TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration
Authors: Cheng Xin
Fan Xu
Xin Ding
Jie Gao
Jiaxin Ding
Venue: ICML 2025 poster (2025)
Keywords: XAI
XGNN
Graph Representation
Graph Neural Networks
Topological Data Analysis
TDA
Persistent Homology
Interpretable GNN
Explainable GNN
Topological Discrepancy
Primary area: deep_learning->graph_neural_networks
TL;DR: A topologically interpretable GNN with a novel topological discrepancy loss is proved to be uniquely optimized by ground truth.

Abstract: Graph Neural Networks (GNNs) have shown remarkable success across various scientific fields,
yet their adoption in critical decision-making is often hindered by a lack of interpretability. Recently,
intrinsic interpretable GNNs have been studied to provide insights into model predictions by identifying rationale substructures in graphs. However, existing methods face challenges when the underlying rationale subgraphs are complex and varied. In this work, we propose TopInG: Topologically Interpretable Graph Learning, a novel topological framework that leverages persistent homology to identify persistent rationale subgraphs. TopInG employs a rationale filtration learning approach to model an autoregressive generating process of rationale subgraphs, and introduces a self-adjusted topological constraint, termed topological discrepancy, to enforce a persistent topological distinction between rationale subgraphs and irrelevant counterparts. We provide theoretical guarantees that our loss function is uniquely optimized by the ground truth under specific conditions. Extensive experiments demonstrate TopInG's effectiveness in tackling key challenges, such as handling variform rationale subgraphs, balancing predictive performance with interpretability, and mitigating spurious correlations. Results show that our approach improves upon state-of-the-art
methods on both predictive accuracy and interpretation quality.

TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration
Cheng Xin
Fan Xu
Xin Ding²
Jie Gao
Jiaxin Ding²
Abstract
1. Introduction
Graph Neural Networks (GNNs) have emerged as pow- erful tool for learning graph-structured data in various sci- entific domains (including chemistry, biology, physics, and materials science), achieving remarkable success in ap- plications of predicting molecular properties (Kamberaj. 2022; Chen et al., 2023), modeling protein-protein interac- tions (Görmez et al., 2021; I; Ravichandran et al., 2024; Li et al., 2023), analyzing phase transitions (Qu et al., 2022), characterizing material characteristies (Hu & Latypov, 2024;
Graph Neural Networks (GNNs) have shown markable success across various scientific fields, yet their adoption in critical decision decision-making is often hindered by a lack of interpretability. Re- cently, intrinsically interpretable GNNs have been studied to provide insights into model predictions by identifying rationale substructures in graphs. However, existing methods face challenges when the underlying rationale subgraphs complex and varied. In this work, we propose TOPING: Topologically Interpretable Graph Learning, a novel topological framework that leverages per- sistent homology to identify persistent rationale subgraphs. TOPING employs rationale filtra- tion learning approach model autoregressive generation process of rationale subgraphs, and introduces a self-adjusted topological constraint, termed topological discrepancy. to enforce per- sistent topological distinction between rationale subgraphs and irrelevant counterparts. We pro- vide theoretical guarantees that our loss function is uniquely optimized by the ground truth un- der specific conditions. Extensive experiments demonstrate TOPING's effectiveness in tackling key challenges, such as handling variform ratio- nale subgraphs, balancing predictive performance with interpretability, and mitigating spurious cor- relations. Results show that our approach im- proves upon state-of-the-an methods on both pre- dictive accuracy and interpretation quality.
Sheriff et al., 2024; Gurniak et al., 2024; Xiao et al., 2024), etc. As GNNs are increasingly applied to critical scientific and decision seccisio--making tasks, there is growing need for inter- pretability and explainability in these models (Zhang et al., 2024a). Scientists and practitioners often ask for not only accurate predictions, but also insights into why and how these predictions are made. This is particularly crucial in scientific applications where understanding the underlying mechanisms and causal relationships is important the predictions themselves.
A recent trend in GNN research focuses on enhancing inter- pretability by developing methods that identify and visual- ize the nodes, edges, subgraphs, or features most influential or causal for given prediction. Existing approaches on GNN interpretation can be broadly categorized into two classes (Zhang al., 2024a): post-hoc explainer meth- ods (Ying et al., 2019; Luo et al., 2020; Schlichtkrull et al., 2021; Wu et al., 2023; Bui et al., 2024) and intrinsically interpretable models (Wu et al., 2022: Miao et al., 2022; Chen et al., 2024). Post-hoc explainer methods analyze a pre-trained GNN model to generate intuitive explanations. These methods enjoy flexibility and be integrated into different kinds of models. But they might provide explana- tions that are suboptimal or inconsistent with the model's actual decision socciiinnmmainng processes (Miao et al., 2022). On the other hand, intrinsically interpretable models incorpo- rate interpretability directly into the model architecture and training process. basic intrinsically interpretable GNN model is built upon the graph attention (Veličković et al., 2018) mechanism. But naïve application of attention weights does not give reliable interpretation for real graph data (Ying et al., 2019; Yu et al., 2021), as attention weights
'Equal contribution 'Department of Computer Science, Rut- gers University. Piscataway, NJ, USA "School of Computer Sci- ence, Shanghai Jiao Tong University, Shanghai, China. Corre- spondence to: Cheng Xin <cx122@nugeersedus edu>> Jiaxin Ding <jiaxingding @ draxiingdnggsjuu.edu... edju..dduecn>.
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration
may not always correlate with actual feature importance. Moreover, the trade-off of interpretability and predictive performance (Du et al., 2019) may not be acceptable in real- world applications. To address these challenges, Miao et al. (2022) proposed a stochastic attention mechanism (GSAT) to use the graph information bottleneck (Wu et al., 2020; Tishby et al., 1999) as the target function, employ attention weights to control the information bottleneck, and sample rationale subgraphs using Gumbel-soffuax reparameteriza- tion. Similarly, Chen et al. (2024) approached interpretation by searching for rationale subgraphs within the framework of subgraph multilinear extension (SubMT) and proposing a graph multilinear net (GMT) for better SubMT approxi- mation. Wu et al. (2022) proposed Discovering Invariant Rationales (DIR), applying interventions on training distri- butions to obtain invariant causal rationales while filtering out spurious correlations.
2021; 2022; Zhao et al., 2020; Immonen al., 2023; Ye et al., 2023; Swenson et al., 2020). Persistent homology studies the dynamics of topological invariants over vari- ous scales through filtrations, allowing us to capture all the changes and persistence of topological features in the data.
Based on this foundation, we introduce a new perspective on the rationale subgraph identification problem. We model the graph attention mechanism as an underlying graph genera- tion process, which ideally constructs the rationale subgraph first, followed by the addition of auxiliary structures. We use persistent homology tools to capture and track the repre- sentations and life cycles of topological features during the generating process. To effectively distinguish the rationale subgraph from the complement subgraph, we optimize the parameterized generation procedure to enhance the stability of the rationale subgraph. Specifically, our goal is to amplify the topological differences between the rationale subgraph and the complement subgraph, creating persistent gap in their topological features throughout the generation process, To achieve this goal, we propose a novel self-adjusting topo- logical constraint, topological discrepancy, which measures the statistical difference between two graphs with respect to their topological structures. Topological discrepancy serves as metric to quantify how well the rationale subgraph is preserved and distinguished from the complement subgraph during the filtration process. We also provide a tractable approximation of topological discrepancy and provide theo- retical guarantees that our models are able to achieve ground truth as the unique optimal solution under our loss function.
Despite these advancements, existing intrinsic methods of- ten assume either explicitly or implicitly that the subgraph rationales are nearly invariant across different instances within the same category of graphs, even a strong one-to- one correspondence between subgraph rationales and pre- dictions. However, this is overly restrictive and unrealistic in many real- world scenarios, where the graph dataset and the downstream tasks exhibit variform subgraph rationales, which can vary significantly in form, size, and topology. even among graphs of the same category. For example, in molecular biology, molecules with the same bioactivity can have different functional groups responsible for that actiy- ity (Patani& & Voie, 1996; Brown, 2012). An aromatic ring, sulfonamide group, or heterocyclic compound can each be the key substructure leading to the same pharmacological effect in different molecules. In social networks, the struc- tural reasons for a user to be influential vary significantly. An influential user might have high degree, high between- ness centrality, or serving as crucial bridge nodes connecting different communities. Our observations, supported by ex- periments on synthetic dataset (see Figure 3 for the results and Appendix E for the dataset construction), show that ex- isting intrinsically interpretable models struggle with such variability. Models obtained under these assumptions may fail to accurately capture the true causal mechanisms under- lying the predictions, resulting in unreliable interpretations and suboptimal generalization performance.
Our main contributions can be summarized as follows:
We introduce TOPING, a novel intrinsically inter- pretable GNN framework that incorporates topologiaal data analysis to identify stable rational subgraphs via persistent rationale filtration learning. We propose new loss function, topological discrepancy, to mea- sure the statistical difference between two graphs with respect to their topological structures.
We provide a tractable approximation of our topolog- ical discrepancy and provide theoretical guarantees that models able achieve ground truth as the unique optimal solution under our loss function. This establishes solid theoretical foundation for the effectiveness of our approach.
To address the above challenges, we propose Topologically Interpretable Graph Learning (TOPING), novel topologi- cal approach to intrinsically interpretable GNNs that lever- ages techniques from topological data analysis to identify stable and persistent rationale subgraphs, effectively han- dling the variability in subgraph structures. Our method is inspired by the concept of persistent homology, originating from algebraic topology and recently applied to data analy- sis and machine learning (Wong& & Vong, 2021; Yan et al.,
We empirically demonstrate that TOPING improves existing methods in both prediction and interpretation tasks multiple benchmark datasets. Additionally, we created a synthetic dataset with variform rationale subgraphs to specifically target challenges faced by previous methods. Our results show that TOPING ef- fectively handles such variability, confirming its ability to address this critical challenge.
2

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TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration
pprior Reg, by petar
U
GNN
Graph Feature
MLP
Lee
Pool
Sampling
G₂
House or Grid?
Input Graph
Filtration Learning
0.999
Topological Feature
Ho
Hg
0.37
G
MLP
H
Hi
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G,
T
L topo (Tx).. (T2))
rett reft We li Q11
Hy
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Figure 1. An overview of TOPING, A GNN parameterized by is used to learn filtration, from which sample the subgraphs G.x and G.. These subgraphs yield topological features through their respective filtrations F. Meanwhile, the combined subgraph Gx L G. processed by the same GNN (sharing parameters with fo) produce graph feature. Finally, the topological features T (which capture global structural information) are combined with the graph feature to form the final graph representation for classification tasks.
The concept of rationalization has also been extensively studied in Natural Language Processing (NLP), where researchers identify text spans as rationales for predic- tions (Yao et al., 2023; 2024; Gurrapu et al., 2023; Liu et al., 2023; 2024; 2025). However, unlike NLP domains where ra- tionales are typically contiguous text spans, graph domains face the challenge with variform rationale subgraphs that vary significantly in size, form, and topology.
sume each graph G' is composed of two edge disjoint sub- graphs G = Gx L G. with vertex correspondence for some Gx E 9x and G, E Ge. 9x and Ge are two families of graphs. 9x is usually a small finite set. Given graph G, Gx S the rationale subgraph that determines the label yG, for some unknown oracle h 9 - [0, 1]. G. is the noisy less relevant part of the graph. Both Gx and G. are unknown, and they have to be learned from the data. The goal is to predict the label ÚG for each graph G and simultaneously identify its rationale subgraphs Gx.
2. Preliminaries
2.1. Graph Neural Networks (GNNs)
2.3. Topological Data Analysis
Graph neural networks are class of neural networks de- signed to operate on graph-structured data. typical message-passing GNN layer updates node representations by aggregating information from neighboring nodes:
Topological Data Analysis (TDA) has emerged as power- ful analytical framework across diverse domains, including machine learning, artificial intelligence, and computational neuroscience. Within graph representation learning specif- ically, TDA has demonstrated significant capacity en- hance GNNs through the systematic incorporation of topo- logical features (Hofer et al., 2017; 2019; 2020; Dehmamy et al., 2019; Carrière et al., 2020; Horn et al., 2022: Zhao et al., 2020; Carrière & Blumberg, 2020; Zhang et al., 2022; 2024b; Yan et al., 2022: Xin et al., 2023; Mukherjee et al., 2024). particularly effective methodological tool in this context persistent homology, which provides a rigorous mathematical framework for analyzing the evolution of topological
((++)) = (ht)). AGG N(t)))
(1)
where 11 is the message representation of node u at layer Nvv) the neighborhood of v, AGG permutation invariant aggregation function, e.g.: sum, mean, max, and & non-linear activation function. Some commonly used GNN architectures include Graph Convolutional Networks (GCN) (Kipf & Welling, 2017), Graph Isomorphism Net- works (GIN) (Xu et al., 2019), Graph Attention Networks (GAT) (Veličković et al., 2018).
such connected components and cycles-throughe throughout graph's construction process. This approach enables the quantitative analysis of structural pat- terns by examining their emergence and persistence across parameterized filtration of the graph, effectively capturing multi-scall topological information that can be integrated into learning framework. We give brief introduction
2.2. Intrinsically Interpretable Graph Learning
Intrinsically interpretable graph learning aims to build model simultaneously targeting both prediction and inter- pretability during the training procedure. Formally, given collection of labeled graphs (g, Y) = {(G,yG)), as-
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TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration
(0))0.000
Ieei- 550
NoP-0.975
Nc/-0.974
1eel-0.22
110000220
H
1.0
0.0
0.974
Persistent Homology Gap 0.65
0.325
G
Topological Discrepancy
0.226
-0.0
0.974
20.975
0.990
Figure 2. The top row is a learned graph filtration an example graph through our method. Red and yellow points correspond to ground truth rationale subgraph G and noisy subgraph G. respectively, Each snapshot is a subgraph of G on edges with filtration values greater than decreasing threshold value indicated on top of each snapshot. Below each subgraph, the number in small box is the size of cycle hasis for the current subgraph, which equals to the dimension of the cycle space, also known as the 1st betti number. The bottom part shows the 1st persistent homology. Each horizontal bar corresponds to topological feature (basic cycle).
the basic concepts of topological data analysis (TDA) and persistent homology. For a more detailed introduction, we refer readers (Edelsbrunner & Harer, 2010; Dey & Wang, 2022).
zero vector in H1 is the empty cycle. This vector space is also called the cycle space and its basis is known as cycle basis, which is well studied in graph theory (Horton, 1987; Kavitha et al., 2009; Jungnickel, 2007). For example, for graph G = (V, E = 0) with edges, the 0-th homology Ho(G) {0, 1 is |VI-dimensional vector space with basis being the set of all isolated vertices, while H1 (G) = 0 is trivial since there are no cycles.
Graph Filtration: For an edge-weighted graph G' (V, E. : E + R), we can create a sequence of nested subgraphs called a graph filtration. For example, assume the edge weight f is normalized, fEE S [0, 1], and repre- sents some "importance scores" of edges. We construct a graph filtration F(G)= == {G<r t 1 }EE) where Gstt is the subgraph on edges with - f(e S t. Essentially, such graph filtration shows how the graph grows edges are included in the decreasing order of importance scores.
Persistent Homology: Starting from graph filtration F(G) given by the reversed order of edge weights:
F(G) : ... S Gggh S Gg1 S GGss S...
if we apply the p-homology functor Hp to the graph fil- tration, each subgraph Gst is mapped to p-th homology vector space (G Each inclusion G S Gev natu- rally induces a linear map Hp(Gst) + HG this way, we get chain of homology vector spaces (FG)) connected by linear maps:
On a graph filtration, one can track all the connected compo- nents and cycles appearing and disappearing (merged with others) during the process. If we encode the lifecycle (birth, death) of each component cycle as an interval on the real line, it turns that there is an essentially unique way to represent such information is topologically stable and equivalent to well-studied alge- braic structure studied in topological data analysis, known as persistent homology. To understand this formally, we first introduce homology vector spaces and then build up to persistent homology.
multi-set of intervals which
0 + HH(Ggn) - Hp(G - .... H,(G).
Such structure Hz(F(G)) is called the p-th persistent ho- mology of the graph filtration F(G). In this work, we only focus on persistent homologies with p = 0, 1, which respec- tively correspond to the lifecycles of connected components and cycle bases of the graph filtration. This algebraic struc- ture is in fact graded module over the polynomial ring F2[4] [4) (Zomorodian & Carlsson, 2004). By the structure the- orem of finitely generated modules over a principal ideal domain, (F))) can be uniquely decomposed into a di- rect sum of cyclic modules. Each indecomposable cyclic module determined by a pair of numbers (11, f2), which
Homology Space:
For given graph G, consider homol- ogy vector spaces over the finite field F2 {0, 1} (homol- ogy spaces in brief). The 0-th homology (G) is defined as the vector space with basis corresponding to all connected components of G. The 1-st homology H, (G') is the vector space defined on the set of all cycles in G, with addition operation defined as the symmetric difference of cycles. The

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TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration
essentially corresponds to the lifecycle of one persistent topological feature. The multiset of all such pairs is called the persistence diagram of the graph filtration, or equiva- lently, it can be viewed as multiset of intervals, which is known as the persistent barcode of the graph filtration.
To address this, we propose to learn a rationale filtration, which represents the importance ordering of edges in process that simulates an autoregressive generation. By learning to prioritize the edges of the rationale, this approach enables us to identify stable and persistent substructures that are most relevant for predictions. We target a generating process for G = =V,, that ideally generates the critical rationale subgraph G first, followed by a less relevant counterpart G² as the complement.
The persistent barcode (also equivalently the persistence diagram) is a complete discrete invariant of the persistent homology, which means it fully encodes all topological fea- tures represented by the persistent homology (Zomorodian & Carlsson, 2004). We refer readers to (Edelsbrunner & Harer, 2010; Zomorodian & Carlsson, 2004; Ghrist, 2008) for further details about persistent homology.
More precisely, we utilize backbone GNN as learn- able filtration functional fo G - [0.1] 1] that, for each graph, produces a filtration function & : E - [0, 1] map- ping edges to their importance score. These scores induce an importance ordering on the edges, and consequently, a graph filtration F(G) {Go, ,I...., G which is an increasing sequence of subgraphs constructed incrementally by adding edges in order of decreasing importance (more important edges introduced earlier in the sequence). By tradition, Go = 0 and GlEy = G. The objective is for the learned filtration functional fo to ideally produce a filtration function on edges whose induced importance ordering is to be consistent with the partitioning into Gi and G such that Ve E &G and e' G., f(e)= > [(e)..
Remark 2.1. In this work, one can treat the persistent homol- ogy as topologically stable representation on sequential input. The representation is differentiable and encodes the evolutions of topological features. See Figure 2 as an ex- ample of a graph filtration (cubes on the top line) and its persistent homology (the multiset of intervals shown as hor- izontal lines in the bottom). We will revisit this example later after our method is introduced.
Comparing Topological Features: One can compare two edge-weighte graphs by computing distances between the persistent homologies of their corresponding graph filtra- tions. A commonly used (pseudo-)metrie is the bottleneck distance de (Definition B.I in Appendix B). Recall that per- sistent homology be represented as a multiset of points in 2 The bottleneck distance is essentially a variant of the Wasserstein distance between these two multisets (persis- lence diagrams) under the -norm. A crucial property of the bottleneck distance is its stability: small perturbations in the input data (e.g., edge weights) lead to small changes in the bottleneck distance, ensuring robustness downstream applications (see Appendix B for more details). Our pro- posed topological discrepancy is constructed based on the bottleneck distance and its stability property.
For notational conciseness. we omit the super- and sub- scripts for f = f and F(G) = Fo when the context is clear. We denote F(Gge) to be the filtration of the sub- graphs on edges with filtration values greater than 1 - t. Symmetrically, let let F(Gz)) be the filtration of the subgraphs on edges with filtration values smaller than or equal to t.
Our approach is founded on ensuring the following property:
Persistent Homology Gap: There is significant difference between the topological features derived from two compo- nents of a graph's 'sph' filtration, F(G and F(Gze), which are well separated at some threshold value E [o, 1]. The persistent homologies computed from these respective parts serve as the topological invariants, denoted as Tx and Te.
3. Method of TOPING
Remark 3.1. The underlying idea of this property is that, when considering the generating process of rationale sub- graphs and their irreverent counterparts, their topological structures follow two distinct evolutionary paths. Our meth- ods are designed to statistically capture such a topologi- cal discrepancy. Figure 2 provides a visual illustration of our method's objective when successfully applied. A well- learned graph filtration (top row) assigns higher importance scores to edges in the ground-truth rationale subgraph G As a result, the 1st persistent homology barcode (bottom row), which tracks the lifecycles of basic cycles, clearly dis- tinguishes the topological features of the learned rationale Gx from those of the complement G.. The annotation "Per- sistent Homology Gap" in the figure exemplifies this desired clear separation, where features identified with Gx (long bars appearing earlier that persist significantly) are distinct
In the following context, for a given G, we denote the ora- cle rationale subgraph and its complement G3 and G respectively. We use Gx and G. to represent a candidate ra- tionale subgraph and its complement respectively predicted by our model.
In contrast to existing methods, we re-examine the prob- lem from a global perspective through the lens of topology. Our central hypothesis is that: if the prediction task de- termined by a core substructure G belonging to a small family of rationale graphs Gx, then the full graph G can be viewed as being "grown" from this core G by attach- ing auxiliary structure G The identification of such core rationale substructures is highly non-trin trivial, as it demands consistency across the generation process and the discovery of common topological patterns across the entire dataset.
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TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration
from features associated with G. (shorter bars introduced later with shorter persistence). Our topological discrepancy measure, introduced subsequently, is designed to quantify
diagrams, dreoo (P((x),, can be lower bounded by:
max NEW
EP (PP) (Q=F(T))((Q))
and optimize such distinguishing topological gap.
Learnable Vectorization for the Lower Bound To prac- tically compute the lower bound of ceepos we need a set of 1-Lipschitz continuous functions V = {3/1, 4/2, 2/,.... , that map persistence diagrams to a Fuclidean space where expectations are tractable. TDA offers well-studied vector- ization methods for this purpose. We adopt the learnable vectorization approach from (Hofer et al., 2019), which rep- resents persistence diagrams as k-dimensional vectors. This is achieved by learning k parameterized kernels, termed structure elements, to capture point distributions on the diagrams. These structure elements are designed to be Lips- chitz continuous with some constant C. Specifically, on a given persistence diagram p. we employ the Rational Hat structure clement with learnable center E IR² and radius E R, which is defined as:
We denote the induced probability distributions of persistent homologies as P(Tx) and P(T.)) respectively.
Definition 3.2 (Topological Discrepancy). The topologcaa discrepancy dopo between P(Tx) and IP(T)) is defined as
dupo (T(x)),((T)) A TEI(((()))))(T)))
inf
(dp.))=lld(P.... Q1)
where ((P(Tx),PPT.)) is the set of all couplings between P(Tx) and P(T.). and dg(P, Q) is the bottleneck distance (Definition B.1 in Appendix B) between the persistent ho- mologies and Q
Essentially, depp is the 1-Wasserstein distance between the distributions of persistent homologies Tx and Te under the metric dg. Now we are ready to introduce the target loss of our model based on our topological discrepancy property.
sppp c,r)
Σ ØEP
1
cl22
1+
|r|
c||2
(())= = Eg (GG. - a (P((x),
(2)
By setting 3/ p, we get 1-Lipschitz continuous repre- sentation function as we want. The expectations (E({)()) (P)] are approximated by empirical means over the data in prac- tice. To select the maximum in the lower bound formulation, in our experiments, instead of a simple softmax, we utilize 2-head attention mechanism to identify and sum the top- 2 maxima from the k: vectorized representations. We use k = 8 Lipschitz continuous representation functions in our experiments. This learnable vectorization, combined with multi-head attention, not only provides an efficient approx- imation of dipp but also facilitates a self-adjusted focus on data-dependent topological features. This mechanism guides the model to learn the most task-relevant topolog- ical information, which we found empirically enhance training stability and improve performance. All topological representations are Lipschitz continuous and differentiable almost everywhere, enabling end-to-end training. Com- putations for these representations and their gradients are performed using the codebase from (Zhang et al., 2022).
The topological constraint term Lupp is realized by the topo- logical discrepancy dropo The prediction loss term Loe is the standard cross-entropy loss between the predicted label gG and the ground truth label ya. The predicted label ůo = hao (G') is obtained by applying prediction network ha on the subgraph Gx extracted through from the filtra- tion (G). hs and fo share the same backbone GN No model, which outputs a permutation equivalent representa- tion (node or edge representation). fo = LL, o GN No applys a simple multi-layer perceptron (MLP) model to get a 1-dimensional edge representation as the filtration function. he MLP,, Pool o GNN& first pools the permutation equivalent presentation of GNN, No to get permutation in- variant graph representation, then applys another MLP to get the final graph representation for predicting KG. Here we omit other details of learnable parameters in the MLP for simplicity. The persistent homologies Tx and T., as permutation invariant graph representations, are also used in the final representation M LP,, through combining with the graph representation Pool GN See Figure high-level illustration of the architecture of our model.
In the rest of the paper, use dhopo to denote the lower bound used in practice. Finally, we give the following theo- rem to show that our loss L(e) with duapo is guaranteed to be optimized by the ground truth.
3.1. Self-adjusted Topological Constraint
Theorem 3.4. Assume VG, |Ex| < [Eel, and Gix is min- imal with respect to yG in the sense that any subgraph Gx C G losses some information of label, then L(d) is uniquely optimized by (e(e) 1ffe E G
In this subsection, we will discuss the construction and prop- erties of our topological features in details. The original construction of depp is intractable in general. Here we pro- vide a tractable lower-bound through Kantorovich duality of Wasserstein distance (Villani, 2009) as follows:
Remark 3.5. Note that our guarantee does not depend on any stability or invariance assumptions on Gx, therefore, it will not be affected by variform rationale subgraphs in theory. The proof deferred to the appendix C.
Proposition 3.3. Given a set of L-Lipschitz continuous func- tions, v (1/1, 8/2, ... ,'k),, } on the space of persistence
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TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration
3.2. Prior Regularization
3.3. Comparing with Related Works
Although we present theoretical guarantees, in practice, sim- ply increasing model capacity does not necessarily lead to better performance, and overfitting may still occur. To miti- gate this, we introduce a prior regularization term on fo that enforces a marginal distribution I' prior over edge filtrations, thereby helping stabilize the training procedure:
Two works most related ours are DIR (Wu et al., 2022) and GSAT (Miao et al., 2022). We briefly compared their work with ours. Compared to DIR, our model also con- siders the distribution of complement graphs of rationale subgraphs, but in a "soft way". Instead of directly storing sample complement subgraphs, our methods can be viewed as storing a distribution of topological summary of comple- ment graphs, which more efficient. Relative to GSAT, our loss can also be seen variational lower bound of the GIB loss. However, we employ a different prior for the rationale subgraph Gx and remove GSAT's hyperparameter r. Our topological loss acts as self-adjusted cut, separating Gx from G. In practice, GSAT's attention can collapse to the constant r if not carefully tuned (Chen et al., 2024). By contrast, our prior performs an unsupervised two-Gawsr -oo-Gassi clustering (akin to k-means), preventing such collapse. Em- pirically, as long as the two means of Gaussian distributions remain distinct, their exact positions have little effect on per- formance. Thus, we do not need to tune the hyperparamters, and simply fix them at 0.25 and 0.75.
(()) + B Cprior (Jo(G), PPG)) prior).
Concretely, we assume each edge filtration 1g E [0,1]
follows two-mixture Gaussian distribution:
prior r1), + (1- w) N(2,,
where w. M41,22, 71, 122 are parameters. Then we can de- fine the prior regularization Cyrior via a Kullback-Leibler (KL) divergence term augmented with a penalty to prevent collapsing to single mode:
Sprior (So(C), P prior) [G]] prior] + 711 +r2?
4. Experiments
= =- PerPP (fo(G)c)
+ (rg(())oo(((((())) +11 + +r2
We evaluate our proposed method in terms of both inter- pretability and predictive performance various bench- mark datasets. Our approach, TOPING, demonstrates signif- icant advantages over state-of-the-art post-hoc interpretation methods as well as intrinsic interpretable models across al- most all datasets. We will provide a brief introduction to the datasets, baselines, and experiment setups, and leave more details in the Appendix E.
eEGE
A key insight is that this two-mixture prior induces clus- tering mechanism on the edge filtrations in [0. 1] for G, and G.. We note that any choice satisfying |يn - p2] > 0 and a reasonable weighting w E (0, 1) can still maintain a suitable two-cluster separation on [0, 1]. The exact cluster centers P1, #2 matter less than their separation since the edge filtration is learned and used in the topological discrep- ancy, thanks to the stability property of persistent homology. This approach fundamentally differs from existing methods such as GSAT (Miao et al., 2022) and GMT (Chen et al., 2024), offering greater stability and reduced sensitivity to hyperparameter choices. In practice, we simply fix w = 0.5, N1 = 0.25, و22 = 0.75, and initialize r1 = r2 = 0.25. We also apply Gumbel-Softman reparameterization trick (Jang et al., 2017) used in (Miao et al., 2022) to sample subgraphs. Remark 3.6. Although we only talk about edge filtra- tions, our methods can be applied to filtrations on nodes, edges, or higher-order simplices (faces, tetrahedrons, etc.). In our experiments, we start with the filtration functions on the nodes and then extend the node filtration to the edge filtration by setting fu, v) min(f(u), f())) or max(f(u), fu)). This type of filtration is known as upper- or lower-star filtration in TDA. It contains less information in general since node filtrations can only represent ((V)) much "information" " but edge filtrations can representup to O(E)) = 0(1V12 "information". However, it provides more computational efficiency.
4.1. Experimental settings
Datasets. We consider eight benchmark datasets com- monly used in the graph explainability literature, catego- rized into three types: Single Motif, Multiple Motif. and Real Dataset. The first two consist of synthetic datasets. Single Motif includes BA-2Motifs (Luo et al., 2020), BA- HouseGrid (Amara et al., 2023), SPmotif0.5 and SPmo- tif0.9 (Wu et al., 2022). These datasets contain graphs with a single type of motif or structural pattern repeated throughout. Multiple Molif includes BA-House BA Bo-Houurrrrrri (Bui et al., 2024), and BA BA-HouseOrrrri- nRnd. The last one is a synthetic dataset we create for veri- fying the variform rationale challenge for existing intrinsic methods (see Appendix E for more details). These datasets involve graphs with multiple types of motifs, thereby in- creasing the complexity and providing more challenging scenario for explanation methods. Real Dataset include Mu- tag (Luo et al., 2020) and Benzene "Sanchez-Lengelinn et al., 2020). Appendix E.5 visually illustrates sample graphs from each dataset and interpretation results of different models.
7

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TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration
Table Interpretation Performance (AUC) on test datasets. The shadowed entries are the results when TOPING outperform the means of the best baselines based on the mean- std of TOPING.
SINGLEMOTIP
MULTIPLMOTTI
REALD
METROD
BA-HoUSEGRID BA-2MOTIFS
SP
SPMOTIF0,9
BA BA-HoUSEANDGRD
BA BA-HOUSZORGRII
MUTAG
BENZENE
GENEXPLAINER
67.35 + 3.29
50.73 + 34
62.62 1.35
58.35 + 1.93
53.04 + 0.38
53.21 + 0.36
61.98 + 5.45
48.72 0.14
PGEXPLAINER
84.59 1: 9.09
50.92 4 1.51
69.54 à: 5.64
72.34 d: 2.91
10.36 4 4.37
3.14 0.01
60.91 17.10
4.26 a: 0.36
MATCHEXPLAINIR
86.06 28.
64.32 32
14.35
13.39
81.67
19.87=10.
55.65 1.16
MAGE
79.81 2.27
82.69 2: 4.78
76.63
0.95
74.38
99.95
99.93 0.07
99.57
0.63
DIR
82.78 :: 10.97
65.50 4: 15.31
78.15 1.32
49.08 3.66
64.96 14.31
59.71 + 21.56
64.44 2: 28.81
54.08 :: 13.75
GIN+
98.85 4: 0.47
98.55 +: 0.59
74.49 +: 4.46
65.25 4: 4.42
92,92 + 2.03
83.56 = 3.57
99.38 = 0.25
91.37 1.48
GIN+GMT-LIN
97.72 0.59
85.68 1: 2.79
76.26 1: 5.07
69 08 5.14
76.12 1: 7.47
74.36 =: 5.41
99.87 =: 0.09
83,90 6.07
GIN+TOPING
99.57 1: 0.60
99.24 1: 0.66
79.50 A: 3.71
80.82 1: 1.22
95.35 4: 0.95
88.56 2 2.04
95.79 =: 1.93
98.22 0.92
CINPE+GSAT
91.12 4.993
91.04 1: 6.59
78.20 4.48
80.24 3.66
95.17 1: 2.46
69.30
2.48
97.27 1: 0.47
95,40 3.05
CINP--GMT-LIN
5.24
76.40
85.08
5.10
97.48 00.81
94.44 2.49
CINPA-TOPING
100.00 0.00
92.82
100.00 0.00
100.00
0.00
96638
100.00
Table Prediction Performance (Acc.) test datasets. The shadowed entries the results when TOPING outperform the means of the best baselines based on the mean-1 1std of TOPING.
REALDATASET
SPURIOUSMOTIP
MODEL
METHOD
MUTAG
BENZENE
b=0.5
h=0.7
b=0.9
DIR
68.72 1 2.51
50.67 1 0.93
45.49 1 3.81
41.13 + 2.62
37.61 +: 2.02
GSAT
98.28 + 0.78
100.00 + 0.00
4 +
43.57 + 2.43
45.39 + 5.02
GIN
GMT-LIN
91.20 + 2.75
100.00 + 0.00
11.16:3..5
53.11 + 4.12
47.60 + 2.06
TOPING
94.20 + 5.61
100.00 + 0.00
52.22 2.07
54.46 + 5.76
50.21 + 3.22
GSAT
96.14 + 0.67
99.43 0.54
74.70 3.37
70.41 + 3.44
65.90 + 4.18
CINPP
GMT-LIN
95.27 + 1.36
98.87 0.92
73.16
3.51
69.11 1 4.12
68.60 1 6.06
TOPING
92.92
100.00 + 0.00
79.30 3.92
75.46 4.62
77.68 + 4.64
Baselines. We evaluate the interpretability of several meth- ods by differentiating between post-hoc and intrinsic in- terpretable approaches. The post-hoc methods we com- pare include GNNExplainer (Ying et al., 2019), PGEx- plainer (Luo et al., 2020), MatchExplainer (Wu et al., 2023), and Mage (Bui et al., 2024). Additionally, we consider the intrinsic interpretable methods DIR (Wu et al., 2022),
protocol to ensure fair comparisons. We employ recom- mended hyperparameter settings on baseline methods.
4.2. Result Comparison and Analysis
Variform Rationale Challenge. As shown in Figure 3, the interpretability of two SOTA intrinsic methods decreases drastically when the complexity of rationale subgraphs in- creases. Our method's performance is much better and more stable on such datasets with variform rationales. See Fig- ure 14 in Appendix E.5 for more visualization results.
GSAT (Miao et al., 2022), and GMT-Lin (Chen et al., 2024), known for their state-of-the-a interpretation capabilities and generalization performance.
Setup. Graph Isomorphism Network (GIN) (Xu et al., 2019) is the default backbone GNN used in baseline models. Be- cause our framework, TOPING, is fundamentally topologi- cal based, we also implement it on the CINPP (Giusti et al., 2023) backbone to showcase its full capabilities. This will al- low our method to naturally extend beyond standard graphs and operate directly on more general filtrations, including those on richer topological domains like simplicial com- plexes (Bodnar et al., 10;b); and hypergraphs.
Interpretation performance. As shown in Table 1, com- pared to the most post-hoc based methods(in the first row), and latest intrinsic interpretable models(in the second/third row), TOPING has shown significant improvement across al- most all datasets. Especially on the Spurious-Mott datasets, which are challenging due to spurious correlations in the training data, we achieve significant improvement over the previous best approach. On the challenging Multiple Mo- tif and Benzene datasets, ToPING even achieves the best performance.
Metries and evaluation. For interpretation evaluation, we report explanation ROC AUC following (Ying et al., 2019; Luo et al., 2020). For prediction performance, we report classification accuracy for real datasets and SPmotif (Wu al., 2022) for generalization performance. All the results averaged over 5 runs tested with different random seeds. All methods adopt the same graph encoder and optimization
Prediction performance. We compare the results of all intrinsic interpretable models training from scratch. Table 2 shows the prediction accuracy on Real Dutaset and Spuri- ous Motif. TOPING significantly outperforms other bascline models on the Spurious- Motif datasets, which exhibit vary-
8

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TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration
5. Conclusion
Comperisons of Performance in the Ba-HoossOOrrrinnnnn
1,00
In this work, we introduced TOPING, a novel intrinsically interpretable GNN framework that leverages persistent ho- mology identify stable rational subgraphs through persis- tent rationale filtration learning. Our approach introduces self-adjusted topological constraint, topological discrepancy, to measure the statistical topological difference between graph distributions. We provided theoretical guarantees that our target function can be uniquely optimized by ground truth under certain conditions. Through extensive experi- ments, we demonstrated that TOPING effectively addresses key challenges in interpretable GNNs, including handling variiform rationale subgraphs, balancing performance and interpretability, and avoiding spurious correlations.
Interpretation ROC AUC
0.60
0.75
0./0
TaplnG
0.65
GSAT
0.60
2
nRnd
Figure 3. In BA-HHousOOrGriddnRid dataset, nRnd increases, the complexity of rationale subgraphs increases. Existing SOTA methods struggles on such datasets, while TOPING's performance is much better and stable.
Acknowledgements
ing degrees of spurious correlations. This supports claim that the model can more effectively focus classifying the optimal stable subgraph through persistent rationale filtra-
We greatly thank the actionable suggestions given by re- viewers. C. Xin and J. Gao acknowledge funding from
IIS- 112-2229,76, DMS-2220271, DMS-231 1064, CRCNS- 2207440, CCF-2208663 and CCF-2118953, F. Xu and J. Ding were supported by NSF China under Grant No. T2421002, 62202299, 62020106005, 62061 146002.
tion learning.
Table Ablation studies. We report both interpretation ROC AUC and prediction accuracy.
BA-2MOTIPS
HOUSEGGRD
METROD
ACC
AUC
ACC
AUC
Impact Statement
TOPING w/o dran
100.00 0.00
97.90 1 1.24
89.24 =: 5.40
92.17 1 6.43
TootuG wio Cyeser
$3.49 + 4.03
20.20-1
4.61
$2.10 - 1.72
98.76 + 1.53
The adoption of powerful but opaque Graph Neural Net- works (GNNs) in critical scientific and decision-making domains is hindered by lack of trustworthy interpretability, as existing methods struggle to explain predictions when the underlying rationales are structurally diverse. This work directly addresses this gap by introducing ToPING, a novel framework that leverages topological perspective to build more transparent AI systems. By learning to identify sta- ble and persistent rationale subgraphs, TOPING enables researchers to clarify the decision-making process of their models, which is crucial in scientific applications like chem- istry and biology where understanding the causal mecha- nism is as important as the prediction itself. This ability to handle complex and varied rationales leads more reliable interpretations and improved generalization, providing new path toward trustworthy AI.
TortsG
100.00 = 0.00 100.00 #: 0.00
100.00 0.00
99.87 + 0.13
Ablation Studies. In addition to the interpretability and generalizability analysis, we also conduct further ablation studies gain deeper understanding of the results. Table 3 illustrates the usefulness of topological discrepancy and the prior regularizer. Topological discrepancy is essential for identifying stable and complex substructures, and the prior regularizer can be useful in partitioning a graph. We also examine the sensitivity of hyperparameters on the BA- Andrrrd dataset. As shown in Fig. 4, TOPING main- tains stable performance on different settings of weights of topological discrepancy and prior regularization. The performance decreases on too large or too small weights.
NAX: 1.00
MAXI 1.00
1.0
5.8
0.8
0.6
0.6
0.4
0.4
0001
6.00
0.3
cos
Yolo Reg Bota
Xpo Rey
Spo Rey Aboa
your Bog
0.05
0.05g
(a) Sensitivity of Interpretation
(b) Sensitivity of Prediction
Figure 4. A sensitivity study BA-HouseAndde AndGrid shows results with the topological constraint coefficient varied from [0.001, 0.005, 0.01, 0.05] and the coefficient of prior regularization term from [0.005. 0.05, 0.5].

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TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration
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Zhao, Q. Ye, Z.., Chen, C., and Wang, Y. Persistence enhanced graph neural network. In Chiappa, S. and Ca- landra, R. (eds.), Proceedings of the Twenty Third Inter- national Conference on Artificial Intelligence and Statis- tics, volume 108 of Proceedings of Machine Learning Research, pp. 2896-2906. PMLR, 2020.
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com/ content/
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A. List of Notations
the following, we list the notations for key concepts involved this paper.
Table 4. List Notations.
Symbol
Description
EV
A graph with vertex set V and edge set E. Candidate rationale subgraph.
Gx
G,
Candidate noise less relevant part of the graph.
G
Oracle rationale subgraph.
G
Oracle noise or less relevant part of the graph.
fe: [0,1]
Filtration functional.
F(G)
Graph filtration.
F(Ggi)
Subfiltration consisting of subgraphs with f(c) S
F(G2()
Subfiltration consisting of subgraphs with f(e) 2
T
Topological invariant (e.g.. persistence barcode). Topological discrepancy.
diopo
dnoute
Bottleneck distance.
dwass
I-Wasserstein distance.
he
GNN model for prediction.
Extraction function separate graph G into Gx and Ge-
Vectorization function for persistence diagrams. Prior distribution on edge filtration.
IP prior
Prior regularization.
a, B,7
Hyperparameters for loss function components.
B. More details of TDA
Bottleneck Distance:
Definition B.1 (Bottleneck Distance). Let Pi and P2 be two persistent barcodes. A partial matching π between P1 and P2 is subset of P1 x P2 such that each point in P1 and P2 appears in most one pair in æ. For any p = (P1, P2) R².
denote p (p2 Pı, P2 p1)- Geometrically, p is the closest point of p to the diagonal line A {(x, x)) RR. The
bottleneck distance between Pi and P2 is defined
da P2) = inf max max lP alloo (p.
max ||p Plooo
max ||g E11o0
(3)
peP (m) 9EPs (ma
where:
π ranges over all partial matchings between P1 and Pa
oloo denotes the oc-norm
mi and T2 denote the projections of ग onto P1 and P2 respectively
Intuitively, bottleneck distance measures the minimum cost of transforming one barcode to another by moving each point to another point in the other barcode. The cost is measured by the maximum distance between matched points partial matching, or the maximum distance between the rest unmatched points to the diagonal line A. The bottleneck distance a metric on the space of persistent barcodes, which is well studied in topological data analysis (Edelsbrunner Harer,
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2010; Zomorodian & Carlsson, 2004; Chazal al., 2014; Dey & Xin, 2018). It is shown in (Edelsbrunner & Harer, 2010) that the bottleneck distance is stable metric on the space of persistent barcodes, which means that small perturbations in the input data will not change the bottleneck distance too much. This property is crucial for the stability of the persistent homology, which is a key property for the robustness and differentiability of all vector representations based on persistent homology (Dey & Xin, 2021; 2022; Xin, 2023).
C. Missing Proofs
Proof of Theorem 3.4
Proof. By the assumption we know that the first term can only be optimized by Gx 2 G'x We just need to show that dipp uniquely maximized by G'x among those Gx 2 Gx In other words, we could assume that we have already restricted f to the region that satisfies foløx 0.5 + (the partition threshold & 0.5 is fixed).
For given G and fixed partition Gx Li determined by some fa. let pu. P1 be the 0-th and 1-st persistence diagrams, and g0,21 be the 0-th and 1-st persistence diagrams. Observe that the bottleneck distance between the 0-th persistence diagrams dg (Po. qu) is maximized when
fe(e) Ife Gx].
(4)
The reason is that since we only care about edge filtrations, the filtration values nodes can be viewed as some global minimum constant value which is commonly set to be time 0 (or more precisely, 1 for Gx and 0.5 for Ge since we build the filtration in the reversed ordering of importance). Then since |E,| > |Ex| = 1ool > |pol. we hope to maximize the death times of points in go and minimize the death times of points in po to maximize dg (Po. qo), which gives us the constant filtration function fe) = Ife E Gx} } on each partition. Then, for constant filtration functions, the induced graph filtrations are essentially reduced to static graphs, and in consequences, persistent homology is essentially reduced to homology. For O-degree homology, we just need to compare the 0-th Betti numbers &i and &6 between G. and Gx. In that case, (p: 9) = C(36- - -8%%) = C(EE. - [xx]] = C(GGgl Exl) for some constant C independent of or G. This is maximized when Gx G
The rest is to check the bottleneck distance du (Pi, @1) on 1-th persistence diagrams. In similar way one can check that (PI, 91) should be maximized for some constant filtration function. Then the problem is again reduced to compare the 1-degree homology between Gx and G.. That is | - -1). However, observe that | 31 I 31 for B1 be the 1-st Betti number of the original graph. By the property of the Euler characteristic on connected graph we know that
31 < [ - +VV+++ |E| < [v/2 Therefore, (g(P1.g) 91) S .M for some large enough M over the whole dataset.
Based on that, since drop is essentially weighted sum of dg on both 0-th and 1-st persistence diagrams, we just need large enough constant scaling factor on O-th persistence diagrams. Then it can been guaranteed that our droo is optimized by G with (e) = 1fe= E }. Such constant factor can be easily learned by our neural networks, or fixed by hand in the model.
D. Limitation
One limitation of our model is the computational cost. Currently the bottleneck is limited by the computation of the topological invariants. In theory, the time complexity of the persistent homology computation is O(n*), where n is the number of simplices (nodes for degree 0, edges for degree 1, and faces for degree 2) and w < 2.371552 is the matrix multiplication exponent (Williams et al.. 2024). Although on graphs, the O-th and 1-st persistent homology can be computed much faster in (n log n) time, the main bottleneck is not in theoretical computational complexity, but in practical implementation.
Practical Runtime Analysis. To provide concrete insights into the computational overhead, we measured the actual training time representative datasets. On BA-2Motifs, each training epoch takes approximately half minute, while on SPMotif (a more complex and larger dataset), the runtime is approximately 10 minutes per epoch. All experiments were conducted on a single RTX 4090 GPU. Importantly, our method consistently converges within 20 epochs across all datasets, in contrast to baseline methods that usually require 50-100 epochs to converge.
While our method is relatively slower per epoch due to TDA incorporation, this overhead is justified by significant performance gains and faster convergence. The reduced number of required epochs (20 vs. 50- 100 for baselines) partially
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compensates for the per-epoch computational cost, making the total training time competitive with baseline methods while achieving consistently superior interpretability performance.
Implementation Bottlenecks and Future Directions. The current software package cannot fully utilize the parallel computation power of GPUs. The data transfer between GPU memory and CPU memory takes much I/O cost. Maybe some system-level optimization based on the CUDA framework can help. Some attempts have been made to use GPU to accelerate the computation of persistent homology (Zhang et al., 2020), but the performance is still not satisfactory enough. Another possible solution is use some approximation algorithms to compute the topological invariants. For example, some efficient sparsification methods (Dey et al., 2019), or pretained NNs for computing persistent homology (Yan et al., 2022). We leave these problems for the future.
E. More Details about the Experiments
E.1. Datasets
Mutag (Kazius et al., 2005): The dataset involves a task of predicting molecular properties, specifically determining whether a molecule is mutagenic. The functional groups -NO2 and -NH2 are regarded as definitive indicators that contribute to mutagenicity, as noted by (Luo al.. 2020).
Benzene (Sanchez-Lengeling et al., 2020): The dataset comprises 12,000 molecular graphs sourced from ZINC15 (Sterling & Irwin, 2015). The objective is to identify the presence of benzene rings within molecule. The carbon atoms in these benzene rings serve as the ground-truth explanations.
BA-2Motifs (Luo et al., 2020): The dataset involves a binary classification task in which each graph combines a Barabasi- Albert base structure with either a house motif or five-cycle motif. The graph's label and ground-truth explanation are based on the motif it includes.
SPmotif ( (Wu et al., 2022): The dataset consists of graphs that merge base structure with motif. Each graph is manually infused with a spurious correlation between the base and the motif. The graph's 's label and the ground truth explanation determined by the motif it contains. Specifically, each graph comprises base graph Gs (tree, ladder, or wheel, encoded as 0. 1, 2) and a motif Gs (cycle, house, or crane, also encoded as O, 1, 2). The label is solely determined by Gs, but spurious correlation is introduced between the label and Gg. During training, G is sampled uniformly, while Gg is sampled with:
b
if GG
P(Gs) =
b-b
otherwise
Here, b E [} 1] controls the degree of spurious correlation; b 3 implies independence. We consider b = 0.5, 0.7, and 0.9. For testing, Gs and Gg are randomly paired to assess overfitting to spurious correlations.
BA-HouseGrid: The house and grid motifs are chosen because they do not have overlapping structures, such as those found in the house and 3 x grid.
BA-HouseAndGria (Bui al., 2024): Each graph is based on Barabasi-Aller structure and may be linked with either house motif or grid motif. Graphs that contain both types of motifs are labeled as while those containing only one type are labeled as 0. Note that each motif appears at most once in each graph.
Brrrod (Bui et al., 2024): Similar to BAnHous,AndGrrd, graphs that contain either house motif or grid motif are labeled as 1, while those containing neither type are labeled as O. Note that each motif appears at most once in each graph.
RA-HouseOrrrii-nnni Similar to BA-Houserrrrid, graphs that contain either n house motifs or n grid motifs are labeled as where n a random integer between (inclusive) and (inclusive). More formally:
Label Assignment:
P(Label = 1) = 0.5, P(Label = 0) = 0.5
For Label 1: Given E Z+, each E (1.2....,.)) n). the three possible manifestations are:
1
P(i x grid iix house) = 6n
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TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration
P(i grid) 1
6n'
P(i x house) 6n
When gr grid and house appear simultancously. their counts are equal. We do not consider cases where houses and grids appear simultaneously in different quantities. To ensure balanced dataset and to avoid potential bias in model training and evaluation, we first guarantee that the number of graphs with label 0 and label 1 is equal. Furthermore, within label 1, we generate an equal number of graphs for the above three manifestations. Therefore, the entire dataset maintains a balanced distribution across subcategories.
E.2. Details on Hyperparamter Tuning
E.2.1. BACKBONE MODELS
Backbone Architecture. We use two-layer GIN (Xu et al., 2019) with 64 hidden dimensions and 0.3 dropout ratio for all baselines. We use three-layer CINpp (Giusti et al., 2023) with 64 hidden dimensions and 0.15/0.3 dropout ratio for TOPING. For all datasets, we directly follow (Giusti et al., 2023) using enhanced Topological Message Passing scheme including messages that flow within the lower neighbourhood, the upper neighbourhood and boundary neighbourhood of the underlying cell complex. Considering that the largest chordless cycle for most interpretable motifs is equal to 5 (the BA-2Motifs dataset includes a 5-cycle, while most of the other motifs have chordless cycles with a maximum length of 4), we lift the maximum length of a chordless cycle to 5 as the cell(dim=2).
Data Splits. For BA synthetic datasets, we follow the previous work (Miao et al., 2022; Chen et al., 2024; Bui al., 2024) to split them into three sets(80%%/10%/100%)) For SPmotifs and real datasets, we use the default splits.
Evaluation. We report the performance of the epoch with the highest validation prediction accuracy and use these models the pre-trained models. If multiple epochs achieve the same top performance, we choose the one with the lowest validation prediction loss.
E.3. More Comparison Results
To further evaluate the effectiveness of TOPING modules and compare with the previous intrinsic interpretable baselines, additionally conduct experiments from the perspectives of constraints and regularization on both GIN and CINpp backbones. The results are given in the table 5. Details are as follows.
Constraint. Central to existing self-interprctabiliiy is the incorporation of the information bottleneck principle into the GNN architecture. We follow previous works to re-implement this infomation constraint under the recomerrecomeedde hyperparameters for fair comparison. The À of information regularizer set to be 1. As for topological constraint, set the coefficient to 0.01 achieve the best performance, which aligning with Figure 4.
Regularization. For marginal prior regularization, the choice of information constraint is a KL divergence regularizer. Specifically, for every graph G ~ I and every undirected edge e in G, we sample ae Bern(r) where r E [0, 1] is hyperparameter. The formulation the mutual information regularizer is:
DEL (Bern(a.) Bern(r))
Σ
ae Oe log
+ +(1 ac) log 1 ag r
r
r is initially set to 0.9 and gradually decay to the tuned value 0.7. We adopt a step decay, where r will decay 0. for every 10 epochs. As it is mentioned in TOPING, we employ a mixture of two Gaussian distributions to establish prior regularization term on Gx and Ge, outlined in Section 3.2. It is noteworthy that our choice aligns with the use of standard Gaussian distribution the latent distribution in variational auto-encoders. For fair comparison, we set the coefficient of Gaussian regularizer to 1.
Different components have different effects. Once could select the best combination and train the new architecture to better extract the subgraph information. Moreover, We provide more discussions and analysis about the the results in the table 5. Specifically.
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Table 5. More Comparison Results
Prediction
Interpretation
Backhone
Constraint
Regularization
BA-2MoliEs
SPMotif.s
BA House OrGrid
Benzene
BA-2Motlik
SPMotirs,
BA-HHaaseOrGrrl.2RW
Benzene
Info.
Bern
10000 = 0.00
47.45 5.87
95:25 1.60
100.00 #: 0.00
98.74 +: 0.55
74.49 +: 4.46
76.02 # 3.64
91.57 1.48
Topo
Garas
100.00 1 0.00
50.22 2.07
91.35 1.83
100.00 0.00
99.57 1. 0.000
79:50
88.74 1.70
98. 1. 0.92
GIN
Topo.
42.80 5.31
88441
3.30
45.95 3.02
92.87
1.81
1.98
86 2.68
Infex
Barn
100.00 + 0.00
63.35 6.06
100.00 + 0.00
100.00 0.00
91.12 4.93
78.30 4.48
75.08 + 7.09
95 40 3.05
CINP
Topo:
Gaass
100000 =: 0.00
79.30 3.92
100000 d: 0.00
100000 0.00
100.00 a: 0.20
95.08 4: 0.82
100000 0.00
10000 à: 0.00
Topo.
53.49 + 4.03
61.18 3.20
91.83 6.30
98.52 + 1.40
93.20 +
92.10 3.32
97.78 + 1.54
98.96 +: 1.66
100000 0.00
79.81
88.84 4.03
100.00 0.00
97.90
98.
94.
When integrating GNN models with enhanced topological message passing scheme, robust explainability methods should be able to adapt accordingly and achieve higher and more stable performance, because stronger expressive power often means that each edge in the graph can be more easily distinguished. However, the performance of graph information bottleneck framework remains highly degenerated simple task BA-2Motifss and complicated task BA--HousOOrGrid-22 2Rnd. Due to the formulation 5, there exists trivial solution where all values of ae converge directly to the given value of r. CINpp employs distinct perceptrons for each layer of the network and each dimension of the complex. It iteratively performs message passing for different types of adjacency. This unique architecture can lead information constraint more easily to zero loss, i.c., r. It will result in the inability to distinguish between interpretable subgraphs and noise subgraphs.
The topological constraint with Gaussian prior distribution significantly outperforms both GIN and CINpp across all tasks, although it occasionally slightly reduces accuracy. Equipped with a more powerful expressive model, ToPING can easily learn persistent rationale filtration through topological discrepancy loss. One surprising result is that using only the Gaussian distribution yields highly competitive results in interpreting spurious motif datasets and the BA-HouseOrGrid-nn -Rnd datasets. Nevertheless, the interpretation performance on Benzene dataset remains highly degenerated.
E.4. Regarding the result of MUTAG
The results of MUTAG can be attributed to the uniquely simple structure of MUTAG's rationale subgraphs. MUTAG's rationale subgraphs consist of just two edges sharing a common node (the functional groups -NO2 and -NH2). This represents the simplest possible non-trivial subgraph structure, lacking the topological complexity present in other datasets. Specifically:
There are no cycles (1-homology features)
The O-homology structure (connectivity) is nearly trivial
The rationale can be identified primarily through node/edge features rather than topological structure
In such cases, our topological discrepancy measure, which excels at capturing complex structural patterns, may introduce unnecessary complexity by analyzing features (like cycle bases) that aren't 't relevant to the true rationale. The model ends up relying more heavily on the prediction loss other than the interpretability regularization.
Following up on our previous response regarding MUTAG performance, we conducted additional experiments that provide compelling evidence for our analysis. Our investigation revealed that the initial lower performance on MUTAG stemmed from incorporating both Oth and 1st dimensional persistent homology features. However, the rationale subgraphs of MUTAG primarily NO2 and NH2 functional groups have relatively simple structures. Therefore, tracking higher-dimensionall topological features like cycles introduced unnecessary complexity that hurt the model's performance.
As shown in the Table 6, TopInG-0 achieves the second- best performance both interpretability (AUC) and prediction (ACC) compared to baseline interpretable GNN models. These results validate our analysis and demonstrate that our approach remains highly competitive when properly configured for molecular datasets with simpler structural patterns.
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Table 6. Comparison of interpretable GNN on MUTAG dataset.
Method
AUC
ACC
DIR
64.44 + 28.81
68.72 + 2.51
GSAT
99.38 + 0.25
98.28 = 0.78
GMT-LIN
99.87 + 0.09
91.20 + 2.75
TopInG
96.38 +: 2.56
92.92 7.02
TopInG-O
99.40 + 0.07
95.18 2.24
E.5. Interpretation Visualization
We provide visualization of the learned interpretabel subgraphs by GSAT, GMT-LIN and TOPING in the different datasets. The transparency of the edges shown in the figures represents the normalized attention weights learned by interpretable method. Note that we longer need min-max normalization like (Miao al., 2022) for better visualization, can directly use edge attention visualize through rational filtration learning, because persistent homology gap has guaranteed that our edge attention is easy to be distinguished.
(a) GSAT
(b) GMT-LIN
(c) ToPING
Figure Learned interpretable subgraphs by GSAT, GMT-LIN and TOPING on BA HouseAndGrid. Figures in cach row belong to the same class, Nodes colored red ground-truth explanations.
Figure 6. Visualizing attention of GSAT(ffrst row), (MTooLNssecoond row) and ToPING (third row) Benzene. Figures in the column represent an identical graph. Nodes colored red are ground-truth explanations.
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Figure? 7. Learned interpretable subgraphs by GSAT (first row). GMM--Ld row) and G row) on BA-HouseOrCrid-
2Rnd. Figures in same column represent an identical graph. Nodes colored red are ground- explanations.
Figure 8. Learned interpretable subgraphs by GSAT (first row). GMT- row) and ToPIN ToPING row) on BA-HouseOrGrid-
4Rnd. Figures in the same column represent identical graph. Nodes colored red ground-truth explanations.
Figure Learned interpretable subgraphs by GSAT (first row), GMT-LLNNsecoond row) and (hhid row) SPmotif0.9 class 0. Figures in the same column represent an identical graph. Nodes colored red are ground-truth explanations.
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Figure 10. Learned interpretable subgraphs by GSAT (first row), GMT-LIN (scoon row) and ThhidG row) on SPmotif0.9 class 1. Figures in the same column represent an identical graph. Nodes colored red ground-truth explanations.
Figure 11. Leamned interpretable subgraphs by GSAT (first row), (ecoodd row) and ToPlN row) on SPmotif0.9 class 2. Figures in the same column represent an identical graph. Nodes colored red ground-truth explanations.
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Figure 12. The rationals of SPmotif0.9 learned by TOPING. Figures in each row belong to the same category. Nodes colored red are ground-truth explanations.
Figure 13. The rationals of Mutag learned by ToPING. Nodes colored red are ground-truth explanations.
Figure 14, We trained TOPING Rand with only, and it on BA-HouseOrGriddnRRm for (2, 3, 5, 6). We still observe high prediction (rediettoonAACC-100))) and Antepretation(AUCC11000 performance on test datasets. The figure illustrates interpretation results RoudeOrGrid-nRnd (2,3, 6). Nodes colored red are ground-trute truth explanations.
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Reviews:
[{"review_id": "pvTE5EbQfJ", "reviewer": "Reviewer_et1a", "summary": "This work proposed a new framework which applies TDA tool to interpret the persistent rationale subgraph in graph learning problems, showing effective result performance on motif classification task, evaluated on Single Motif, Multiple Motif and Real Dataset.", "questions": "", "limitations": "", "rating": 3, "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": "We sincerely appreciate your thorough review and invaluable feedback. Below, we provide detailed responses to all your comments or concerns.\n\n\n### Q1: Evaluation on Datasets with Hierarchical Rationales \n\n\nWe appreciate the reviewer's insight into the importance of hierarchical rationales in interpretable graph neural networks. Indeed, this is an emerging area with significant potential. \nHowever, graph interpretability requires ground-truth rationale annotations, and such comprehensively labeled datasets are scarce (Zhang et al., 2024). Our experiments follow established benchmarking practices. Furthermore, we tested out-of-distribution (OOD) generalization by training TopInG on BA-HouseOrGrid-nRnd with fixed motif count (n=4) and testing on varying counts (n=2,3,5,6). TopInG maintains stable performance across these distribution shifts, as shown in Figure 14.\"\n\n**Regarding more real-word dataset**\n\nTo further demonstrate the practical applicability of TopInG on real-world tasks, we conducted additional experiments on the MNIST-75sp dataset [1], which is widely used for evaluating graph-based models on visual tasks. In this dataset, each MNIST image is converted into a superpixel graph, where nodes correspond to superpixels and edges represent spatial adjacency. Nodes with nonzero pixel values provide ground-truth explanations, making it suitable for evaluating interpretability methods.\n\n\nThis dataset consists of 60,000 graphs for training and 10,000 for testing, with an average of 70.57 nodes and 590.52 edges per graph. Notably, the ground-truth explanatory subgraphs vary in size across samples, adding to the challenge. We trained all models for 30 epochs on a single RTX 4090 GPU (over 26 hours). For TopInG, we used default hyperparameters and only reduced the coefficient of the topological loss, without extensive tuning.\n\n\nThe table below presents both interpretation and prediction performance:\n|method| Interpretation Performance (AUC)| Prediction Performance (Acc)|\n| -- | -- | -- |\n|DIR| 32.35| 88.51|\n|MAGE| 67.50| 89.43|\n|GSAT| 80.47| 96.20|\n|GMT-Lin| 82.98| 96.01|\n|TopInG| 84.50| 95.20|\n\n\nAs shown, TopInG achieves the best interpretability performance (AUC = 84.50) while maintaining competitive prediction accuracy. This result demonstrates TopInG's strong potential for real-world graph-based tasks involving complex, variable-sized explanations.\n\n\n[1]Knyazev, B., Taylor, G. W., and Amer, M. R. Understanding attention and generalization in graph neural networks. In Advances in Neural Information Processing Systems, pp. 4204–4214, 2019.\n\n\n### Q2: Runtime Efficiency of the Proposed Framework \n\n\nWe appreciate the reviewer's concern regarding the runtime efficiency of our framework. In Appendix D, we provided a theoretical analysis of the runtime complexity. In brief, the time complexity can be as fast as O(nlogn), without considering GPU acceleration. Practically, we provide representative results on two datasets to give a sense of our method's efficiency.\n\n\nOn BA-2Motifs, each training epoch takes approximately 0.45 ± 0.12 minutes, and on SPMotif (which is a more complex and larger dataset), the runtime is approximately 9.20 ± 2.35 minutes per epoch. All experiments were conducted on a single RTX 4090 GPU, and importantly, our method consistently converges within 20 epochs across all datasets.\n\n\nAlthough our method is relatively slower per epoch due to the incorporation of TDA, this added cost is justified by the significant performance gains. Unlike many baseline models that typically require 50 to 100 epochs to converge, TopInG achieves convergence within 20 epochs, while consistently achieving perfect or near-perfect AUC scores on multiple datasets.\n\n\nWe hope this additional information clarifies the practical efficiency of our approach. \n\n### Regarding PersGNN:\nThank you for providing the reference. We will mention this work in the revised version. As a related work, PersGNN applies TDA to analyze the structure of protein graphs by combining persistent homology with GNNs to capture both local and global structural features for improved protein function prediction. While our work focuses on leveraging TDA to enhance the interpretability of GNN models through persistent rationale filtration, PersGNN primarily aims to improve prediction performance on protein structure-function relationships via a novel TDA-enhanced GNN architecture. This distinction highlights complementary applications of topological methods in graph learning.", "claims_and_evidence": "The authors provide experiments, theoretical analysis, ablations and visualizations to validate the proposed framework.", "methods_and_evaluation_criteria": "The proposed method applies a good design on TDA based GNN interpretation framework. However, it would benefit more from further real-world generality evaluation.", "theoretical_claims": "The proofs in the appendix are checked, no significant issues were found.", "experimental_designs_or_analyses": "The experimental designs look valid, but additional datasets for validating the Theorem 3.5 would be desired, e.g. datasets where rationales are hierarchical.", "relation_to_broader_scientific_literature": "Relation to interpretable graph neural networks (GNNs), topological data analysis (TDA), and robust learning with variform rationales.", "essential_references_not_discussed": "PersGNN[1], related to TDA with GNN task, should be added in references.\n[1] Swenson, Nicolas, et al. \"PersGNN: applying topological data analysis and geometric deep learning to structure-based protein function prediction.\" arXiv preprint arXiv:2010.16027 (2020).", "other_strengths_and_weaknesses": "Strengths:\nThis work proposed a new framework of interpretable graph neural networks integrated with TDA tool to interpret the persistent rationale subgraph in graph learning problems. The experimental results and theoretical analysis sound robust, evaluated on motif-like datasets.\n\n\nWeaknesses:\n1. Evaluation on datasets where rationales are hierarchical is missing.\n\n2. The runtime efficiency of the proposed framework seems missing.", "other_comments_or_suggestions": "It'll be more persuasive to provide the evaluation on datasets where rationales are hierarchical.", "questions_for_authors": "The evaluation of the runtime efficiency of the proposed framework seems missing. It is desired to know the efficiency and comparison with related methods.", "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": "PgcdmnQRsy", "reviewer": "Reviewer_H6hM", "summary": "The paper introduces TopInG, a topologically interpretable graph neural network (GNN) framework leveraging persistent homology to identify stable rationale subgraphs for model explanations. Key contributions include: (1) a novel *rationale filtration learning* technique that models graph generation processes to capture persistent topological features (0th/1st homology) across scales, and (2) a *topological discrepancy* loss measuring structural differences between rationale and non-rationale subgraphs. The method addresses limitations of prior interpretable GNNs in handling variable subgraph structures (e.g., differing sizes, topologies) by encoding multi-scale topological persistence. Experiments show TopInG outperforms state-of-the-art baselines (e.g., GSAT, DIR) in prediction accuracy (e.g., 50.21±3.22 Acc on MUTAG) and interpretation quality (e.g., 100% AUC on synthetic datasets), particularly in scenarios with diverse rationale subgraphs. Theoretical guarantees ensure the learned rationale subgraphs align with ground-truth causal mechanisms under optimal conditions.", "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": "# Clarifications and Corrections\nWe thank the reviewers for their detailed feedback and appreciate the opportunity to clarify a few misunderstandings and highlight concerns already solved in our Appendix:\n\n* BA-2Motifs do include cycles in their rationale subgraphs. All datasets used in our work contain complex topological structures.\n* MUTAG graphs contain cycles, even if the rationale subgraphs may not. Appendix E.5 contains illustrative examples of every dataset we used. \n* \"overfitting\" refers to poor generalization from training to test data, which does not fit the content since all scores we reported are based on test data. We discuss a more related topic called \"out-of-distribution (OOD) generalization\" in Q1 below. \n* The formula \"\\mathcal{L}{\\text{topo}} \\leq W{\\text{GH}}(\\mathbb{P}, \\mathbb{Q})\" is not from our submitted manuscript.\n\n\n## Methodological Notes\n* GSAT’s hyperparameter r is tunned following author-recommended strategy: initially set to 0.9 and gradually decay to 0.7 (Appendix E.3, line 892).\n* Additional ablation studies are provided to support the impact of topological discrepancy vs. Gaussian(Appendix E.3, Table 5).\n* Backbone choices have been discussed in Appendix E.3 line 914.\n\n\n## Q1: To evaluate on more Real-World Datasets\n\nWe acknowledge requests for more real-world datasets. However, graph interpretability requires not just graph labels, but also ground-truth rationale annotations identifying causal subgraphs. The scarcity of such comprehensively labeled datasets is a recognized challenge in this field (Zhang et al., 2024). Our experiments follows established benchmarking practices. \nFurthermore, we have tested OOD generalization by training TopInG on BA-HouseOrGrid-nRnd with fixed motif count (n=4) and testing on varying counts (n=2,3,5,6). Despite the distribution shift, TopInG maintains stable prediction and interpretability performance without degradation, as shown in Figure 14 (Appendix).\n\n\n\nTo further mitigate the concern, we conducted an additional experiment on MNIST-75sp [1], which contains noisy rational subgraphs with richer topologies.\n\n[1]Knyazev, B., Taylor, G. W., and Amer, M. R. Understanding attention and generalization in graph neural networks. In Advances in Neural Information Processing Systems, pp. 4204–4214, 2019.\n\nOur results (table below) show that TopInG achieves the best interpretation and good prediction scores compared to existing SOTA methods. This suggests our topological signal contributes meaningfully to explanation quality, even when tested on a real-dataset with more complex and noisy varing rationale subgraphs.\n\n|method| Interpretation Performance (AUC)| Prediction Performance (Acc.)|\n| -- | -- | -- |\n|DIR| 32.35| 88.51|\n|MAGE| 67.50| 89.43|\n|GSAT| 80.47| 96.20|\n|GMT-Lin| 82.98| 96.01|\n|TopInG| 84.50| 95.20|\n\n\n\n# Q2: Test on Graphs with Conflicting Topologies\n\nTopInG is evaluated on the two datasets, SPMotif (synthetic) and MUTAG (real-world),  which include non-informative or spurious cycles irrelevant to labels. The experiment results (Table 1) confirm that our model is robust. \n\n# Q3: Regarding Theorem 3.5\n\nOur theoretical assumptions in Theorem 3.5, while idealized, significantly relax the strict conditions required by previous works, making our method theoretically better suited for handling variable rationale structures. Empirical evidence strongly supports the practical validity of these assumptions.\nSpecifically, previous works (e.g., GSAT, GMT, DIR) assume that within each category, the corresponding rationale substructure is globally invariant across the entire data distribution—a much stronger and less practical condition compared to ours. This stricter requirement explains why existing methods struggle with variform rationales.\nWe would greatly appreciate it if the reviewer could clarify the intended meaning of \"overfitting risks\" in the context of |E_X| \\ll |E_\\epsilon|. As we mentioned before in \"Clarifications\", \"overfitting\" does not fit the content. Additionally, the inequality |E_X| \\ll |E_\\epsilon| is covered by the weaker assumption |E_X| < |E_\\epsilon| used in our theorem.\nMoreover, in our experiments, nearly all datasets tested exhibit the property that rationale subgraphs are significantly smaller than the full graphs (e.g. |E_\\epsilon| > 10*|E_X|). TopInG consistently performs well across these scenarios, further supporting that the assumptions underlying Theorem 3.5 do not limit its practical utility.\n\n# Q4: How Sensitive are TopInG’s Gains to Backbone Choices?\n\nAs disucussed in Appendix E.3, TopInG performs more robustly with backbones that preserve topological structures (e.g., CIN). Some backbones like GCN and GAT can struggle with interpretability or prediction, as noted in (Bui et al., 2024).\n\n# Q5:   What Specific Strategies Would Address the Scalability Challenges?\n\nFor larger graphs, we have mentioned several promising directions for speedup the computation, in Appendix D line 807.", "claims_and_evidence": "**Key claims lacking convincing evidence:**  \n1. **Handling variable subgraphs via topological persistence**: While persistent homology captures multi-scale features, the paper’s synthetic datasets (e.g., BA-HouseOrGrid-nRnd) exhibit controlled variability. Real-world datasets like MUTAG lack complex topological structures (e.g., cycles), limiting validation of 1st homology utility. The claim is **not fully supported** for diverse real-world graphs.  \n\n2. **Superior interpretation quality (100% AUC on synthetic data)**: Perfect AUC scores suggest overfitting to synthetic tasks. For example, BA-2Motifs has trivial 0th homology (no cycles), making topological analysis redundant. The **evaluation lacks noisy or adversarial examples** to test robustness.  \n\n3. **Theoretical guarantees**: The topological discrepancy loss is framed as a lower bound of the Wasserstein distance under Gromov-Hausdorff metric:  \n   $\\mathcal{L}_{\\text{topo}} \\leq W_{\\text{GH}}(\\mathbb{P}, \\mathbb{Q})$\n\n   However, the proof assumes optimal filtrations and ignores graph noise, limiting practical relevance.  \n\n**Methodological flaws**:  \n- **Dataset limitations**: Experiments rely heavily on synthetic graphs with simplistic topologies (e.g., MUTAG lacks cycles). The variiform rationale challenge is tested on BA-HouseOrGrid-nRnd, which may not generalize to real-world heterogeneity.  \n- **Baseline comparisons**: GSAT and GMT-LIN are compared under constrained backbones (GIN/CINpp), but their hyperparameters (e.g., GSAT’s $r$ ) are not exhaustively tuned, risking unfair comparisons.  \n- **Ablation gaps**: The impact of topological discrepancy vs. standard regularizers (e.g., Gaussian) is not rigorously isolated.", "methods_and_evaluation_criteria": "The proposed methods (topological filtration, discrepancy loss) align with the goal of capturing stable rationale subgraphs but **rely heavily on synthetic datasets (e.g., BA-2Motifs) with simplistic topologies (no cycles, trivial 0-homology)**, limiting validation of 1st-homology utility. Evaluation metrics (AUC/ACC) are standard but **perfect synthetic AUC scores suggest overfitting**, and **real-world datasets (MUTAG) lack topological complexity** to test claims. Backbone choices (CINpp) suit topological data, but comparisons to GIN-based baselines may not fully stress-test scalability.", "theoretical_claims": "**Theorem 3.5** assumes minimal subgraphs $G^*_X$ and ignores graph noise, making its uniqueness guarantee **theoretically valid but practically fragile**. The proof relies on idealized filtrations and does not address overfitting risks when \\(\\|E_X\\| \\ll \\|E_\\epsilon\\|\\) (common in real graphs), weakening its real-world applicability.", "experimental_designs_or_analyses": "See comments below", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "See comments below", "other_comments_or_suggestions": "- **Content Ambiguities**:  \n  - \"variiform rationale subgraphs\" lacks clear prior definition. (abs, and remark 3.6) \n\n\n- **Dataset Descriptions**:  \n  - SPmotif0.5/0.9 and BA-HouseOrGrid-nRnd lack clarity on construction/split details.", "questions_for_authors": "1. The paper emphasizes handling variiform rationale subgraphs but relies heavily on synthetic datasets (e.g., BA-2Motifs) with trivial 0-homology and no cycles. How would TOPING perform on real-world datasets with richer topological structures (e.g., molecular graphs with complex cycles)? If no such experiments were conducted, does this not undermine the claim that 1st-homology analysis is critical for real-world interpretability?  \n\n2. The perfect AUC scores on synthetic datasets suggest potential overfitting, especially since BA-2Motifs lacks cycles. Did the authors test on synthetic graphs with adversarial noise or conflicting topological features (e.g., cycles irrelevant to labels)? If not, how can the method’s robustness to such challenges be assured?  \n\n3. Theorem 3.5 assumes minimal subgraphs (\\(G^*_X\\)) and ignores graph noise. How does the method handle real-world graphs where noise or spurious edges dominate (\\(\\|E_\\epsilon\\| \\gg \\|E_X\\|\\))? Does the bimodal prior for edge filtration collapse under such conditions, as observed in GSAT?  \n\n4. The comparison with GSAT and GMT-LIN uses constrained backbones (CINpp/GIN). How sensitive are TOPING’s gains to backbone choice? For instance, would the method maintain performance on Transformers or MPNNs, which lack explicit topological inductive biases?  \n\n5. The paper mentions future work on efficient persistent homology computation but does not address scalability challenges (e.g., large graphs). What specific strategies would address this, given that persistent homology’s \\(O(n^3)\\) complexity (Hofer et al., 2020) could limit practical deployment?", "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": "1gtJZW0iA7", "reviewer": "Reviewer_obEh", "summary": "This paper presents a framework, TopIng, designed to enhance the interpretability of GNNs by enabling them to automatically identify key subgraphs that influence prediction outcomes. It uses TDA to characterize the growth process of key subgraphs and employs topological distance loss to help the GNN distinguish between relevant and irrelevant parts. Overall, the framework makes the GNN more transparent, stable, and trustworthy, in addition to achieving accurate predictions.", "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": "We sincerely appreciate your thorough review and invaluable feedback. Below, we provide detailed responses to all your comments or concerns.\n\n\n### Regarding the weakness \"Persistent homology computation is computationally intensive on large-scale graphs.\"  \n\nFor larger graphs, we have mentioned several promising directions for speedup the computation, in Appendix D. \nFor example: (1) Development of efficient GPU implementations. (2) Using carefully designed GNN models to approximate persistent homology computations. \n\n### Regarding more real-world dataset evaluation\nWe acknowledge requests for more real-world datasets. However, graph interpretability requires not just graph labels, but also ground-truth rationale annotations identifying causal subgraphs. The scarcity of such comprehensively labeled datasets is a recognized challenge in this field (Zhang et al., 2024). Our experiments follows established benchmarking practices. \nFurthermore, we have tested out-of-distribution (OOD) generalization by training TopInG on BA-HouseOrGrid-nRnd with fixed motif count (n=4) and testing on varying counts (n=2,3,5,6). Despite the distribution shift, TopInG maintains stable prediction and interpretability performance without degradation, as shown in Figure 14 (Appendix).\n\n\n\nTo further mitigate the concern, we conducted an additional experiment on MNIST-75sp [1], which contains noisy rational subgraphs with richer topologies.\n\n[1]Knyazev, B., Taylor, G. W., and Amer, M. R. Understanding attention and generalization in graph neural networks. In Advances in Neural Information Processing Systems, pp. 4204–4214, 2019.\n\nOur results (table below) show that TopInG achieves the best interpretation and good prediction scores compared to existing SOTA methods. This suggests our topological signal contributes meaningfully to explanation quality, even when tested on a real-dataset with more complex and noisy varing rationale subgraphs.\n\n|method| Interpretation Performance (AUC)| Prediction Performance (Acc.)|\n| -- | -- | -- |\n|DIR| 32.35| 88.51|\n|MAGE| 67.50| 89.43|\n|GSAT| 80.47| 96.20|\n|GMT-Lin| 82.98| 96.01|\n|TopInG| 84.50| 95.20|\n\n### Regarding \"The method requires simultaneous optimization of multiple loss components and the adjustment of several hyperparameters, which may increase engineering complexity and tuning burden in practical applications.\"\n\n\nWe appreciate the reviewer's concern regarding the potential complexity introduced by optimizing multiple loss components and tuning associated hyperparameters. While this may appear to increase the engineering burden in theory, in practice, we fixed most hyperparameters across all datasets without extensive tuning. Despite this, TopInG consistently achieved strong performance, suggesting that the method is not heavily sensitive to hyperparameter choices, and the additional loss components can be integrated in a stable manner.\n\n\nWe believe this demonstrates that while the design includes multiple components, the practical tuning burden is low, and the framework remains accessible for real-world applications.\n\n\n### Regarding \"There are also studies about rationalization in the field of NLP. Would it be possible to discuss some of them in the related work?\"\n\n\nWe appreciate the reviewer's suggestion to incorporate discussions on rationalization studies from the Natural Language Processing (NLP) domain into our related work section. Incorporating insights from NLP rationalization studies inspire advancements in graph-based interpretability methods.​\n\nRationalization has been extensively studied in NLP literature, emphasizing causal feature selection, faithful rationale-input alignment, and stable training dynamics. Recent advances include alternatives to the maximum mutual information (MMI) criterion, highlighting input utilization and causal feature isolation [1,2,4]. Other studies enhance rationale fidelity by enforcing semantic alignment with original inputs [3], and propose methods to stabilize joint rationale-predictor training through asymmetric learning rates, multi-generator ensembles, and unified encoders [5,6,7]. However, unlike NLP domains where rationales often manifest as contiguous text spans, graph domains often exhibit \"variform rationale subgraphs\" that can differ significantly in form, size, and topology even among instances of the same class.\n\nWe will include a brief discussion in the related work section to acknowledge related NLP studies.", "claims_and_evidence": "The authors make several key claims that are partially supported by experimental results and theoretical analysis, but there are still some issues.\nTopIng can effectively handle subgraphs with variable structures: Although the authors tested the robustness of the method using the synthetic dataset BA-HouseOrGrid-nRnd, this dataset only simulates a limited range of variations and does not fully explore more complex real-world scenarios.", "methods_and_evaluation_criteria": "The TopIng method proposed by the authors is based on Topological Data Analysis (TDA) and uses persistent homology to identify stable rationale subgraphs, incorporating topological differences as a loss function. This approach is innovative theoretically, and experimental results show that it performs well across multiple datasets.\nHowever, there are some limitations: On the MUTAG dataset, TopIng's prediction performance is lower than that of GSAT. The paper identifies the reasons for this and improves performance using TopIng-0, but does not fully discuss the limitations and applicability of the method on simpler structured datasets. Additionally, there are shortcomings in the evaluation of explanation quality. Using AUC to measure explanation quality is limited in scope and fails to comprehensively reflect the accuracy of the explanations.", "theoretical_claims": "The TopIng framework shows some potential in handling subgraphs with variable structures. The theorem assumes that |Eε| > |EX| (the number of edges in the rationale subgraph) is always smaller than that of the complement graph. This may not hold true in certain real-world scenarios. Additionally, the authors mention that computational complexity limits the use of higher-order homology (beyond the second order), but they do not analyze the impact of this limitation on the theoretical guarantees.", "experimental_designs_or_analyses": "The experimental design is reasonable, and the choice of datasets, baseline comparisons, and result analysis support the effectiveness of TopIng. However, there is a lack of systematic transfer learning validation across different types of datasets, making it difficult to assess the method's adaptability in diverse application scenarios.", "relation_to_broader_scientific_literature": "The contributions of TopIng are closely related to existing research in the fields of TDA, explainable graph learning, and optimal transport.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths:\n\n- The proposed topological discrepancy has a strong mathematical basis, elevating the comparison of graph structures to the topological level. This is an important contribution to the theory of graph learning.\n- The method is successfully applied to different backbone networks (GIN and CINpp), demonstrating the framework's nature and independence from specific neural network architectures.\n\nWeaknesses:\n\n- Persistent homology computation is computationally intensive on large-scale graphs. Nevertheless, the paper acknowledges this issue.\n- The method requires simultaneous optimization of multiple loss components and the adjustment of several hyperparameters, which may increase engineering complexity and tuning burden in practical applications.", "other_comments_or_suggestions": "There are also studies about rationalization in the field of NLP. Would it be possibile to discuss some of them in the related work? \n\n[1] Breaking Free from MMI: A New Frontier in Rationalization by Probing Input Utilization  [2] Is the MMI Criterion Necessary for Interpretability? Degenerating Non-causal Features to Plain Noise for Self-Rationalization [3] Enhancing the Rationale-Input Alignment for Self-explaining Rationalization [4] D-Separation for Causal Self-Explanation [5] Decoupled Rationalization with Asymmetric Learning Rates: A Flexible Lipschitz Restraint [6] MGR: Multi-generator Based Rationalization [7] FR: Folded Rationalization with a Unified Encoder", "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": "OQkPzopueu", "reviewer": "Reviewer_Nxw2", "summary": "In this paper, the authors propose a GNN interpretation framework named TopInG, which applies topological data analysis and persistent homology to separate the graph into rationale and noise subgraphs. The method is theoretically sound, and the experiments show improvements in interpretation performance.", "questions": "", "limitations": "", "rating": 3, "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": "We sincerely appreciate your thorough review and invaluable feedback. Below, we provide detailed responses to all your comments or concerns.\n\n\n### Q1: BA-HouseOrGrid-nRnd Generated is a Bit Confusing\n\n\nThank you for pointing this out! We clarify the label definitions as follows:\n\n\n*  Label 0: The graph contains neither houses nor grids.\n*  Label 1: The graph contains (a) only houses, (b) only grids, (c) an equal number of houses and grids.\n\n\nWe do not consider cases where houses and grids appear simultaneously in different quantities, as this would introduce significant combinatorial complexity.\nTo ensure a balanced dataset and avoid potential bias in model training or evaluation, we first guarantee that the number of graphs with label 0 and label 1 is equal. Furthermore, within label 1, we generate an equal number of graphs for conditions (a), (b), and (c), so as to maintain a stable and unbiased distribution across subcategories.\n\n\n### Q2: Are the Assumptions in Theorem 3.5 Realistic for Real-world Graphs?\n\n\nWe consider the assumptions made in Theorem 3.5 to be reasonably realistic for a range of real-world graphs, especially in structured domains such as molecular graphs. For instance, in datasets like MUTAG and BENZENE, there often exists a relatively small rationale subgraph (e.g., a specific functional group) embedded within a larger graph, which aligns well with our assumption of a distinguishable and minimal rationale G_X​.\n\nOur theoretical assumptions, while idealized, significantly relax the strict conditions required by previous works, making our method theoretically better suited for handling variable rationale structures. Empirical evidence strongly supports the practical validity of these assumptions. Specifically, previous works (e.g., GSAT, GMT, DIR) assume that within each category, the corresponding rationale substructure is globally invariant across the entire data distribution—a much stronger and less practical condition compared to ours. This stricter requirement explains why existing methods struggle with variform rationales.\n\n\n**Regarding more real-word dataset**\n\nWe acknowledge requests for additional real-world datasets. However, graph interpretability requires not just graph labels, but also ground-truth rationale annotations identifying causal subgraphs. The scarcity of such comprehensively labeled datasets is a recognized challenge in this field (Zhang et al., 2024). Our experiments follows established benchmarking practices. \nFurthermore, we have tested out-of-distribution (OOD) generalization by training TopInG on BA-HouseOrGrid-nRnd with fixed motif count (n=4) and testing on varying counts (n=2,3,5,6). Despite the distribution shift, TopInG maintains stable prediction and interpretability performance without degradation, as shown in Figure 14 (Appendix).\n\nTo further mitigate the concern, here we conducted an additional experiment on MNIST-75sp [1]. This dataset contains noisy rational subgraphs with more complex and varied topologies.\n\n[1]Knyazev, B., Taylor, G. W., and Amer, M. R. Understanding attention and generalization in graph neural networks. In Advances in Neural Information Processing Systems, pp. 4204–4214, 2019.\n\nOur results, with details in the table below, show that TopInG achieves the best interpretation and good prediction scores compared to existing SOTA methods. This suggests that our topological signal contributes meaningfully to explanation quality, even when tested on a real-dataset with more complex and noisy varing rationale subgraphs.\n\n|method| Interpretation Performance (AUC)| Prediction Performance (Acc.)|\n| -- | -- | -- |\n|DIR| 32.35| 88.51|\n|MAGE| 67.50| 89.43|\n|GSAT| 80.47| 96.20|\n|GMT-Lin| 82.98| 96.01|\n|TopInG| 84.50| 95.20|\n\n\n\n\n\n**Regarding the typo of 'variiform'**\n\n\nThank you for pointing this out. We will correct the spelling.\n\n**Post-hoc methods missing for Fig3 experiment**\n\nWe prioritized comparing interpretable methods (DIR, GSAT, GMT) since our work specifically addresses the variform rationale challenge observed in these approaches. Post-hoc methods show diverse performance on BA-HouseOrGrid dataset, suggesting this is not neccessarily a common challenge across post-hoc methods. We tested MAGE on the dataset BA-HouseOrGrid-nRnd for n=(2,3,4), and observed that MAGE has consistent performance close to 100%. It indicates that post-hoc methods like MAGE are not very sensitive to the variation of rationale structures. \n\n\n**Regarding the notations in Fig.1**\n\n\nTo improve its readability and self-containment, we will revise the figure caption to explicitly define all notations used, including:\n\n\n* F, which denotes the filtration applied to the graph.\n\n\n* T, which refers to the resulting topological invariant (e.g., persistence diagram).", "claims_and_evidence": "The claims in the paper are generally well supported. However, the claim that TopInG performs better on tasks with variable rationale subgraphs is only empirically verified on a synthetic dataset. This claim would be more convincing if more diverse real-world datasets were included, and the post-hoc interpretation methods are also compared.", "methods_and_evaluation_criteria": "Yes the methods and evaluation criteria make sense. The datasets selection can be improved by adding more diverse real-world dataset.", "theoretical_claims": "No I didn't check the correctness proofs or theoretical claims, as I'm not familiar with the TDA theory.", "experimental_designs_or_analyses": "Overall, the experimental design is sound. The datasets used include both synthetic and real-world datasets. The evaluation metrics, including ROC-AUC and accuracy, are common and appropriate choices.\n\nHowever, in the experiment shown in Fig. 3, it is questionable why post-hoc methods were not included.\n\nAdditionally, the real-world datasets are quite similar. Adding more diverse datasets could be beneficial.", "relation_to_broader_scientific_literature": "The paper is closely related to GNN interpretability and TDA. Applying persistent homology from TDA to GNN interpretability is novel.", "essential_references_not_discussed": "no", "other_strengths_and_weaknesses": "Strengths:\n1. The paper opens a new direction in the area of GNN interpretability by applying topological data analysis.\n2. The theoretical foundation of the proposed method is solid, and the authors provide theoretical guarantees that their method can find a unique optimum.\n\nWeaknesses:\n1. The experiments are limited to synthetic datasets and simple real-world datasets and do not demonstrate the method’s potential for real-world applications.", "other_comments_or_suggestions": "1. ‘variiform', 'varriform' seem like typos, should it be 'variform'?\n2. Fig. 1 is hard to understand, even with the caption. Ideally, it should convey the core idea of the proposed method, and the caption should include all necessary definitions of the notations used in the figure. For example, $\\mathcal F$ and $\\mathcal T$ are not defined in either the figure or the caption.", "questions_for_authors": "1. The description of how the dataset BA-HouseOrGrid-nRnd is generated is a bit confusing. Is it correct that label 1 means the graph contains (a) only houses, (b) only grids, or (c) an equal number of houses and grids? Does label 0 mean the graph contains (a) neither houses nor grids or (b) an unequal number of houses and grids?\n2. Are the assumptions in Theorem 3.5 realistic for real-world graphs?", "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": {}}]

Author rebuttal:

Meta-review: The paper presents an approach that leverages PH in order to identify persistent rationale subgraphs. Considering the initial reviews and the author's rebuttal, my assessment is that all reviewers agree that the work is interesting and novel with largely sufficient evaluation. The author's did a good job in addressing the major concerns raised by the reviewer, and in my opinion, the paper also needs only minor adjustment to incorporate these issues. Hence, I think this is a solid contribution to ICML and I am recommending acceptance; also, I would like to encourage the authors to take all comments seriously in their preparation of the camera-ready version.
Decision: Accept (poster)