Diss-l-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms
Authors: Julius von Rohrscheidt
Bastian Rieck
Venue: ICML 2025 poster (2025)
Keywords: Euler Characteristic Transform
Topology
Geometry
Topological Data Analysis
Topological Deep Learning
Graph Learning
Primary area: general_machine_learning->representation_learning
TL;DR: This paper presents an alternative to message-passing for node classification by utilizing Euler Characteristic Transforms.

Abstract: The Euler Characteristic Transform (ECT) is an efficiently computable geometrical-topological invariant that characterizes the global shape of data. In this paper, we introduce the local Euler Characteristic Transform ($\ell$-ECT), a novel extension of the ECT designed to enhance expressivity and interpretability in graph representation learning. Unlike traditional Graph Neural Networks (GNNs), which may lose critical local details through aggregation, the $\ell$-ECT provides a lossless representation of local neighborhoods. This approach addresses key limitations in GNNs by preserving nuanced local structures while maintaining global interpretability. Moreover, we construct a rotation-invariant metric based on $\ell$-ECTs for spatial alignment of data spaces. Our method demonstrates superior performance compared to standard GNNs on various benchmarking node classification tasks, while also offering theoretical guarantees of its effectiveness.

Diss-I-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms
Julius von Rohrscheidt
Bastian Rieck 123
Abstract
on (weighted) sums, the ECT can be computed efficiently, making it a powerful tool for representation learning (Röell & Rieck, 2024). Moreover, the ECT is known to be invertible for data in IK", ensuring that the original data can always be reconstructed (Curry et al., 2022; Ghrist et al., 2018). In this paper, we extend the ECT local neighborhoods, presenting the Local Euler Characteristic Transform (ÉECTT), a novel method designed to preserve local structure while retaining global interpretability. The (EECT captures both topological (i.e., structural) and geometrical (i.e., spatial) information around each data point, making particularly advantageous for graph-based or higher-order data.' The (ECCT thus becomes an expressive fingerprint of local neighborhoods, specifically addressing the challenge of neighborhood aggregation in featured graphs while ensuring the lossless representation of local node neighborhoods. We theoretically investigate how E-ECTs maintain critical local details, and therefore provide a nuanced representation that can be used for downstream graph-learning tasks such as node classification. Our method is highly effective, particularly for tasks where node feature aggregation may obscure essential differences, such as in graphs with high heterophily. Additionally, the EEET framework's natural vector representation makes it compatible with a wide range of machine-learning models, facilitating both performance and interpretabilityy
The Euler Characteristic Transform (ECT) an compuutblee geometrical-topoloocca invariant that characterizes the global shape of data. In this paper, we introduce the Local Fuler Characteristic Transform (f-FCT)), novel extension of the ECT particularly designed to enhance expressivity and interpretability in graph representation learning. Unlike traditional graph neural networks (GNNs), which may lose critical local details through aggregation, the É-FCT provides lossless representation of local neighborhoods. This approach addresses key limitations in GNNs by preserving local structures while maintaining global interpretability. Moreover, we construct a rotation-invariant metric based on fEECTs for spatial alignment of data spaces. Our method exhibits superior performance compared to standard GNNs variety of benchmark node- classification tasks, while also offering theoretical guarantees that demonstrate its effectiveness.
1. Introduction
Traditionally, graph neural network (GNNs) rely on message-passing schemes to aggregate node features. While effective for many tasks, this approach often leads to the loss of critical local information, as the aggregation process can diffuse and obscure the original node vector representations (Topping al., 2022). This limitation makes it challenging to preserve local characteristics that may be essential for some applications. To address this limitation, we harness the Euler Characteristic Transform (Turner et al., 2014, ECT), an expressive geometrical-tooologgee invariant that can be computed efficiently. Relying only
As our main contributions, we (i) construct éEECTs in the context of embedded simplicial complexes (and graphs) and theoretically investigate their expressivity in the special case of featured graphs. (ii) empirically show that this expressivity positions E-FCT's as a powerful general tool for interpretable node classification, often superior to standard GNNs, and (iii) introduce an efficiently computable rotation- invariant metric based on f-ECTs that facilitates the spatial alignment of geometric graphs
2. Background
'Institute of AI for Health, Helmholtz Munich, Germany Technical University of Munich, Germany Unniversity of Fri- bourg, Switzerland. Correspondence Julius von Rohrscheidt cjulius..ohrscheidd @ helnhoott- munich.de>> Bastian Rieck <bas- tian, laan.groseenaacher @un.hhhhhheniiitt.h......
We define our method in the most general setting, i.e., that of a simplicial complex, while also providing a brief introduction graph neural networks.
'Our experiments deliberately focus on graphs, but we note that the method can be extended to novel higher-order datasets based on simplicial complexes, for instance (Ballester et al., 2025).
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Diss-I-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms
Simplicial Complexes A simplicial complex K is a mathematical structure that generalizes graphs model higher-order (non-dyadie) relationships and interactions. While graphs model pairwise (dyadic) connections between entities using nodes and edges, simplicial complexes extend this representation to higher dimensions by including triangles (2-simplices), tetrahedra (3-simplices), and their higher-dimensional analogues. Let oo..... affinely independent points. The (geometric) k-simplex determined by these vertices is the convex hull
by the model. The AGG function combines information from neighboring nodes, while the UPDATE function re- fines node embeddings. Popular choices for these functions include mean, sum, and attention mechanisms. Through multiple layers of message passing, GNNs aggregate information from larger neighborhoods, capturing both local and global graph structure (Veličković, 2023).
:......Uk.. Uk E R be
3. Related Work
GNNs have revolutionized the field of graph representation learning by enabling end-to-end learning of node/graph embeddings through message passing (Kipf & Welling, 2017). However, traditional GNNs face theoretical limitations that pose fundamental obstructions to learning expressive and general representations of graph data (Xu et al., 2019). Related to the latter phenomenon. GNNs known to suffer from issues like oversmoothing (Rusch et al., 2023; Zhang et al., 2024) and oversquashing (Di Giovanni et al., 2023). Hamilton et al. (2017) and Veličković et al. (2018) have addressed these issues by incorporating sampling and attention mechanisms into the message- passing paradigm. However, even these advancements often show limited performance, particularly in graphs with high heterophily, and there is no "general" GNN capable of handling both heterophilous and homophilous graphs. Recent work in graph machine learning thus started incorporating additional inductive biases into architectures, such as geometric information (Joshi et al., 2023; Pei et al., 2020; Southern al., 2023) or topological information (Horn al., 2022; Verma et al., 2024), with the ultimate goal of improving the expressivity of model, i.e., its capability to distinguish between non- non-isomorphic families of graphs (Morris et al., 2023).
σ a= [o0000 uk.
UR] ==
Σ Aaoy
A4
0,
Σ-=1
The points ......, U2 are called the vertices of a. Simplices determined by subset of Uo..... 2k called faces of a. Formally, a simplicial complex is a finite collection of simplices such that every face of a simplex in the collection is also in the collection, and the intersection of any two simplices is either empty or a common face.
Euler Characteristic The Euler characteristic x is a topological invariant that provides a summary of the "shape" or structure of a topological space, such as simplicial complex. It is defined as the alternating sum of the number of simplices in each dimension, i.c.,
X(K)=
(-1)) k=
(1)
where oie (K) is the number of k-dimensional simplices in the simplicial complex K, and d is the dimension of K. As topological invariant, the Euler characteristic remains unchanged under transformations like homeomorphisms, making it-despite its simplicity-a fundamental tool for distinguishing topological spaces.
Many such endeavors arise from the field of topological deep learning (Papamarkou et al., 2024), which aims to develop models that are "aware" of the underlying topology of space, and thus also capable of handling data with higher-order relations. Other constructions include special architectures for heterophilous tasks that are not based on message-passing (Lim et al., 2021), or modifications of the graph itself to improve predictive performance. Suresh et al. (2021), for instance, use edge rewiring to raise graph assortativity and thus gain accuracy under low homophily, whereas Luan et al. (2022) mix feature channels during aggregation to obtain state-of- the-art results on heterophilous benchmarks. Finally, as Rampášek et al. (2022) show, hybrid model, combining message- passsag-passing (local information) with attention (global information) via structural encodings, may exhibit high expressivity and high scalability. Subsequently, Müller et al. (2024) extended these results by providing a taxonomy of elements related to the "design space" of graph transformers.
Graph Neural Networks and Message Passing Graph neural networks (GNNs) are a class of neural network models designed to operate on graph-structured data. They extend neural networks by incorporating the relational structure inherent to graphs, enabling the learning of tasks such as node classification. The core mechanism of many GNNs is message passing, an iterative procedure that propagates information through the graph to update node representations based on their local neighborhood. Given a graph G (V, F.), where V is the set of nodes and F. is the set of edges, each node E V is associated with feature vector x,. At each layer I, i.e., each message-passing step, a new node embedding h
(t+1) is calculated via
UPDATE (n. AGG (nL)) E N(w))))
(2)
where (V) denotes the set of neighbors of node v, and AGG and UPDATE are learnable functions parameterized
2

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Diss-I-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms
As a geometriiall-ooooogiaall invariant, the ECT is poised to contribute to more expressive architectures. Contributing Being already popular tool in topologiaal data analysis (Ghrist et al., 2018; Turner et al., 2014), recent extensions started tackling the integration into deep-learning architectures (Röell & Rieck, 2024) or the incorporation of additional invariance properties (Curry et al., 2022; Marsh et al., 2024). Despite advantageous performance in shape- classification tasks, however, all existing contributions solely focus on global ECT's and do not discuss any local aspects, which are crucial for our approach. In addition to the ECT, some works also use other topology-basee -based tools in graph learning, primarily persistent homology (PH), an expressive but computationally expensive geometrical- topological invariant. Examples of this approach include Rieck et al. (2019) and Hofer et al. (2020), who use PH for graph classification, or Zhao & Wang (2019), who learn weighted kernel on topological descriptors arising during PH computations, Closest to our approach in spirit is Zhao et al. (2020), who include topological features of graph neighborhoods into a GNN, again leveraging PH. However, to the best of our knowledge, ours is the first work to develop local variants of the ECT for graph-learning tasks, analyze the theoretical properties of such local variants, and finally show their empirical utility for node classification.
((log/mmm)4 (Beck, 1987), and that the equidistant partitioning of a compact interval scales as O())), leading to O ((1oog (logm/m)) (n-)))))-m///))))))))))/)-)))))) for the total approximation error of the domain of (CT(X). Curry et al. (2022) prove that the aforementioned approximation actually determines the true value, provided that m, are sufficiently large. We notice that both translations and scalings of X in the ambient space lead a reparametrization of ECT(X). Hence, (XT( remains essentially unaltered (up to parameter change) under these two types of transformations.
Local ECT (EEECT) Given geometric simplicial complex X C R" and a vertex E X, we define the local ECT of x with respect to k 2 0 as
(-ECTx(E;X) = ECT(N((I;X)),
(4)
where (2; X) denotes an appropriate local neigh of x in X, whose locality scale is controlled by a parameter k. Usually, .Nx (2; X) will be either the full subcomplex of X, which is spanned by the k-hop neighbors of x, or the full subcomplex of X, which is spanned by the k- nearest vertices of z. The first important special case arises when X is O-dimensional simplicial complex, i.c., point cloud. In this case, the full subcomplex of X, which is spanned by the k-nearest vertices of æ, N₂ X), is given by the k-nearest neighbors of x. Being based on the Euler Characteristic, the construction of (-ECTs appears to be purely topological at first glance. However, in light of the invertibility theorem, we note that (FF:T( X) can be interpreted fingerprint of local neighborhood of x in X. The upshot is that this fingerprint can be well approximated in practice, making it possible to obtain local representations of combinatorial data embedded in Euclidean space. Similar to the approximation of the ECT, this approximation works by sampling ui..... Um S"-1 and f1, 14,...44 .4 instead of (EECT&(;; X). The latter quantity is well- computable in practice, and the approximation error can be controlled by the sample sizes m and I, as we discussed above. Again, this approximation has a natural representation as a vector of dimension m L, enabling us to encode local structural information of point neighborhoods in an approximate lossless way that can readily be used by machine- learning algorithms for downstream tasks.
4. Methods
Euler Characteristic Transform (ECT) The Euler Characteristic Transform (ECT) of a simplicial complex X C R. is a function EXT(X): 5n-1 R + Z. given by
(CT(()((,))) == x({=> E X x (x ()),
(3)
where x denotes the Fuler characteristic and denotes the Euclidean dot product. The interpretation of ECT(X) is that it scans the ambient space of X in every direction and records the Huler characteristic of the sublevel sets. The ECT(X) is invertible, meaning that X can be recovered from ECT(X), as long as X is a so-called constructible set (Curry et al., 2022; Ghrist et al., 2018). The main focus of this work are compact geometric simplicial complexes (like geometric graphs), which are constructible, and thus the invertibility theorem applies in our setting. Note that in practice, we approximate (CT(X) via ECT(X)(m.)) == ECT(X), (X) uniformly-diistiibueed directions UI,..., Um E filtration steps . t E R. Since X is compact, t,....,t can be chosen to lie in a compact interval [a, b] with f1 and t; b, and so that the sequence {t,}, forms a uniform partition of (a. b). We note that this approximation is efficiently computable and has a natural representation as a vector of dimension m I. Regarding the choice of the magnitudes of m, I we notice that the expected nearest- neighbor distance for uniform samples on S scales as
IR and considering ECT(N& (m,)
[(m......mm) x ()
for ) and
4.1. Properties of 4ECTTs
Our formulation of (EECTs provides natural representation of local neighborhoods of geometric simplicial complexes. One important special case is that of featured graphs, meaning graphs in which every node admits a feature vector. The latter data structure forms the basis of many modern graph-learning tasks, such as node classification, graph classification, or graph regression.
3

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Diss-I-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms
The predominant class of methods to deal with these graph learning problems are message-passing graph neural networks. We develop alternative procedure for dealing with featured graph data, built on EEECTss and we show that (-ECTs provide sufficient information to perform message passing, which we explain in the following.
In addition essentially subsuming the information from one message-passing step, we can also show that the EEFCT is "aware" of local structures like subgraphs. As shown by Chen et al. (2020), pessage-passingg graph neural networks cannol perform counting of induced subgraphs for any connected substructure consisting of or more nodes. By contrast, we will now show that ECTs for featured graphs and their local variants can indeed be used to perform subgraph counting. We start with the definitions of the necessary concepts.
Definition 1. A featured graph is a (non-directed, graph G such that every node
admits a feature vector x((v) E R.. We denote the set of nodes of G by V (G), and the set of edges by E(g).
Definition 2. Two featured graphs Gr and 92 are isomorphic if there is a bijection T: V(G1) - V(G2), such that (v. w) E F((g)) if and only if (((v);((w) E F((22) and so that for all v E G1 one has æ(v) ((T(()))( for the respective feature vectors.
We notice that a featured graph G can naturally be interpreted as graph embedded in IR", by representing each node feature vector as a point in R", and by drawing an edge between two embedded points if and only if there is edge between the underlying nodes in G. This construction yields a graph isomorphism between G and the embedded graph if and only if for any pair of nodes U, w E G with U o w we have z(v) o z(w) for their associated feature vectors. In practice, the latter assumption can always be achieved by adding an arbitrarily small portion of Gaussian noise each feature vector, and we therefore may restrict ourselves to featured graphs that yield an isomorphism their Euclidean embeddings. 2 We now show that {-FCTs are in fact expressive graph-lcarning representations.
A featured graph G3 is called subgraph of g if VGSs C (g) and F(Gs) C E(G), such that the respective node features remain unaltered under the induced embedding. A featured graph Gs is called an induced subgraph of g, if Gs is subgraph of G, and if E(Gs) = E(g) n Gs. For two featured graphs G and Gs. we define Csab (G: Gg) to be the number of subgraphs in G that are isomorphic to Gs. Similarly, we define (9a Gs) to be the number of induced subgraphs in G which are isomorphic to Gs.
Theorem 2. Two featured graphs Gn and G2 are isomorphic if and only ECCT()) = ECT((G).
Theorem 1. Let G be a featured graph and let (E-ECT, (x:; 9) be the collection of local ECTs with respect to the 1-hop neighborhoods in G. Then the collection {&-ECT, (x:CTT(;;9))) }) provides the necessary )oo-learrabbee) information for performing single message-passing step on G. in the sense that for a given vertex x E G one can reconstruct the feature vectors of its 1-hop neighborhood from (ECT((x;G))
An immediate consequence of the previous Theorem is: Corollary 1. ECTs can perform subgraph counting.
We therefore conclude that F.CT methods for graph- representation learning can be more powerful than message- passing- based approaches, suggesting the development of hybrid architectures, making use of both message passing and ECT variants.
Theorem tells us that for a featured graph G, the collection {&-ECT, (ECTT(2;9)))) F) already contains sufficient information to perform single step of message passing. The advantage of using é-ECTs instead of message passing to represent featured graph data lies in the possibility to additionally use {E-ECTa (z; be for k 2 2. which contain both structural and feature vector information of larger neighborhoods of nodes in the graph. This type of information is typically not explicitly available through message passing since passing messages to non-direct neighbors depends on prior message passing steps, which solely produce an aggregation of neighboring feature vectors.
4.2. Rotation-Invarian Metric based on Local ECTs
The aforementioned invariance properties of ECTs with respect to translations and scalings naturally raise the question if (-ECTs may be used to compare the local neighborhoods of two distinct points/vertices. However, the ECT is sensitive to rotations since rotating the underlying simplicial complex leads to misalignment of the respective directions in 5 Because a local comparison should not depend on the choice of a coordinate system, this property is a fundamental obstruction of using {-ECT as a local similarity measure. We therefore construct a novel rotation- invariant metric as follows. Let X.Y Y C IR." be two finite geometric simplicial complexes. Since X, Y are finite, ECT(X) and ECT(Y)) only take finitely many values, and we may define a similarity measure dECT(X,)) Y) as
*Alternatively, we can drop the requirement of an embedding by noting that featured graph can be considered as abstract simplicial complex G with an arbitrary function f: G + I= defined on its vertices and edges. In this case, we may define the ECT as ECT(X)(,, f)= = x(1f-(x E R" x v < t)). This formulation, developed by Marsh & Beers (2023), demonstrates that the ECT is generally applicable and daes not require node features to provide an embedding of graph.
ABCT(X, (X,Y) = inf ||ECT(X) - ECT(PY)) (5) pESO(n)
4

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Diss-I-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms
We first prove that this similarity measure satisfies the definitions of a metric.
models to several standard GNN models, namely graph attention networks (Veličković et al., 2018, GAT), graph convolutional networks (Kipf & Welling, 2017, GCN), graph isomorphism networks (Xu et al., 2019, GIN), as well as a heterophily-speeifee architecture (Zhu et al., 2020, H2GCN), Furthermore, we showcase how the rotation- invariant metric from Section may be used for spatial alignment of graph data.
Theorem 3. dECT is a metric on the collection of rotation classes finite simplicial complexes embedded in IR..
Theorem ensures that we may use dECT as a metric that measures the similarity between embedded simplicial complexes up to rotation. In particular, for simplicial complex X we have a rotation-invarian
IR" and z. E X, measure to compare local neighborhoods of x and y by defining decn (z, y; X) as inf peSO(x) [|--ECT (z; X)- - (U)CA((;)X))) practice, we approximate dECT(X, by
5.1. EECTSs in Graph Representation Learning
The link between message passing and (EECTs (cf. Theorem 1) encourages us to empirically validate the expressivity of &-FCT's for node-classification tasks. Given featured graph G and fixed k > 0, we assign (=; to every node v E G. We then use the EEETT corresponding to node together with the respective node feature vector as the input to classification models. Subsequently, we focus on XGBoost (Chen & Guestrin, 2016), as we found to outperform more complex models. However, our E-ECT can be used with any model. Notice that experiments are not about claiming state-of-the-ar performance but rather about showcasing that an approach based on EEECT yields results that are on a par with and often superior to more complex graph-learning techniques based on message passing, while at the same time working well in both heterophilous and homophilous settings.
In
ECT (X)(())m.)) ECT (mp())mmll PESO(N)
inf
for a choice of samples U1,..., Um E S and fi....,tt E R: this works analogously for the local version de (E. y; X). As discussed before, the approximations of the ECTs used in Eq. (6) have a natural vector representation, so that the ||~ in Eq. (6) is in fact the maximum of the entry-wise absolute differences between the two respective representation vectors. Hence, the approximation shown in Eq. (6) is efficiently computable. However, our experiments in Section 5 use the Euclidean metric for differentiability reasons.
4.3. Limitations
Implementation details We assume that we are given a featured graph G such that there is an assigmment V(G) + y. with V(G) being the node set of G and Y' being the space of classes w.r.t. the underlying node- classification task. For fixed k 2 0, E V(G) and (z; being the k-hop neighborhood of x in G, we then approximate &-ECT (z; 9) via (NT (m..))) for sampled directions and filtration steps, as explained in Section 4. We use m = 64 (but the number of samples may be tuned in practice) and use the the resulting l-dimensional vector(s) ECT ()) together with the feature vector of as additional inputs for the classifier. The architecture of our baseline models includes a two-layer MLP after every graph-neighborhorr aggregation layer, as well as skip connections and layer normalization. We train each model for 1000 epochs and report the test accuracy corresponding to the state of the model that admits the maximum validation accuracy during training. This makes the predictive performance of our baseline models directly comparable with Platonov et al. (2023).
While {-ECTs present clear advantages in preserving local details, there are some trade-offs to consider. In certain cases, messagg-passing GNNs, which aggregate information across neighbors, may be preferable, in particular for tasks where global context is more important than local details (see Coupette et al. (2025) for a recent analysis of graph-learning datasets under different perspectives). Furthermore, while our method is computationally feasible on medium-sized datasets (as demonstrated in our experiments), the complexity of "naively" calculating {-ECTs increases for larger is and with the size and density of the graph, suggesting a need for improved methods (see Section A.2 for extended discussion).
5. Experiments
In this section, we present experiments to empirically evaluate the performance of the E-ECT-based approach in graph representation learning, focusing on node- classification tasks. We aim to demonstrate how (-ECT representations can capture structural information more effectively than traditional message-passing mechanisms, especially in scenarios with high heterophily (even though we consider other scenarios as well). Our experiments compare the performance of --ECT-based
WebKB Datasets For all datasets of the WebKB collection (Pei et al., 2020), our E-ECT-based approach outperforms the baseline GNNs by far (cf. Table 1; GraphSAGE results from Xu et al. 2024). While the combination of both (EEC,, and EFEC, performs best for
5

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Model Diss-I-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms
Table 1:
Performance (aceuracy, in percent) of graph- learning models on WebKB datasets (5 training runs).
Table 3: Performance (accuracy, in percent) of graph-
learning models on Amazon datasets (5 training runs).
Model
Cornell
Wisconsin
Texas
Model
Computers
Photo
GCN
4.2
47.3 4..2 3.27 2,9
1.5
GCN
91.6 1.6
93.6 1.7
GAT
2,
48.2
GAT
92.4 1.3
94.8 1.1
GIN
4.1
2.5
GIN
1.5
82.2 1.3
GraphSAGE
3.50
1.
71.8
GraphSAGE
8.92
9.7
H2GCN
6.52
7..
7..
H2GCN
8.4
92.8
&-RCT,
6.2
21.2
0.5
(FCC
0.3
9..1
4ECT,
4,9
76.1
2.,
(EET,2
9..1
94.4
&ECT, +EECT2
67.1
78.5
74.8 3.1
(EECT, + &EEC,,
92.2 0.66
94.9
Table 2: Performance (accuracy, in percent) of graph-
Table 4: Performance (accuracy, in percent) of graph- learning models on heterophilous datasets (5 training runs).
learning models on heterophilous datasets (5 training runs).
Model
Amazon Ratings
Roman Empire
Actor
Squirrel
Chameleon
GCN
0.7
73.3 = 0.8
GCN
30.7 + 2.1
28.9 1.4
42.8 1.8
GAT
4.
76.4 1.2
GAT
31.1
47.3 1.3
GIN
4.8
56.8 1.0
GIN
2..5
1.5
43.1 1.7
GraphSAGE
0.5
GraphSAGE
1.22 0.7
1.3
H2GCN
0.7
64.2
H2GCN
1.9
1.4
1.67
(ECT,
0..
80.4
AFEC,
1,9
4.7 3.7
(ECT,
0.3
78.0 = 0.3
(ECT,,
30.1 1.3
40.4 35.6
4ECT 2ECT,
49.8 0.3
81.1 = 0.4
(ECTT, + &
30,9 35.
4.7
"Texas," using only (-ECT, leads to best performance for "Wisconsin." " However, for the two aforementioned datasets, the combination of (EECT, and éEECT, only slightly improves the performance in comparison to (EECT,, suggesting that 1-hop neighbor information is already sufficiently informative here.
Actor/Wikipedia Datasets Moving to additional heterophilous datasets with high feature dimensionality, we compare predictive performance on "Actor" (Pei et al., 2020) as well as "Chameleon" and "Squirrel" (Rozemberezki et al., 2021); ef. Table 4. For "Actor", the (-ECT, model achieves the highest accuracy, while 4EECT, + &EEC, performs slightly worse. <ECT,, performs the lowest on this dataset, suggesting that larger neighborhoods are detrimental here. For the other datasets, the heterophily- specific model H2GCN performs best. However, (-RCT, and (EECT, and (-ECT, exhibit similar (or even better) performance as all other standard baselines, showing the advantages of {-ECT methods in the absence of hyperparameter tuning.
Heterophilous Datasets Platonov et al. (2023) introduced several heterophilous datasets; we validate our method on "Amazon Ratings" and "Roman Empire," again observing that the combination of (-ECT, + 2-ECT2 performs best, substantially outperforming baseline models (ef. Table 2). The results are closely aligned with findings by Platonov et al. (2023), i.e., that specialized architectures like H2GCN often perform less well than "standard" architectures. Moreover, (-FCT, outperforms (-ECT, on "Roman Empire," while &EEC,, outpertonm Ratings." " We interpret this as 1-hop neighborhoods being particularly informative for "Roman Empire," " while 2-hop neighborhoods are more informative for "Amazon Ratings."
outperforms ÉEEC,, on "Amazon
Table 5: Performance (accuracy, in percent) of graph- learning models on "Planetoid" datasets (5 training runs).
Model
Cora
CiteSeer
PubMed
GCN
88.1 1.2
74.6
8.73
Amazon dataset The Amazon dataset (Shchur et al., 2018) consists of the two co-purchase graphs "Computers" and "Photo." While GAT outperforms all methods on "Computers," the combination of &ECT,, and É-ECT, performs best on "Photo" (cf. Table 3). Overall, --ECT- based methods exhibit competitive performance here, given that they are not based on message passing.
GAT
88.3
7,5
85.7
GIN
85,0
1.7
87.0
GraphSAGE
8.2
6.21
8.73
H2GCN
85.4
1.94
8..4
(EEC
0.6
7.6
90.2
AECT,
87.2
7.83
0..
2-ECT, +EECT,
8.6
7.75
0.5
6

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Diss-I-ECT:
Dissecting Graph Data with Local Euler Characteristic Transforms
Planetoid Datasets We also analyze node-classfficttonn performance on datasets from the "Planetoid" collection (Yang et al., 2016), comprising "Cora," "CiteSeer," " and "PubMed." We trained all models using a random 75-25 split; cf. Table 5. Although GCN and GAT perform slightly better than EEECT methods for "Cora" and "CiteSeer," the gap is surprisingly small. For "PubMed," the E-ECT-based models even outperform both all comparison partners. These findings suggest that the theoretical expressivity of EECT's, which we formally established in Section 4, also has practical implications, providing an alternative way of dealing with graph data that is not restricted by the underlying model architecture and therefore allows for interpretability.
the context of message passing). In this sense, E-ECTs naturally overcome fundamental limitation inherent to message-passing methods. Another advantage of (EECTs is that they are agnostic to the choice of the downstream model. This permits us to use models that are easy to tune, enabling practitioners to make use of their graph data without necessarily having specialized knowledge in GNN training and tuning while at the same time also working well in the small-sample regime. Moreover, it permits using models that are interprelable, making our method well- suited for domains where regulatory demands often ask for levels of interpretability that cannot readily be achieved by (graph) neural networks. In fact, by using feature importance values (which are directly available for tree- based algorithms like XGBoost) and since the entries of the EECT vectors that are used as the input for the model can be linked to the directions in the calculation of the E-ECTs, one may obtain a deeper understanding of how the model arrives at predictions (see Section A.3 for more in-depth discussion and an ablation on the number of directions used to calculate (EECTs).
Post-hoc Evaluation To evaluate our methods across datasets, we used critical difference diagrams, enabling us to compare the performance of various models on both homophilic and heterophilic graph datasets (ef. Section A.7). The results highlight the superior performance of E-ECT-based approaches over standard baselines architectures such as H2GCN.
and heterophily-speciie Notably, the E-FCT, + 8-FCT, method achieved the best average rank of 2 and even the least effective (-ECT-based model (f-ECT2) outperformed all non- b-seT- methods, including GAT. Further evaluation against heterophily -specfifi models reported in the literature corroborates these findings. In comparison state-of-the- art methods such as H2GCN, UPR-GNN, and GloGNN, (EECT, + E-ECT, achieved a competitive rank, matching GloGNN and surpassing both GAT and GT. Despite being a generul-purpose approach not specifically designed for heterophilic graphs, &- -bsed methods demonstrated exceptional adaptability and robustness across diverse graph structures. These results establish 6-ECT-based architectures as a versatile and high-performing solution for node classification tasks, suitable for tackling challenges across a wide range of graph data.
5.2. Learning Spatial Alignment of Geometric Graphs
In the following, we use the approach described in Section 4 in order to learn the spatial alignment of two data spaces by re-rotating one into the other. We start by showing that synthetic data, which only differs up rotation, can be re-aligned using EEECTs. Moreover, we show that this alignment is stable with respect to noise, making it a robust measure for the comparison of local neighborhoods in data. In comparison to other spatial alignment methods like the iterative closest point algorithm, ours does not necessitate the computation of all pairwise distances between points in the respective spaces. The latter is often a computational bottleneck, especially for large datasets, thus positioning a computationally
our method for spatial alignment more efficient method in practice. While alignment methods like Procrustes alignmenl are restricted to point- cloud data, we observe that our approach is also capable of aligning embedded graph data (or, more generally, simplicial complexes). This makes it particularly useful for dealing with geometric graphs, constituting highly- efficient alternative to more involved machine-learning models like geometric GNNs (Joshi et al., 2023).
Summary of Mode-Classification Experiments We find that 8-FCC's work particularly well in situations where aggregating neighboring information is inappropriate, such as when dealing with graphs that exhibit high degree of heterophily. In such contexts, our approach may outperform message- saassagg-asssin--tee based methods. The upshot of our method is that local graph information can be incorporated without the architectural necessity to diffuse information along the graph structure, as it the case for message-passing pass-ag-assseeeeee models. While this discrete diffusion process induced by message passing is useful for a plethora of graph-learning tasks, it can also be an obstruction in learning the right representation for tasks where node features of neighbors in the graph should not be aggregated (cf. Coupette et al. 2025 for a recent analysis of graph-learning datasets in pESO(n)
For the subsequent learning problem (see Section A.5 for an example of how to align point-cloud data), we assume that we are given two embedded simplicial complexes X,Y Y c R". In light of Section 4, the metric properties of dEcT ensure that decr Y) = 0 if X and Y only differ up to rotation. We therefore approximate dECT(X, Y) via
min
ECT (CT(()(m.)) ECT mm..l12
(7)
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Diss-I-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms
(a) Examples geometric graphs
2-star graph (line)
3-star graph
5-star graph
11-star graph
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Hausdorff distance
(b) Alignment results
Figure I:
A comparison of the Hausdorff distances of aligned graphs. The black dots represents the Hausdorff distance between the original graph and mudomll-rotateed version of itself. Our (-ECT-based alignment always results in substantially lower distances, with median distance close to zero.
for a choice of directions U1,..... E Sn~1 and filtration steps f,,...,4 l E R. As explained in Section 4, the ECT approximations are given by vectors, making it feasible to approach the above learning problem by any gradient-basse based learning algorithm. The advantage of this formulation is loss and the rotation
200 times, maintaining consistent initializations for both the graph and the rotation matrix. Figure shows the results; we observe that our realignment procedure consistently achieves small Hausdorff distances with medians near zero, indicating the successful recovery of the original graph's orientation. By contrast, the Hausdorff distances between the original graph and its perturbed version are significantly larger. While equivariant GNNs have been shown to struggle with distinguishing the orientation of rotationally-symmend structures (Joshi et al., 2023), our method generates graph representations that are initially sensitive rotations but can be made rotation-invariam through alignment of the underlying ECTss.
that it yields both the that leads to this minimum loss.
Geometric graphs provide a compelling example of the utility of our method, particularly in addressing the challenging problem
graph re-alignment. For this analysis, we focus on a specific type of geometric graph known the k-star graph. A k--tarr graph is defined tree with one internal node and & leaves, i.e., simple structure with relevant geometric properties (ef. Figure la). To embed such graphs into 2D space, we assign a unique 2D vector to each node, ensuring the assigned vectors are equidistant to maintain structural symmetry. Furthermore, introduce variability and assess robustness, we subject each embedded graph random 2D rotation, simulating realistic perturbations encountered in practical settings. The central goal is to recover the original graph's 's orientation by learning the rotation matrix using the metric defined Section 4. We evaluate the performance of our approach by measuring the similarity between the original graph and re-rotated version using the Hausdorff distance, metric that quantifies the maximal deviation between two of points. To ensure significance, repeated the learning procedure
These findings highlight the potential of RCT--based metrics for robust geometric graph alignment, paving the way for broader applications in domains requiring precise graph- based comparisons such as the analysis of geometric graphs with constrained parameter budgets (Maggs al., 2024). This result is particularly notable since the (-ECT easily outperforms more involved architectures, pointing towards its overall utility as alternative to message- pessaagepassing graph neural networks. At the same time, we believe that the EEETT could also help in aligning higher-order data like geometric simplicial complexes.
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6. Discussion Diss-I-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms
Future Work: Hybrid Models Beyond representation learning on graphs, &EECT framework also opens up new applications in domains where local structure is critical, such point-cloud analysis (including sensor data or computer-graphics data), 3D shape analysis, or data from the life sciences (like molecular data or biological networks). Future work could thus explore more efficient algorithms for computing ECTs and EEECTs at scale, as well as hybrid approaches that balance local and global information more effectively. A highly-relevant direction would be the integration of {-FCTs into existing paassagg-pssingg neural networks, similar to recent work that leverages persistent homology (Verma et al., 2024). Moreover, heterophily- specific mechanisms such as a separation of neighborhood aggregation (as used in specialized GNN architectures) may be incorporated into our 4-ECT-based framework to further strengthen its expressivity in the presence of high- heterophily graphs.
We introduced the Local Euler Characteristic Transform (LEETT) providing a novel approach to graph-representation learning that preserves local structural information without relying on aggregation. Our method addresses fundamental limitations in message-passing neural networks, particularly in tasks where aggregating neighboring information is suboptimal, such as in graphs with heterophily. By retaining critical local details, &ECTTs enable more nuanced and expressive representations, offering significant advantages in node classification tasks, One key strength of our approach is its model-agnosti nature, allowing it to be paired with interpretable machine learning models (in our experiments, an XGBoost model was used to provide feature importance values, for instance). This is particularly useful in domains such as healthcare, finance, and legal applications, where regulatory frameworks demand high levels of transparency and interpretability that are often difficult to achieve with black- box neural networks. By leveraging EEECT', we can satisfy these requirements while maintaining the high expressivity and high predictive performance required for graph-learning tasks. In this way, 4-FCT- representations offer a novel pathway toward interpretable machine learning on graph data: they yield setoooiicalllggouneeeeee vectorized encodings of local structure that not only retain predictive power but also support downstream diagnostics.
Software and Data
Our code is available under aidos- LibosslabbDiss-111 henchmarking datasets, loaded and processed via the PyTorch Geometric library (Fey & Lenssen, 2019).
com/ We make use of standard
Acknowledgments
The first author acknowledges the use of ChatGPT for grammar suggestions. Both authors are very grateful for the discussions with the anonymous reviewers, in particular reviewer nYTM, and the area chair, who also believed in the merits of this work. This work has received funding from the Swiss State Secretariat for Education, Research, and Innovation (SERI).
Higher-Order Domains Being situated at the intersection of geometry and topology, our EECC method is part of the nascent field of topological deep learning (Papamarkou et al., 2024), which aims to develop novel inductive biases that are capable of leveraging additional structural information from data, both in the context of graphs and in the context of higher-order domains like simplicial complexes. It is in this context where we believe that future work could be beneficial, in particular since recent research (Ballester et al., 2025) showed that tasks on such domains are highly challenging for existing GNNs. Given the advantageous scalability properties of the ECT (Röell& & Rieck, 2024; Turner et al., 2014), we believe that this constitutes a useful avenue for future research.
Future Work:
Impact Statement
This paper presents work whose goal is to advance the field of graph machine learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
References
Future Work: Comparing Representations Containing both geometrical and topological components, we also believe ECT-based methods to be suitable in navigating different representations.
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A. Appendix
A.1. Proofs
We briefly restate all theorem from the main text for the reader's convenience before providing proofs.
Theorem 1. Let G be a featured graph and let {6-ECT, (I; g) be the collection of local ECTs with respect to the 1-hop neighborhoods in G. Then the collection {f-ECT, (z; }) provides the necessary (non-learnable) information for performing a single message- passsing step on 9, the sense that for given vertex 9 one can reconstruct the feature vectors of its 1-hop neighborhood from (=;CT1(<;;))
Proof. By the remark in Section 4 (in the paragraph right above the original statement of this Theorem), we may assume that the natural embedding of & into R is graph isomorphism. Then, making use of the invertibility theorem, the 1-hop neighborhood of point in the embedding of G can be reconstructed from E (z: 9). Therefore, the feature vectors of x and its 1-hop neighbors can be deduced from E-ECT (2;; 9), which is the only non-learnable information one needs to perform a message-passeaee passing step.
Theorem 2. Two featured graphs 91 and G2 are isomorphic if and only if EFCCT()) = ECT(()).
Proof. When two featured graphs are isomorphic in the sense of Definition 2, their respective Euclidean embeddings produce equal ECT's by construction because the node feature vectors of two corresponding points under the isomorphism are equal, By contrast, let us assume that EC) = (G2) Then by the invertibility theorem, the Euclidean embeddings of G and G2 are equal. Therefore, the only information that may tell apart the two graphs are their node labels, but this means that 91 and G2 are isomorphic.
Theorem 3. dECT is a metric on the collection of rotation classes of finite simplicial complexes embedded in R".
Proof. Xx) = 0 holds for p being the identity. Now assume that dgcr (X, Y) = 0. Then there exists E SO(n) with (XECT(X) - (CT( = As || a norm, follows that ECT(X) ECT(pY), and by the invertibility theorem we obtain X pY, This shows the first property of metric (note that positivity follows from ||=]].c). ). For symmetry, note that (X ECT(pY)) inequality, let Z be another finite simplicial complex. d (X, Z) then reads inf pnfpp) [|(ECT(X) - ECT( + +I(EET((p')) ECC(()
(ECT(p- XX) ECT(Y)) since rotations are invertible. For the triangle
EC) This term, which is less than or equal to ( ECT + (ECT((Y - ECT((P')~ pZ)) xc)) which is equal however, is equal to inf p.o' (II(EC)( - to inl (E)((n) [I(EC(()) ECT (CC()))))l )llso + +in (IECC(() ( )CC But this final term is precisely the definition of dEcT (X, Y) + dECT(Y, (Y, Z).
(II(EC((
A.2. Computational Complexity
For fixed node the computational complexity of (EECT((x) is O(m - [Nx(z))), where: (i) m is the number of sampled directions, (ii) l is the number of filtration steps, and (iii) [N)( is the number of vertices (or simplices) in the k-hop neighborhood of æ.
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Diss-I-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms
Feature Importances
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
Figure 2: Feature importance scores of an XGBoost model for the "Coauthor Physics" dataset (using 2-ECT,). Only small number of features admit high importance scores.
A.3. Ablation on Directions and Interpretability
Coming back to our approximation of ECT(X) via ECT(X) (m..) == ECT(X) )x for uniformly- distributed directions U1, B1....,P1 Um 5 and filtration steps 11,...,4 R. we notice that the (I (j-jj-1 - 1) + 1)-th till (I -))-th entries of ECT(X) (m.1) correspond the direction v.. The latter gives the opportunity get a deeper understanding of how the model predicts its outcome, by analyzing its feature importance values (which available for tree-based algorithms like XGBoost). Therefore, approach enables to analyze which features, i.c. directions, of the underlying ECT vector are most important. In practice, we often observe that small number of features admits high feature importance with respect to the corresponding model (ef. Figure 2). This raises the question if we may use smaller random collection features and still obtain reasonably useful results. We therefore ran experiments for collection of (EECT, vector; ef. Table 6. Here, 4096 corresponds to
datasets for a varying number of randomly-sample entries the whole vector. We observe that for certain datasets, such as "Coauthor CS," "Coauthor Physics," and "Amazon Ratings," the performance of the model only slightly changes when using a reduced version of the 4-ECT, vector. In light of the results by Curry al. (2022), this observation is not entirely surprising- one main claim therein is that the ECT can be determined using small number of directions.
Table 6: Mean accuracy (in percent, 5 runs each) for different node-classification tasks, and varying numbers of randomly- sampled entries of the corresponding EETT,, vectors.
Dataset
50
100
500
1000
4096
WikiCS
67.8
69.2
70.5
71.3
72.7
74,6
Coauthor CS
92.1
92.3
92.4
92.5
92.6
92.6
Coauthor Physies
95.2
95.6
95.6
95.8
95.9
96.1
Roman Empire
73.7
75.8
78.3
79.7
80.4
Amazon Ratings
47.9
47,9
48.2
48.4
48.2
48.4
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Diss-I-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms
Table 7: Performance (accuracy, in percent) of graph-learning models on WikiCS dataset (5 training splits).
Model
WikiCS
GCN
75.2 =
GAT
78.7 = 1.2
GIN
74.2 1.7
GraphSAGE
73.4 = 1.5
H2GCN
75.3 = 1.4
&ECT
74.6 0.5
aligned
non-aligned
0
100
150
200
250
300
350
400
Squared L distance
Figure 3: A comparison of the squared L distances of (-ECT's of aligned and non-aligned MNIST digits of "1," respectively.
A.4. Additional Node Classification Experiment
WikiCS Dataset To further validate the effectiveness of our approach, we consider the WikiCS dataset (Mernyei & Cangea, 2020), a medium-sized co-occurrence graph derived from Wikipedia articles on computer science topics. Nodes represent articles, and edges reflect mutual links between them. Each node equipped with a 300-dimensional embedding, and the task classify articles into one of several predefined categories. As shown in Table 7, our ( method achieves competitive performance compared message- pagge-gggnnngg baselines. While GAT obtains the best accuracy overall, é-ECT, performs par with H2GCN and GCN, despite not relying on neighborhood aggregation. This supports the idea that &-ECT-based representations can serve effective input features in classification settings, even for graphs with moderately homophilous structures. The small standard deviation further illustrates the stability of our method across splits.
A.5. Spatial Alignment of High-Dimensions Data
Following our previous observations that dECT enables to align two spaces, we now use it to investigate its effect on nigh-dimensional data. We start this discussion with the well-known MNIST benchmark dataset, following an analysis of local maneetricalltpologiiccoo structures that we performed previously (von Rohrscheidt & Rieck, 2023). We thus first represent each (gray-scale) image in the dataset as 784-dimmnnioonl vector, by flattening the image. In this way, we obtain high-dimensional point cloud corresponding to the dataset. Subsequently, we sample 300 points of digits of "1" and calculate the pairwise distances of their respective EEECT (with respect the whole point cloud), for k 10. Finally, calculate the pairwise distances of the respective aligned E-ECTs (by using the approach of Eq. (7) with & 10). Figure 3 shows the results; we observe that the aligned (-FCTs have significantly lower squared 7,2 distance (with median of x 112) than the non-aligned ones (with median of RJ 224), showcasing that rotations cause dissimilarity between small neighborhoods of points, in many cases.
A.6. Homophily Scores of Mode-Classificatte Experiments
Table 8 reveals that our benchmark suite spans the entire range from extreme homophily to extreme heterophily. At the homophilic end lie the Amazon co-purchase graphs "Computers" and "Photo." together with the "Planetoid" citation graphs "Cora," "CiteSerr." and "PubMed." In each of these networks, at least seven of every ten edges connect nodes with identical class labels, replicating the conditions under which early message-pase paaagggg GNNs achieved their seminal successes. Near the middle of the spectrum, "Cornell" and the "Amazon Ratings" datasets exhibit mixed behavior, with roughly one third of their edges being heterophilous. Thus, neighborhood aggregation still conveys useful class-specific information, but the signal is noticeably diluted. The lower end is populated by the remaining datasets, where fewer than one edge in three is homophilic, and by "Roman Empire," the most extreme case in our experimental suite, where only about one edge in twenty links same-label endpoints.
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Diss-IEET:: Dissecting Graph Data with Local Euler Characteristic Transforms
Table 8: Fdge-homophily ratios for every dataset used in our node-classification experiments, sorted in ascending order. As the table shows, our experiments comprise a wide variety datasets.
Because six datasets heterophilic, five strongly homophilic, and two occupy the transition zone, believe experimental suite to be effectively balanced. A model must therefore operate reliably across sharply different structural regimes to achieve consistently high average rank. Classical message-passing architectures depend on homophily and tend to deteriorate as the ratio falls, whereas our empirical results from Section demonstrate that our proposed EEETT representations retain competitive and often even superior- accuracy regardless of homophily level. The most conspicuous gains appear precisely on the graphs where neighbor aggregation is least informative, namely "Roman Empire," "Texas," and "Wisconsin," confirming that (-ECT features capture structural cues that message passing alone fails to exploit. Hence, the numerical landscape mapped
Dataset
Hegge
Roman Empire
0.047
Texas
0.110
Wisconsin
0.210
Actor
0.217
Squirrel
0.220
Chameleon
0,230
Cornell
0,300
Amazon Ratings
0.380
CiteSeer
0.736
Amazon Computers
0.777
Table 8 substantiates the claim that our experimental design both stresses the limits of common GNNs while the same time showcasing the robustness of (-F-CT approaches in settings where label agreement along edges is sparse.
PubMed
0.802
Cora
0.810
Amazon Photo
0.827
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Diss-I-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms
13X
100K
500
Critical difference diagram showing the ranks of different models across all node-classífication tasks from Section 5. Even the worst- pest-performing é-ECT-bbuse approach (f-ECT,) exhibits superior performance to all other methods, when averaged across all tasks.
Figure 4:
A.7. Post-hoe Evaluation of Node-Classification Experiments
A critical difference diagram arranges the average ranks of multiple models across a set of datasets in order to facilitate overall performance comparisons between the model performances. Such diagrams are commonly used when comparing a suite of models different datasets (ef. Borgwardt al. (2020) for similar plots in the context of graph kernels).
Figure 4 shows the results for all node-classification results from Section 4, including both homophilic and heterophilic graph datasets. We observe that the 2-ECT-basse approaches outperform standard methods and the reterophily-speccnee architecture H2GCN by far, when averaged over all datasets." 3 'Te best-performing method (FCC + & E-ECT, exhibits an average rank of 2, while the worst performing method GIN with an average rank of 5.7. Even the worst-performing method, i.e., GAT. However, the most
(-ECT-based method (4-ECT2) performs better than the best Con-é-ECT-bee interesting fact that can be gleaned from the diagram involves the statistical significance of the results. Methods connected the same bar not performing statistically significantly differently. This seemingly negative result has positive implication: Despite being orders of magnitude more complex, even specialized graph neural networks do not perform statistically significantly better than --ECC--base methods. Given that our results are based on standard XGBoost model without any task-specific hyperparameter tuning, we believe that this demonstrates the potential and practical utility of our proposed methods.
To further evaluate the performance of our methods in comparison to those reported in the literature, we also included comparison with the results presented by Platonov et al. (2023), using the ranks of the respective models as the basis for evaluation; cf. Table 9. Among the listed methods, several, such as H2GCN, CPGNN, and GPR-GNN, are explicitly designed for heterophilic graph settings, leveraging specialized architectures to handle the challenges posed by such data. contrast, our + method, despite being general-purpose approach not tailored specifically for heterophilic settings, achieves competitive rank of 11. This performance is par with other top-performing heterophily-specte as GloGNN, and outperforms well-established architectures like GT and GAT by a significant margin. Overall, these results highlight the robustness and adaptability of our method, demonstrating its ability handle diverse graph structures effectively without requiring customization for heterophilic scenarios. In consideration of the results given in Figure 4, this makes 4ECT-basek approaches versatile general-purpose solution for node classification tasks.
Table 9: Ranks (lower is better) of models from Platonov et al. (2023) across the heterophilic datasets therein, in comparison to our methods. Notice that our method is a ayneral-purpose method for node classification and neither geared towards heterophily nor homophily.
Model
Rank
H2GCN (Zhu al., 2020)
18.3
CPGNN (Zhu al., 2021)
16.8
GPR-GNN (Chien et al., 2021)
15.3
ResNet (He et al., 2016)
13.8
models, such
1-ECTI
12.4
1-ECT2
12.4
GAT (Veličković et al., 2018)
12.3
GT (Shi et 202))
11.0
I-ECT1 + 1-ECT (ours)
11.0
GloGNN (Li al., 2022)
11.0
ResNet+SGG (Wu et al., 2019)
10.8
FAGCN (Bo et al., 2021)
10.0
JacobiCony (Wang & Zhang, 2022)
9.8
GCN (Kipf & Welling, 2017)
9.6
GBK-GNN (Du et al., 2022)
9.0
ResNettadj (Zheleva & Getoor, 2009)
7.3
SAGE (Hamilton et al., 2017)
5.9
GAT-sep (Veličkovié et al., 2018)
5.5
GT-sep (Shi al., 2021)
5.3
FSGNN (Maurya al., 2021)
3.0
/tt aiggaaa for the creation of the critical difference diagram.
We used wwwphhttttthhhbb.ee.ct wwwp:/tthhwbbccccooaaaaaaaaaaaaa
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Diss-I-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms
aligned
non-aligned
0
6
8 10
12
14
16
18
20
22
24
26
28
30
32
34
36
Squared L distance
Figure 5: comparison of the squared T,2 distances of the ECTs aligned and non-aligned wedged spheres, respectively. We see that alignment results in median loss of zero, thus effectively showing that the two spaces are the same.
Original point cloud Rotated point cloud
Original point cloud Re-Rotated point cloud
y
Figure 6: A comparison of two wedged spheres with one being rotated around the wedge point and the points being perturhed by Gaussian noise (left) and the learned re-rotated sphere that is aligned with the original data (right).
A.8. Spatial Alignment of Wedged Spheres
We approximate the optimization problem from Eq. (7) to show that we can learn spatial alignment of two data spaces, while the distance between ECTSs of non-aligned spaces that only differ up to a rotation will generally be high. We start with so-called wedged sphere, meaning two 2-dimensional spheres which are concatenated at gluing point (cf. Figure and von Rohrscheidt & Rieck 2023). We use 2000 uniformly-sample the squared 2 loss between the ECTss of this sample and rotation the same data space. We repeat this procedure 500 times, where each step both the sample of the wedged sphere and the rotation matrix which yields the rotated version of the same space are sampled randomly. We notice that the L losses between the non-aligned spaces are high (with median of around 19), whereas the 2 losses of the non-aligned spaces are significantly lower, with a median loss close to zero (cf. Figure 5). Moreover, we observe that the ECT of the same space significantly changes when the coordinate system transformed, which corroborates the necessity of a rotation-invan -natiinninvariiant metric for the comparison of ECTss. We conclude that an alignment of the ECTSs the two underlying data spaces fact leads to an alignment of the data spaces itself, as promised by the theoretical results in Section
points from such wedged sphere, and compare
Robustness Figure and Figure 8 show that the spatial aligmment wedged spheres still works satisfactorily, even in the presence of outliers and noise. This property is an important feature when dealing with real-world data, which is often noisy, and enables to align spaces that only approximately differ up to rotation. By contrast, the Hausdorff distance, i.e., widely-used metric between point clouds is (by definition) highly sensitive to outliers. We therefore conclude that the proposed metric based on ECTs is a robust metric to compare point clouds of potentially different cardinalities.
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Diss-I-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms
Original point cloud Rotated point cloud
Original point cloud
Re-Rotated point cloud
Y
Figure 7: A comparison of two wedged spheres, with one being rotated around the wedge point and added 200 outliers (left) and the learned re-rotated sphere that is aligned with the original data (right).
Original point cloud Rotated point cloud
Original point cloud Re-Rotated point cloud
V
Figure 8: A comparison of two wedged spheres, with one being rotated around the wedge point and the points being perturbed by Gaussian noise (left) and the learned re-rotated sphere that is aligned with the original data (right).
20


Reviews:
[{"review_id": "1rSZs9qZxM", "reviewer": "Reviewer_vCjY", "summary": "The paper introduces a novel method called the Local Euler Characteristic Transform (ECT), designed to enhance graph representation learning by preserving critical local structures while maintaining global interpretability. ECT provides a lossless representation of local neighborhoods around graph nodes. This method, grounded in topological principles, uses Euler characteristic transformations to capture both structural and spatial information around each node, making it particularly useful for tasks like node classification. The authors argue that ECT can overcome limitations found in GNNs, especially in graphs with high heterophily, where aggregating neighboring information could obscure crucial differences. The paper also introduces a rotation-invariant metric based on ECT, which helps in spatially aligning data spaces, offering a practical advantage in comparing graph data.", "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": "Dear Reviewer,\n\nwe sincerely thank you for the thoughtful and constructive feedback. \nWe are pleased that you found our paper to be well-written and our proposed method—the Local Euler Characteristic Transform (l-ECT)—to be novel and well-motivated. Below, we address your main concerns point-by-point and outline the changes we will make in the revised version.\n\n> The Texas and Wisconsin datasets are relatively small, and previous studies [1] have indicated that the Chameleon and Squirrel datasets contain significant amounts of duplicate data, which undermines their reliability. So the above datasets are not suitable for evaluating heterophilic GNNs any more, and I suggest that authors should consider more datasets proposed by [1] such as minesweeper, tolokers, questions.\n[1] Platonov et al. A critical look at the evaluation of GNNs under heterophily: Are we really making progress? ICLR 2023.\n\nThank you for pointing this out. We agree with your assessment of Chameleon and Squirrel and appreciate the recommendation. While our focus was to provide a broad comparison with commonly used benchmarks in heterophilic GNN evaluation, we recognize the importance of newer, cleaner datasets. **In the revised version, we will include results on datasets like Minesweeper, Tolokers, and Questions as proposed in [1].**\n\n> More heterophilic GNN baselines such as spectral-based models [2] [3] [4] and spatial-based models [5] [6] should be considered in the section 5. Besides, GraphSAGE is also a powerful baseline for heterophilic graphs. [...]\n\nWe appreciate the reviewer’s recommendation and agree that these are strong and relevant baselines in the context of heterophily-specific architectures. However, we wish to emphasize that our approach is not only proposed as a specialized architecture for heterophilic graphs, but rather as a general-purpose representation method that provides a robust and interpretable alternative to neighborhood aggregation. Our focus is on illustrating the effectiveness of l-ECT as a complementary or standalone representation, rather than competing directly with task-specific model architectures. That said, we agree that GraphSAGE is a widely adopted and illustrative baseline for generalization across homophilic and heterophilic settings. **We will include GraphSAGE in our revised experiments, and clarify the scope of our contribution with respect to architecture design for heterophilic graphs.**\n\n> In Table 7 of Appendix A2.3, all the listed models should be accompanied by their corresponding references.\n\nThank you for spotting this oversight. **We will revise Table 7 to include full citations for each model.**\n\nWe thank the reviewer again for acknowledging the **novelty, motivation, and clarity of our work.** We believe that the l-ECT offers a fundamentally new perspective on graph representation learning—particularly by preserving geometrical-topological information at the local scale, a signal that is often diluted by aggregation-based models.\nIf our revisions adequately address your concerns, we would be grateful if you would consider raising your overall score. We are happy to address any additional questions.\n\nBest regards,\n\nThe Authors", "claims_and_evidence": "Claims made in the submission are clear and convincing.", "methods_and_evaluation_criteria": "The Texas and Wisconsin datasets are relatively small, and previous studies [1] have indicated that the Chameleon and Squirrel datasets contain significant amounts of duplicate data, which undermines their reliability. So the above datasets are not suitable for evaluating heterophilic GNNs any more, and I suggest that authors should consider more datasets proposed by  [1] such as minesweeper, tolokers, questions.\n\n[1] Platonov et al. A critical look at the evaluation of GNNs under heterophily: Are we really making progress? ICLR 2023.", "theoretical_claims": "I am not sure the correctness of the proposed proofs for the theorems.", "experimental_designs_or_analyses": "More heterophilic GNN baselines such as spectral-based models [2] [3] [4] and spatial-based models [5] [6] should be considered in the section 5. Besides, GraphSAGE is also a powerful baseline for heterophilic graphs.\n\n[2] Bo et al. Beyond low-frequency information in graph convolutional networks. AAAI 2021.\n\n[3] He et al. BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation. NeurIPS 2021.\n\n[4] Luan et al. Revisiting Heterophily For Graph Neural Networks. NeurIPS 2022.\n\n[5] Li et al. Finding global homophily in graph neural networks when meeting heterophily. ICML 2022.\n\n[6] Want et al. Powerful graph convolutional networks with adaptive propagation mechanism for homophily and heterophily. AAAI 2022.", "relation_to_broader_scientific_literature": "This paper builds on limitations of losing local information by introducing the Local Euler Characteristic Transform (ECT), a method inspired by topological data analysis, which provides a topological fingerprint of local graph neighborhoods. The use of Euler characteristic transformations is rooted in previous works like Turner et al. (2014), who utilized Euler characteristics for global shape classification. However, unlike prior work, which focuses on global properties, this paper introduces a local variant, ECT, that preserves detailed neighborhood information while still maintaining global interpretability. This extension is crucial, especially for graphs with high heterophily, where local node characteristics may be more important than aggregated neighbor features.", "essential_references_not_discussed": "Please refer to my comment in Experimental Designs Or Analyses section.", "other_strengths_and_weaknesses": "Strengths:\n\n1. The paper is well-written, and the motivation behind the proposed approach is sound and clearly articulated.\n2. The idea of extending the Euler Characteristic Transform to capture local graph structures is novel and intriguing.\n\nWeaknesses:\n\n1. Please refer to my comments in the sections above.\n2. In Table 7 of Appendix A2.3, all the listed models should be accompanied by their corresponding references.", "other_comments_or_suggestions": "NA.", "questions_for_authors": "NA.", "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": "0WZprctICE", "reviewer": "Reviewer_DDk2", "summary": "The paper introduces the Local Euler Characteristic Transform (l-ECT), a novel approach for graph representation learning that extends the Euler Characteristic Transform (ECT) to local neighborhoods. The key innovation is capturing local structural information in graphs without relying on conventional message-passing aggregation schemes used in Graph Neural Networks (GNNs). The authors theoretically show that l-ECTs can preserve local neighborhood information without loss and provide a rotation-invariant metric for spatial alignment. They empirically demonstrate that l-ECT-based approaches outperform standard GNNs on various node classification tasks, particularly on heterophilous graphs where traditional aggregation methods struggle.\n\n## update after rebuttal\nAfter checking rebuttal from different reviewers, I incline to keep my score.", "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": "Dear Reviewer,\n\nwe sincerely thank the reviewer for the careful and thoughtful evaluation of our work. We appreciate your recognition of our contributions, particularly the novelty of integrating topological data analysis into graph representation learning, and your acknowledgment of the soundness of our theoretical and empirical work. Below, we address your comments and questions in detail.\n\n\n> Limited exploration of computational efficiency—l-ECT computation could be expensive for large graphs.\n\nFor a fixed node $x$, the computational complexity of its $\\ell ECT_k(x)$ is:\n\n$$\nO\\big(m \\cdot l \\cdot |N_k(x)|\\big),\n$$\n\nwhere:\n\n- $m$ is the number of sampled directions,  \n- $l$ is the number of filtration steps,  \n- $|N_k(x)|$ is the number of vertices (or simplices) in the $k$-hop neighborhood of $x$.\n\nIn other words, the complexity scales linearly with neighborhood size and sampling resolution. In practice, as shown in Appendix A.2.1, even a small subset of directions suffices for high accuracy, reducing runtime *considerably*. For large graphs, parallelization of l-ECT computation provides scaling of the procedure. **We will include a thorough discussion on computational complexity and improved implementations in our revision.**\n\n> Relationship between l-ECT parameters and performance is not thoroughly analyzed\n\nIn Appendix A.2.1, we provide an ablation study on the number of sampled directions, showing how performance varies when using a smaller portion of directions. That said, we agree that further exploration with respect to other parameters such as the number of filtration steps l would be valuable. **We are happy to include these additional ablations in the revised version to provide a more complete picture of the trade-offs involved!**\n\n> LINKX (Lim et al., 2021) - A method that also bypasses message passing for heterophilous graphs WRGAT (Suresh et al., 2021) - Addresses heterophily with rewiring techniques ACM-GCN (Luan et al., 2022) - Reports state-of-the-art results on heterophilous benchmarks Papers on graph topological operators like GraphGPS (Rampavsek et al., 2022)\n\nWe appreciate the references and agree these are valuable works that provide additional context. **We will discuss these references in our revised manuscript for an additional comparison!**\n\n> Actually, I think the experiments should pay more attention to the special properties the proposed method can demonstrate.\n\nThis is an insightful observation. Our motivation for including heterophilous benchmarks was to highlight the method’s robustness in the absence of homophily, where traditional GNNs often degrade. We agree that l-ECT’s real strength lies in capturing local structure independent of homophily assumptions, yield broad utility. **In the revision, we will reframe the narrative to better emphasize the general-purpose nature of l-ECT.**\n\n> How does the computational complexity of computing l-ECTs scale with graph size and neighborhood size? This seems critical for practical applications.\n\nPlease refer to the response above.\n\n> Could you clarify your notion of graph embedding using node features? How do you handle cases where the embedding might not preserve edge relationships?\n\nThe embedding is created by assigning a node to the point in Euclidean space which is specified by its respective node feature vector. Edges are drawn between nodes if and only if there's an edge in the original graph. Since nodes may appear multiple times, edges might get lost by using this embeddings (see the paragraph before Thm.1). Therefore, in practice we use virtual edges to prevent such cases (i.e. an embedded node may occur multiple times as a neighbor). **We will clarify this subtlety in our revised paper, thanks for spotting that!**\n\n\n> What is the theoretical justification for l-ECTs working well on heterophilous graphs beyond the empirical results?\n\nThe core limitation of message passing in heterophilous settings lies in its aggregation mechanism, which tends to blur dissimilar feature signals. Our method bypasses aggregation entirely, instead constructing a vectorized topological summary of a node’s neighborhood. From a theoretical standpoint, Thm. 1 shows that l-ECT_1 encodes sufficient information to reconstruct the feature vectors of a node’s neighbors without aggregation. This property is critical in heterophilous settings where the difference between features (not their average) is informative.\n\n> How do you address the scenario where nodes have identical feature vectors but different structural roles in the graph?\n\nPlease see our response above.\n\n**We appreciate your detailed review and the encouragement regarding our theoretical and empirical contributions.** If our revisions adequately address your concerns, we would be very grateful if you would consider raising your overall score.\nThank you again for your time and constructive feedback! Please let us know if you have any other questions or concerns.\n\nBest regards,\n\n The Authors", "claims_and_evidence": "Most claims are supported by theoretical analysis and empirical results.", "methods_and_evaluation_criteria": "The proposed methods are generally sound:\n#### 1. Using l-ECTs to capture local neighborhood information is a novel approach that addresses limitations in traditional GNNs.\n#### 2.The evaluation on both homophilous and heterophilous graph datasets is appropriate.\n#### 3. The comparison with standard GNNs (GCN, GAT, GIN) and a heterophily-specific model (H2GCN) provides good context.", "theoretical_claims": "Most claims are supported by theoretical analysis and empirical results.", "experimental_designs_or_analyses": "The experimental design is generally sound.", "relation_to_broader_scientific_literature": "I am not sure about that.", "essential_references_not_discussed": "LINKX (Lim et al., 2021) - A method that also bypasses message passing for heterophilous graphs\nWRGAT (Suresh et al., 2021) - Addresses heterophily with rewiring techniques\nACM-GCN (Luan et al., 2022) - Reports state-of-the-art results on heterophilous benchmarks\nPapers on graph topological operators like GraphGPS (Rampavsek et al., 2022)", "other_strengths_and_weaknesses": "### Strengths:\n\n#### 1. Novel integration of topological data analysis techniques with graph learning\n#### 2. Theoretically grounded approach with provable properties\n#### 3. Strong performance on heterophilous graphs without specialized architecture\n\n### Weaknesses:\n\n#### 1. Limited exploration of computational efficiency - l-ECT computation could be expensive for large graphs\n#### 2. Relationship between l-ECT parameters and performance is not thoroughly analyzed\n#### 3. Limited comparison to recent heterophily-specific approaches", "other_comments_or_suggestions": "Actually, I think the experiments should pay more attention to the special properties the proposed method can demonstrate. I don't think heterophilous graph is a good target.", "questions_for_authors": "1. How does the computational complexity of computing l-ECTs scale with graph size and neighborhood size? This seems critical for practical applications.\n\n2. Could you clarify your notion of graph embedding using node features? How do you handle cases where the embedding might not preserve edge relationships?\n\n3. What is the theoretical justification for l-ECTs working well on heterophilous graphs beyond the empirical results?\n\n4. How do you address the scenario where nodes have identical feature vectors but different structural roles in the graph?", "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": "sux2WA3Db8", "reviewer": "Reviewer_nYTM", "summary": "The paper introduces a local Euler Characteristic Transform (l-ECT), a local topology measure. l-ECT is an application of ECT for analysis of a neighborhood. Then, author apply it to enhance expressivity and interpretability of graph representations (mostly graph's nodes classification).\nAuthors identify crucial issues where GNN fail while l-ECT are useful. A rotation invariant l-ECT is provided. Theoretical guarantees are provided.", "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": "Dear Reviewer,\n\nwe sincerely thank you for the thoughtful and detailed feedback! **We are glad that you found our contributions novel and our empirical and theoretical results sound.** We especially appreciate your engagement with the proofs and your recognition of l-ECT’s usefulness for node classification. Below, we address your comments point by point.\n\n> The examples of interpretability are not provided. \n\nThank you for raising this point. A discussion on the interpretability of l-ECT is included in Appendix A.2.1, where we show how feature importance of the underlying model naturally gives rise to interpretability. However, we agree that the main paper could more clearly showcase this aspect. **In the revised version, we will better illustrate how l-ECT enables interpretation of node roles and neighborhood structures.**\n\n> The details on evaluation of rotation-invariant metric is not provided. \n\nSection 5.2 evaluates our rotation-invariant metric for aligning spatial graph representations, showing it captures structural similarity under random rotations. **We’ll clarify this purpose in the text and add more details.**\n\n> How do you perform differentiation since the Euler characteristic is not differentiable?\n\nThe Euler Characteristic Transform is made differentiable by embedding it into a continuous pipeline using soft (smoothed) filtrations. It doesn’t change the discrete nature of EC itself but creates a differentiable approximation around it. **We will clarify this in our revision.**\n\n> The performance of l-ECT on WebFB dataset is better that on common Planetoid. Do you have an explanation?\n\nThe stronger performance of l-ECT on WebKB over Planetoid datasets reflects structural differences: WebKB graphs are heterophilic, where local structure matters more than feature similarity—making l-ECT well-suited—while Planetoid graphs are homophilic, favoring message-passing GNNs. **We’ll clarify this in the revision.**\n\n> The proposed method outperformed standard GCN, GAT, GIN but is still worse that modern methods dedicated to hetero-graphs (see Appendix). \n\nWe agree that recent heterophily-specific methods are highly competitive on heterophilic benchmarks. However, our goal with l-ECT is not to design a specialized architecture optimized solely for heterophily, but rather to propose a general-purpose, interpretable representation that captures rich local structural information without message passing. What’s notable is that l-ECT achieves competitive performance despite being a non-aggregating and model-agnostic approach, and in many cases closes the gap with specialized models. We believe this highlights the complementary strengths of topological representations, particularly when combined with other methods. **We will emphasise this aspect further in our revision and substantiate it using additional experiments.**\n\n\n> If I understood correctly, l-ECT method is not applicable for graphs without features associated with nodes.\n\nYou're right—l-ECT, as formulated, assumes node features or geometric embeddings. This is a common assumption in graph learning, where such features are typically available and important for downstream tasks like node classification. Our method therefore aligns well with standard benchmarks and real-world settings. **We'll clarify this assumption in the paper.**\n\n> The definition of a simplical complex is quite clumsy mixing together geometrical and abstract simplical complexes. \n\nThank you for pointing this out—we completely agree. Our current presentation conflates geometric and abstract simplicial complexes. Since our method relies on geometric realizations, we’ll restrict the discussion to geometric simplicial complexes in the revised version. **We’ll clarify the terminology and focus solely on the relevant setting for l-ECT.** We appreciate your attention to this—it helps improve the clarity and precision of the exposition.\n\n> Hyperlinks to proofs of theorems in Appendix is not provided in main text\n\nThanks for this suggestion. **In the revised version, we will include hyperlinks and explicit references from each theorem in the main text to its corresponding proof in the appendix.**\n\n> I assume that l-ECT is defined for a geometric simplical complex, otherwise the definition (3) doesn't make sense. Why you still use term \"abstract simplicial complex\" in line 113? Consequently, l-ECT makes sense for graphs only if each vertex has attributes and a graph is mapped onto R^n. If I am right, this shoud be stated more directly.\n\nThis is a mistake, please see response above. **We will fix this in our revision.**\n\nWe are grateful for your recognition of our contributions and your close reading of the manuscript. If the improvements we propose address your concerns, we would be very grateful if you would consider raising your score.\nThank you once again for your constructive and thoughtful review, we’d be happy to address any other questions you might have.\n\nBest regards,\n\n The Authors", "claims_and_evidence": "Main claims are: \n\n> As our main contributions, we (i) construct l-ECTs in the context of embedded simplicial complexes (and graphs) and theoretically investigate their expressivity in the special case of featured graphs, (ii) empirically show that this expressivity positions l-ECTs as a powerful general tool for interpretable node classification, often superior to standard GNNs, and (iii) introduce an efficiently computable rotation-invariant metric based on l-ECTs that facilitates the spatial alignment of geometric graphs.\n\nClaims are mostly convincing, except the \"interpretability\". The examples of interpretability are not provided.\nThe details on evaluation of rotation-invariant metric is not provided. How do you perform differentiation since the Euler characteristic is not differentiable?", "methods_and_evaluation_criteria": "Benchmarks are fine.\nThe performance of l-ECT on WebFB dataset is better that on common Planetoid. Do you have an explanation?", "theoretical_claims": "Yes, I have checked most of the proofs.", "experimental_designs_or_analyses": "Experiments are sound. See questions above.", "relation_to_broader_scientific_literature": "Authors cite broader scientific literature, for example SOTA studies on topological deep learning and learning with hetero-graphs.", "essential_references_not_discussed": "--", "other_strengths_and_weaknesses": "**Strengths**\n\n1. The paper introduced a novel concept, l-ECT and empirically proved its usefulness for node classification in graphs.\n\n**Weaknesses**\n\n1. The proposed method outperformed standard GCN, GAT, GIN but is still worse that modern methods dedicated to hetero-graphs (see Appendix).\n2. If I understood correctly, l-ECT method is not applicable for graphs without features associated with nodes.", "other_comments_or_suggestions": "1. The definition of a simplical complex is quite clumsy mixing together geometrical and abstract simplical complexes.\n2. Hyperlinks to proofs of theorems in Appendix is not provided in main text, \n3. I assume that l-ECT is defined for a __geometric__ simplical complex, otherwise the definition (3) doesn't make sense.\nWhy you still use term \"abstract simplicial complex\" in line 113?\nConsequently, l-ECT makes sense for graphs only if each vertex has attributes and a graph is mapped onto R^n.\nIf I am right, this shoud be stated more directly.", "questions_for_authors": "See above.", "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": "OP54o3QTCr", "reviewer": "Reviewer_DNCh", "summary": "In their paper \"Diss-l-ECT: dissecting graph data with local Euler characteristic trasnform\", the authors suggest a local version of Euler characteristic transform (ECT) that, given a graph with node features, assigns to each node an additional feature vector containing Euler characteristics of local subgraphs with nodes selected by a range of feature-based thresholds. The authors then show that XGBoost on the concatenation of original features and l-ECT features performs very well in various node clasification tasks, in particular for heterophilic graphs.", "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": "Dear Reviewer,\n\nwe thank you for the constructive and thoughtful feedback. We’re especially grateful for recognizing the clarity of our exposition, the soundness of our method, and the convincing experimental results. Below, we respond to all concerns.\n\n> Question regarding illustration and intuition\n\nThank you for the helpful suggestion. To clarify, we are not computing the local Euler characteristic, but rather the local Euler Characteristic Transform (l-ECT). Unlike the raw Euler characteristic, l-ECT scans the local neighborhood in multiple directions, computing Euler characteristics of sublevel sets. This directional process, known to be invertible under mild assumptions, provides a rich fingerprint of local geometry and topology. While the Euler characteristic is a topological invariant, the collection of directional profiles captures detailed geometric structure. **We'll add more explanations and illustrations in the revision.**\n\n> “No intuition about why/when l-ECT works. No baseline comparison with XGBoost on the node features.”\n\nRegarding intuition, see the response above; we also plan on illustrating the computation better. Regarding the baseline, we’re happy to include an **additional comparison against XGBoost** for the respective node classification tasks.\n\n> Request for XGBoost on raw node features\n\nWe agree that this baseline is important and are happy to **include the corresponding results in our revision.** In the meantime, Table 6 already shows that l-ECT features are informative: as the number of directions used in the transform is reduced, classification performance often degrades substantially. This suggests that the l-ECT encodes nontrivial structural information not already present in the raw features, and that this information contributes meaningfully to the model’s performance.\n\n> Question about spatial alignment using l-ECT\n\nThank you for raising this. While the experiment in Section 5.2 does serve as a sanity check, it also demonstrates that l-ECT provides a scalable alternative to alignment methods like Procrustes, which rely on costly SVD. Our approach scales linearly with n, l, and m, and allows control over approximation quality via m and l. **We'll revise the text to better explain this motivation and highlight the practical benefits of l-ECT-based alignment.**\n\n\n> Question on the definition of simplicial complexes\n\nYou're right—thanks for catching this. We meant geometric, not abstract, simplicial complexes, and **will revise the text to reflect this consistently.**\n\n> Question on l-ECT and point clouds\n\nTo clarify, we don’t compute a single Euler characteristic, but use the local Euler Characteristic Transform (l-ECT), which builds a vector from directional filtrations of the local neighborhood. Point clouds are treated as edge-free geometric graphs (0D complexes in R^n. Since the ECT is invertible, it serves as a rich fingerprint of local structure (in this case the point cloud). **We’ll revise the text to make this clearer.**\n\n> Request of homophily score reporting\n\nThank you for the suggestion. We will add a standard homophily measure (edge homophily ratio) for each dataset and indicate whether each dataset is homophilic or heterophilic. For example, the WebKB datasets exhibit low edge homophily (~0.2) and are thus considered heterophilic, while citation graphs like Cora are homophilic (e.g., ~0.8).\n\n> Request for using bold for all results within error margins\n\nThank you for pointing this out. **We will update all result tables to use boldface for all values within one standard error of the best performance.** We believe this makes the comparisons clearer and more meaningful.\n\n> “\"Planetoid datasets\" -- what is \"planetoid\"?”\n\n“Planetoid” refers to the data introduced in the benchmark (Yang et al., 2016), containing the citation graph datasets Cora, Citeseer, and Pubmed. The dataset is introduced in l.353 of our manuscript. **We will clarify this.**\n\n> Questions about Table 7\n\nThank you for pointing this out. Table 7 uses the ranking setup from Platonov et al. (2023), with lower mean ranks indicating better performance. We added l-ECT post hoc using the same protocol. FSGNN, while strong, was excluded from our benchmarking due to its specialized design. As the comparison focuses on heterophilic graphs, such methods are favored—but our goal is to show that l-ECT performs well without architectural tuning, **which we’ll clarify in the revision.**\n\n**We are grateful for your recognition of our contributions and your close reading of the manuscript.** If the improvements we propose address your concerns, we would be very grateful if you would consider raising your score.\nThank you once again for your constructive and thoughtful review, we are happy to address any other questions you might have!\n\nBest regards,\n\n The Authors", "claims_and_evidence": "One important baseline is missing.", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "Didn't check", "experimental_designs_or_analyses": "Seems sensible", "relation_to_broader_scientific_literature": "Not sure what one is supposed to say here. This paper suggests l-ECM, clearly building up on ECM etc.", "essential_references_not_discussed": "Not that I know of", "other_strengths_and_weaknesses": "Strengths: The paper is very clearly written. l-ECT definition makes sense. Experimental results are convincing.\n\nWeaknesses: No intuition about why/when l-ECT works. No baseline comparison with XGBoost on the node features. \n\nOverall, borderline paper but I tend to acceptance.\n\n\nMAJOR COMMENTS\n\n* The main argument of the paper is that local Euler characteristics can contain information useful for node classification. What would be very helpful is to have some illustration of how this can be the case. Can you show an example of a small toy graph with nodes from two classes and some node features, where local Euler characteristic in 1-hop neighborhood predicts the class? Currently, the paper provides very little intuition for why/when this should work. The authors emphasize heterophilic graphs, so it would be great if the toy graph is heterophilic. For a graph, Euler characteristic is simply the number of nodes minus the number of edges. Maybe the toy graph can have 2D node features and a particular 2D direction and a particular threshold value that would make 1-dimensional l-ECT (m=l=1) perfectly predictive of class? Something along the lines.\n\n* In every table I am missing one IMHO crucial comparison: XGBoost performance on node features without any l-ECT features. Please add this to Tables 1--5 as additional row. Currently it is impossible to say if l-ECT performance is due to XGBoost on original node features performing well or due to l-ECT features actually providing some additional information. If for some of the graphs the performance of XGBoost on original features is as good as for l-ECT, it would mean that l-ECT features are useless for that graph.\n\n* I did not understand the point of section 5.2. Randomly rotated graphs shown in Figure 1 could be aligned using a trivial application of Procrustes rotation. Why does one need l-ECT machinery for that? Is this whole section supposed to be a sanity check that l-ECT alignment works correctly? If so, it should be presented as a \"merely\" sanity check, and not as a important result that takes 1 page of text and 1 figure. The result seems trivial.", "other_comments_or_suggestions": "MINOR COMMENTS\n\n* On page 2, simplicial complex is defined abstractly, without any node features. Then page 3 starts with simplicial complex X \\in R^n. But what does it mean for X to be in R^n? This was never defined. Featured graphs and feature vectors are only introduced later. This makes it confusing for the reader.\n\n* Several times the authors refer to \"point clouds\", even calling them an \"important special case\". I don't understand how l-ECTs would be applicable to point clouds, where the (local) Euler characteristic is just the number of points. This seems to carry no useful information. Please clarify what is meant.\n\n* In section 5.1, for all graphs please specify where they are heterophilic or homophilic (e.g. WebKB datasets are homo- or hetero- philic? it's not stated). Perhaps use some measure of homophily and report the value for each graph.\n\n* \"Amazon dataset\" paragraph: \"combintaion of l-ECT1 and l-ECT2 outperforms on Photo\" -- this statement is misleading, the difference is 0.1% with errors being ~1%, clearly not significant. In fact, I think GAT should be bold for Photo in Table 3, because it's not meaningfully different from the top value in the Photo column. I suggest to use bold in all such cases in all tables (e.g. l-ECT2 for Amazon Ratings in Table 2 etc.)\n\n* \"Planetoid datasets\" -- what is \"planetoid\"?\n\n* I did not understand Table 7. Does it take ranks from Platonov et al.? If yes, how are your methods added there? Also, is H2GCN the best or the worse? It has the highest rank value, meaning it's the worst? As in Figure 4 (higher value => worse). If FSGNN is the best, why did not you use it in your benchmarking? If H2GCN is the best, then why did you find it so much worse then l-ECT in Figure 4? Very confusing.", "questions_for_authors": "See above", "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: This paper introduces the local Euler Characteristic Transform (Diss-l-ECT), a novel, and theoretically grounded approach to graph representation learning that enhances node classification performance, especially on heterophilic graphs. The method is well-motivated, clearly presented, and shows strong empirical results across diverse benchmarks, outperforming standard GNNs in several settings. While the paper would benefit from deeper analysis and intuition about why Diss-l-ECT is effective and a direct comparison with XGBoost on raw features, the core contribution is technically sound and compelling. I recommend acceptance.
Decision: Accept (poster)