a short sentence in the beginning to give this context would be helpful", "questions_for_authors": "How would RISE deal with long-range and non-local interactions?\nAre there any non-trivial explanations found by RISE beyond covalent bonds that can be seen as a known physical interaction?\nWhat is the overall vision with these explanation approaches? What are they useful for? What kind of actionable insights are we hoping to gain?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "N/A"}}]

Author rebuttal:

Meta-review: This paper introduces RISE, a method to explain the decision of 3D GNN by locating the subgraphs which their decision is based on. The reviewers are essentially split regarding this paper, with two reviewers supporting acceptance and two rejection. Overall, while the reviewers agree the paper is well executed, some of the reviewers are skeptical regarding (a) the importance of the problem in general and (b) the need for methods targeting 3D GNN specifically, in comparison with existing "2D GNN" explainability methods. The comparisons in the paper show some improvement but perhaps not substantial enough

I tend to support acceptance. Re (a) I know that chemist *are* interested in explainable models, and re (b) comparisons do show a consistent advantage in terms of accuracy. The proposed method seems to be the first explainability method proposed for 3D GNNs, so potentially this could be a nice contribution to the community.
Decision: Accept (poster)