SCNode: Spatial and Contextual Coordinates for Graph Representation Learning
Authors: Md Joshem Uddin
Astrit Tola
Varin Singh Sikand
Cuneyt Gurcan Akcora
Baris Coskunuzer
Venue: Accepted by TMLR ()

Abstract: Effective node representation lies at the heart of Graph Neural Networks (GNNs), as it directly impacts their ability to perform downstream tasks such as node classification and link prediction. Most existing GNNs, particularly message passing graph neural networks, rely on neighborhood aggregation to iteratively compute node embeddings. While powerful, this paradigm suffers from well-known limitations of oversquashing, oversmoothing, and underreaching that degrade representation quality. More critically, MPGNNs often assume homophily, where connected nodes share similar features or labels, leading to poor generalization in heterophilic graphs where this assumption breaks down.

To address these challenges, we propose *SCNode*, a *Spatial-Contextual Node Embedding* framework designed to perform consistently well in both homophilic and heterophilic settings. SCNode integrates spatial and contextual information, yielding node embeddings that are not only more discriminative but also structurally aware. Our approach introduces new homophily matrices for understanding class interactions and tendencies. Extensive experiments on benchmark datasets show that SCNode achieves superior performance over conventional GNN models, demonstrating its robustness and adaptability in diverse graph structures.

Published in Transactions on Machine Learning Research (11/2025)
SCNode: Spatial and Contextual Coordinates for Graph Representation Learning
Md Joshem Uddin Deportment of Mathematical Science The University of Teras Dullas
mdjoshem. wwdsn@rtdalllsscccomm
Astrit Tola
atola &fsu edu
Department Mathematics
Florida State University
Varin Singh Sikand
wwrin.skannd@uudall.. edu
Department of Computer Science The University of Texas at Dallas
Cuneyt Gurcan Akcora
cuneyt..kcoom@@cffccl edu
AI Instiative
Universily of Central Florida
Baris Coskunuzer
coshunns@uudalllas.ei edd.aaalaaasduaa
Department of Mathematical Science The University of Texas at Dallas
Reviewed OpenReview: https: // opwnreviee. net/ net/forue forum? id=wdcdKeFbf0
Abstract
Effective node representation lies at the heart of Graph Neural Networks (GNNs), as it directly impacts their ability to perform downstream tasks such as node classification and link prediction. Most existing GNNs, particularly message passing graph neural networks, iteratively compute node embeddings. While pow- erful, this paradigm suffers from well-known limitations of oversquashing, oversmoothing, and underreaching that degrade representation quality. More critically, MPGNNs often sume homophily, where connected nodes share similar features labels, leading to poor generalization in heterophilic graphs where this assumption breaks down.
rely on neighborhood aggregation
To address these challenges, framework designed to perform consistently well in both homophilic and heterophilic set- tings. SCNode integrates spatial and contextual information, yiclding node embeddings that are not only more discriminative but also structurally aware. Our approach introduces new homophily matrices for understanding class interactions and tendencies. Extensive ex- periments on benchmark datasets show that SCNode achieves superior performance over conventional GNN models, demonstrating its robustness and adaptability in diverse graph structures.
propose SCNode, Opatial-Contertunn Node Embedding
1
Introduction
Over the past decade, GNNs have made a breakthrough in graph machine learning by adapting several deep learning models originally developed for domains such as computer vision and natural language processing the graph setting. By using different approaches and architectures, GNNs produce powerful node embed- dings by integrating neighborhood information with domain features. While recent GNN models consistently

---

Published in Transactions on Machine Learning Research (11/2025)
outperform the state-of-the-art (SOTA) results, in the seminal papers (Xu et al., 2019; Morris al., 2019). the authors showed that the expressive power of message passing GNNs is about the same as well-known former method in graph representation learning, ie., the Weisseiler-Lhhman algorithm (Shervashidze et al., 2011). Furthermore, recent studics show that GNNs suffer from significant challenges such as oversquash- ing (Topping et al., 2021), oversmoothing (Oono & Suzuki, 2020), and underreaching (Barceló et al., 2020).
Another major issue with MPGNNs is their reliance on the (useful) homophily assumption, where edges typically connect nodes sharing similar labels and node attributes (Sen et al., 2008). However, many real- world scenarios exhibit heterophilic behavior, such in protein and web networks (Pei et al., 2019), where conventional GNNs (Hamilton et al., 2017; Veličković et al., 2018a) may experience significant performance degradation, sometimes even underperforming compared to a multilayer perceptron (MLP) (Zhu et al., 2020; Luan et al., 2022; 2024). Therefore, there a pressing need to develop models that function effectively in both homophilic and heterophilic settings.
We altribute current GNN shortcomings to their inability to capture two complementary sources of informa- tion in graphs: spatial information, which reflects the class distribution in a node's 's immediate neighborhood, and contertual information, which measures node's relation to class representatives across the entire graph. To address this, we position each node relative to all classes. In the attribute space, we introduce class land- marks, reference embeddings that act as beacons for each class, and record the distances from each node's feature vector to these landmarks as its contextual coordinates. In the graph topology, we encode spatial coordinates by computing the distribution of neighboring node labels within each node's local subgraph.
Our landmark scheme establishes an implicit coordinate system: by triangulating distances class land- marks, locate each node in attribute space much like a GPS. These relative coordinates capture global patterns that standard pressage-passingg GNNs often miss due to underreaching, i.e., the inability to prop- agate sufficient information from distant but relevant nodes within a limited mumber of layers. Spatial embeddings encode local label distributions in multi-hop neighborhoods, enriching high-frequency structural signals, while contextual embeddings inject low-frequency global components through landmark- based coor- dinates. From a spectral perspective, concatenating these embeddings expands the effective receptive field by combining complementary frequency bands. This joint representation mitigates the 1-WL locality bottleneck and enhances exprossivity by unifying local structural distributions with long-range semantic context.
By leveraging relative positioning, we generate feature vectors that seamlessly blend spatial and contextual cues. The resulting SCNode model achieves state-of-the-art performance on node classification and link prediction across both homophilic and heterophilic benchmarks. Furthermore, SCNode embeddings can be dropped into any GNN as plug-and-play components, remedying underreaching and consistently boosting performance diverse graph settings.
Our contributions can be summarized follows:
We propose novel approach to capture the relative posilioning of nodes with respect to node classes, incorporating both spatial and contextual perspectives.
We introduce class-awore homophily matrices, providing detailed insights into homophily tendencies and enabling deeper understanding of class interactions in the graph.
Our SCNode model effectively integrates spatial and contextual information, delivering state-of the- art performance in both node classification and link prediction tasks.
We demonstrate that SCNode vectors can be easily integrated into any GNN model as plug-and- play components, significantly boosting performance in both homophilic and heterophilic settings, showeasing their versatility across diverse real-world applications.
2
Related Work
Over the past decade, GNNs have dominated graph representation learning, especially excelling in node classification tasks (Xiao et al., 2022). After the success of Graph Convolutional Networks (GCN) (Kipf
2

---

Published in Transactions on Machine Learning Research (11/2025)
& Welling, 2017), using neighborhood aggregation strategy to perform convolution operations on graph, subsequent efforts focused on modifying, approximating, and extending the GCN approach, e.g, involving attention mechanisms (GAT) (Veličković et al., 2018a), sampling and aggregating information from a node's local neighborhood (GraphSAGE) (Hamilton et al., 2017). However, a notable scalability challenge arose due to its dependence on knowing the full graph Laplacian during training- To address this limitation, numerous works emerged to enhance node classification basod on GCN, such as methods, importance sampling node (FastGCN) (Chen et al., 2018), adaptive layer-wise sampling with skip connections (ASGCN) (Huang al., 2018), adapting deep layers to GCN architectures (deepGCN) (Li et al., 2019), incorporating node-feature convolutional layers (NFC-GCN) (Zhang et al., 2022).
Two fundamental shortcomings of MPGNNs are the loss of structural information within node neighbor- hoods and the difficulty in capturing long-range dependencies (Khemani et al., 2024; Corso et al., 2024). To address these inherent issues, numerous studies conducted in the past few years have introduced various enhancements, including diverse aggregation schemes, such as skip connections (Chen al., 2020), geometric methods (Pei et al., 2019), aiming to mitigate the risk of losing crucial information. Additionally, advance- ments like implicit hidden layers (Geng et al., 2021) and multiscale modeling (Liu et al., 2022) have been explored to augment the GNN's capabilities.
Another major issue with MPGNNs is their reliance on the homophily assumption, leading to poor per- formance in heterophilic networks (Zhu et al., 2020; Luan et al., 2024). Recent efforts have focused on designing GNNs that perform well in heterophilic settings (Lim et al., 2021a; Luan et al., 2022; Zheng et al., 2022; Xu et al., 2023; Zhao et al., 2024). While prior works analyze class-wise relations via neighborhood label distributions (Ma et al., 2021; Zhu al., 2023) class-comaatibilityy matrices (Zhu et al., 2021), SCNode introduces coordinate-based formulation that combines local structural distributions with global class-aware landmark distances, providing backbone-agnostii feature augmentations that capture long-range class relations.
In addition, graph kernels, such as Weisfeiler-Lehmann (Shervashidze et al., 2011), shortest-path Borgwardt & Kriegel (2005), and graphlet kernels (Shervashidze et al., 2009), characterize similarity through handcrafted structural mappings (Vishwanathan al., 2010). SCNode related in spirit, but it differs in two important ways. It combines local structural distributions with global class-aware relational signals, providing richer contextual information than fixed kernel sinilarities. Moreover, SCNode is computationally more efficient, avoiding the quadratic complexity of kernel evaluations by generating embeddings in linear time that can be directly used within GNNs or graph transformers.
3 SCNode Framework
In the following discussion, we use the notation G (V, E, W.X) the vertices (nodes), E {ei} represents edges, W = (wig C R+} represents edge weights, and ===fX { C R" represents node features. If no node features are provided, X = 0. Consider a graph where nodes are categorized into N classes. For each node u, we aim to construct an embedding i(u) of dimension q x N, where q is determined by specifics of the graph such as its directedness,, weighted na- ture, and the format of domain features.
W,
where V
{or Jm
represents
Spatial Limbelding
Input Graphs
Contestual Libeediin
SCNode Embedding
Specifically, we generate spatial embeddings (w), az(w),...) derived from local neighbor- hood information, and contertual embeddings (Bi (w),, (2().....) derived from node properties specific to the domain. Each embedding, au) and (2),,, is N-dimensional, with one entry cor- responding to each class (SCNode coordinates).
② Nod: Clasifictioo
Praliction
② Clustoring
Figure 1: SCNode framework. Spatial and con- textual embeddings are incorporated in class-aware
contrast- based learning.
These embeddings then concatenated to form the
3

---

Published in Transactions on Machine Learning Research (11/2025)
final embedding 7(u) of size q N. For simplicity, focus applications in an inductive setting. In Appendix B.2, we discuss adaptations of our methods to accommodate transductive settings.
3.1 Spatial Node Embeddings
Spatial node embedding leverages the structure of graph to measure the proximity of a node to various known classes within that graph. For each node u in the graph 9 with vertices V, the k-hop neighborhood of u, denoted as Nu), includes all vertices v such that the shortest path distance d(u, v) between u and v is at most hops: (u(()) = fu E V d(u, v) S k). Here, d(u, v) represents the shortest path length between u and v. If no path exists, then d(u, v) = co. In the case of a directed graph 9, edge directions are disregarded when calculating distances.
Assume that there are N classes of nodes, represented as C1, C2,..., Cv- Let Verain be the set of nodes with known labels in the training dataset. The feature vector (u) for node is initialized as the distribution of class occurrences: dg (u) = [ax1, a22...., [gN]] where asj counts how many neighbors of u belong to class C, within the k-hop neighborhood intersecting with Visin 0kj = Hy E C, v (x) n V. Consider the toy example of Figure 2(a), where the (k=1)-hop neighborhood of node u contains node from class 1 and node y from class 2. As result, āi = 1 0]. We extend the spatial embeddings to directed and weighted graph cases by incorporating edge directions and weights in additional dimensions (see Appendix Section B.3) for the definitions.
3.2 Contextual Node Embeddings
In this section, we focus on extracting the most relevant information from the node attribute space by utilizing class-level reference points. Instead of relying solely on local neighborhood information, we assess each node's position relative to a set of representative class embeddings, or class landmarks, defined in the attribute space. These landmarks serve as stable points that reflect the typical feature profiles of different classes.
To compute contextual embeddings, we mcasure the similarity or distance between node's attribute vector, Xu, and these class landmarks. This approach enables the model to capture how closely a node aligns with each class, regardless of its immediate neighbors.
We use the term contertual node embeddings to emphasize the integration of node attributes with relational and structural information from the graph. Unlike traditional alfribute embeddings, which focus solely on embedding node characteristics, contextual embed- dings account for both the intrinsic properties of node and its position within the graph structure. This approach is especially important in scenarios where node features and class behavior are inter- twined, such as when the topic of an article influ- ences its citation links. By combining attribute and relational context, our embeddings provide a richer representation of the graph, enabling improved per- formance in classification tasks.
KM0
(a)
(b)
In the following, we define class representatives (landmarks) in the node attribute space for each class. Let (C,15= denote the set node classes. For each class CG we identify the cluster of points Z, = (X()) | for this cluster.
Figure 2: Computing spatial (a) and contextual (b) embeddings for node the graph.
E C within the node attribute space RT, and establish a representative landmark & E E
4

---

Published in Transactions on Machine Learning Research (11/2025)
Definition 3.1 (Class Landmark). Given node class C and corresponding cluster of node embeddings 2,. the landmark &j is computed as the centroid of the points in 2j, normalized by the number of nodes in Cy:
f
Σ X(u) DEC,
Cg
Definition 3.2 (Distances to Class Landmarks). Let d be a metric that measures the distance in R" For node with an embedding (u), the distance to the class landmark E is defined as: dj = d()()), for each class Cy. These distances dj help in assessing how similar distinet the node is from each class represented by the landmarks.
Definition 3.3 (Contextual Embodding). Given a node u and a set of class landmarks {fý} S RR, one for each class Cj. the contextual embedding of node u defined by a vector of distances from the node's embedding each class landmark:
((u) id1, d2, d dxl
where d,- &)) for each class C (see Figure 2(b)). This vector A(u) encapsulates the node's position relative each class within the attribute space.
Class Landmarks. To extract more detailed information about the domain attributes of nodes for each class, multiple landmarks can be defined. In many cases, the node attribute space lacks an inherent metric, and using different metrics can result in diverse landmarks and contextual embeddings. For example, the first. landmark { E I may represent the center the cluster Zj using the Euclidean metric, while a second landmark e E IR" for the same class can be based on Jaccard similarity, capturing the most frequent attributes within 2j (see Appendix C).
For each type of landmark, a corresponding distance or similarity measure (...) is defined, such as Euclidean or cosine distance for real-valued attribute vectors, or Jaccard similarity for categorical attributes (Niwat- tanakul et al., 2013).
By leveraging different metrics and their corresponding landmarks, we compute an N-dimensional vector
(u) = [dk1, d22, dra dia...., dEN]
where dæj =x)) represents the distance (or similarity) between node embeddings X(u) and the landmark zJ of class Cy.
3.3 SCNode Embedding
We may expand spatial neighborhoods and extend landmark sets arbitrarily. However, for exposition pur- poses, we will define SCNode embeddings over a directed graph where each node's neighborhood is considered up to two hops and each class C, has two landmarks. We define the SCNode Embedding of a node u by concatenating spatial and contextual embeddings. Specifically, we consider:
Spatial embeddings from incoming and outgoing 1-hop neighborhoods (āie(u) and (n)( and the 2-hop neighborhood (áz(u)).
Contestual embeddings based on distances two class landmarks (1) and B2(u)), representing domain- specific characteristics.
5

---

Published in Transactions on Machine Learning Research (11/2025)
Definition 3.4 (SCNode Embedding). The final em- bedding 7(u) for node u a concatenated vector of these embeddings as shown in the right. For example, if the graph is directed, and want to utilize two landmark types, {C)) and {()), the resulting SCNode vector will be in a 5- N-dimensional vector where N is the number of classes. For clarity, we represent 7(u) in a 2D format (m x N), where each column corre- sponds to one class and each row represents one type of SCNode vector.
d:(u)
Spetial
Incoming 1-ngbd
(1o(w)
Spatial
Outgoing I-ngbd
7()-
do(u)
Spatial
2-ngbd
B,(x)
Contextual
(d(X... 59)
B2(u)
Contextual
(d(X...
Relation label-structure methods. SCNode augments each node with two complementary signals: contextual coordinates given by distances to class landmarks in the embedding space, and spatial class- distribution histograms over k-hop ncighborhoods. This encodes label structure over nodes without modify- ing the graph introducing virtual nodes. In contrast, techniques such as LEGNN (Yu et al., 2022b) and GraphGPS (Rampášek et al., 2022) inject graph-level signals or virtual nodes derived from class labels. Con- ceptually, the class landmarks play the role of prototypes: distances to class-conditoneed centroids provide barycentric coordinates label space (Snell et al., 2017), but unlike prototypical networks (Hou & Sato, 2022), our prototypes are computed from the training graph and kept fixed during message passing, which decouples metric learning from the backbone and enables transductive inference. Our approach is backbone agnostic and operates at the feature level, it pairs naturally with standard GNNs and graph transformers.
3.4 Homophily and SCNode
In this section, we demonstrate the effectiveness of the SCNode framework in analyzing homophily be- havior within the graph from multiple perspectives and gaining deeper insights into class interactions. In recent years, several metrics have been introduced to study the effect of homophily on graph representation learning (Lim et al., 2021b; Jin et al., 2022; Luan et al., 2022; Platonov et al., 2023a) (see overview in Appendix A.1). A widely used metric, the node homophily ratio, is defined (g) [ Evev dey(u)* where ππ(1) represents the number of adjacent nodes to v sharing the same class. A graph G is termed homophilic if (9) 2 0.5, and heterophilic otherwise.
v(v)
Although homophily measures similarity across an entire graph, individual groups within a graph may display different homophily behaviors. Our SCNode approach leverages class interactions and introduces class-aware homophily score through non-symmetric measures:
Definition 3.5 (SCNode Homophily Matrices). Let (G, V) be graph with node classes {C1, C2, Cn Let ā be a spatial or contextual embedding. We define the homophily rate between classes and j KG = C. Evec, Jawl where at, is the jth entry of ā.. Considering pairwise homophily rates of all classes, we create the N x N matrix M [b3]] as the Homophily Matriz of G.
Table 9 in the Appendix presents examples of SCN homophily matrices. These matrices olfer detailed insights into intra-class (homophily) and inter-class (heterophily) interaction ions across spatial and contextual contexts, with the diagonal elements representing the likelihood of nodes connecting within their own class.
Definition 3.6 (a Homophily Ratio). For a given Homophily Matrix M Homophily Hy is defined the average of the diagonal elements. i.e, Ho N EN has
For example, the Spatial-1 Homophily ratio reveals a node's likelihood to connect with nodes of the same class within its immediate neighborhood. Homophily matrices not only introduce new ways to measure homophily but also relate to various existing homophily metrics:
Theorem 3.7. For given G = (V, let (1) be SCN spatial-1 vector. Let āi (v) be the vector where the entry corresponding to the class of v is set to 0. Then, 1 Hnode 14. Evey [v]ll]]
[
6

---

Published in Transactions on Machine Learning Research (11/2025)
The proof of the theorem and further discussions on how our SCNode homophily matrices relate to other forms of homophily metrics are detailed in the Appendix A.
4 Experiments
We evaluate SCNode in two tasks: node classification and link prediction. We share our Python implemen- tation at pettp://githutt http///tttuccoojjsaaeeeee
Datasets. We use three widely-used homophilic datasets, which are citation networks, CORA, CITE- SEER, and PUBMED (Sen et al., 2008), two Open Graph Benchmark (OGB) datascts: OGBN-ARXIV and OGBN-MAG (Hu et al., 2020), and ten heterophilic datasets, including TEXAS, CORNELL, WISCON- SIN, and CHAMELEON (Pei et al., 2019), as well as filtered versions of SQUIRREL and CHAMELEON, AMAZON-RATINGS, TOLOKER, and QUESTIONS (Platonov et al., 2023b). The datasets are described in Appendix and their statistics are given in Table 1.
Table 1: Benchmark datasets for node classification.
Datasets
Nodes
Edges
Classes
Features
Tr/Val/ Vesstt/Tst (%)
Hom.
CORA
2,708
5,429
7
1,133
18/32/20
0.83
CITESEER
3,312
4,732
3,703
48/32/20
0.72
PUBMED
19,717
44,338
3
500
48 48/32/20
0.79
OGEN-ARXIV
169,313
1,166,243
40
128
OGB
0.65
OGBN-MAG
1,939,743
21,017,007
349
128
OGB
0.30
TEXAS
183
309
5
1,703
48 18/32/20
0.10
CORNELL
183
295
5
1,703
48 18/32/20
0.39
WISCONSIN
251
499
1,703
48/32/20
0.15
CHAMELEON
2,277
36,101
2,325
48 48/32/20
0.25
SQUIRREL'
2,223
46,998
2,089
48 /3//2220
0.19
CHAMELEON
890
8,854
5
2,325
48/32/20
0.24
ROMAN-EMPHN
22,662
32,927
18
300
50/26/25
0.05
AMAZON-RATINGS
24.492
93,050
300
50/25/25
0.37
TOLOKERS
11,758
5,19,000
2
10
50/25/25
0.63
QUESTIONS
48,921
153,540
2
301
50/25/25
0.89
denotes the filtered version of the datasct.
Experimental Setup. We evaluate SCNode three citation networks and four heterophilic graph datasets, following the experimental protocol established by Bodnar et al. (Bodnar et al., 2022). For these datasets, we randomly split the nodes of each class into 48% for training, 32% for validation, and 20% for testing, and report the average accuracy over 10 independent runs. For the OGB datasets, OGEN-ARXIV and OGBN-MAG, we use the official train/vall aaaln/valdaatio//ten (OGB) (Hu al., 2020), and report the classification accuracy accordingly. For the remaining datasets, we adopt the same splitting strategy as outlined in (Platonov et al., 2023b). We give the details of SCNode embeddings for each dataset in Appendix C. The dimensions of the embeddings used for each dataset are given in Table 11.
splits provided by the Open Graph Benchmark
For classification, we employ Multi-Layer Perceptron (MLP) with a single hidden layer consisting of 100 neurons. We use the ReLU activation function, a learning rate of 0.001, and the Adam optimizer for training up to 500 epochs. To mitigate overfitting, we apply L2 regularization with a weight decay of 0.0001.
All noB experiments were conducted on local machine equipped with an Apple M2 chip (8-core CPU, 10-core GPU, 6-core Neural Engine) and 16 GB of RAM. OGB experiments were run on Google Colab, using system with an Intel(R.) 2.20GHz CPU, NVIDIA V100 GPU, and 25.5 GB of RAM.
Runtime. SCNode is computationally efficient. SCNode requires approximately 4 hours for OGBN- ARXIV and 10 hours for OGB-MAG to create all embeddings, and about 10 minutes to train for all OGB datascts. End-to-end end, our model processes three citation network datasets and three WebKb datasets, in- cluding the CHAMELEON dataset, in under minute. PUBMED takes about 20 minutes. For comparison,
7

---

Published in Transactions on Machine Learning Research (11/2025)
Table 2: Node Classification Performance. Accuracy results for node classification tasks. Baseline rosults up to Gen-NSD are sourced from (Bodnar et al., 2022; Li et al., 2022), while the remaining results are taken from their respective original papers. The best results are shown in bold, while the second-best results are underlined. The last column reports the average performance gap (%) relative to the best result across all datasets.
Dataset
CORA
CITESEER
PUBMED
TEXAS
CORNELL
WISC. 0.15
CHAM. 0.25
Avg. Gap %
Nodo Homophily
0.83
0.72
0.79
0.10
0.39
GCN [Kipf& & Welling, 2017)
86, 86.14=1.30
75.51=1.25
87.22=0.a7
56 56.2226.81
60.51=5.00
51 96
65.91=a.25
19.9
GraphSAGE (Hamilton al., 2017)
1.55
76.04=1.90
88.45
75. 75.95=6.01
75. 75.95=5.01
$1.8855.0
58.73=1.66
10.4
GAT (Veličlovíé et al., 2018a)
15.0
1..55
1.13
54.32
61.89
49.41
2..26
20.1
Geom-GCN (Pei et al. 2019)
85.35 1.55
18.0211.35
80.951..47
76.972
00.54...07
64.51
60.00
15.8
H2GCN (Zhu es al., 20200)
1.
77.11
89. 29.49
84. 18.8667722 2:22
82. a.
87. 47.65544499
60.11
8.9
GPRGCN (Chien ol., 2020)
87.95
17.13:1.47
19.5
61.35
78. 18..5
82. $2.55.16
1.75
GONII [Chen al., 2020)
85.37
1..3
15.15
77.57
77.86137
80.3913.40
63.86.33.0
8.2
WRGAT (Suresh et al., 2021)
88.20
76.81:1.89
88.32
83. 62 +5.50
81.62
86.98
65.24
8.1
LINKX (Lim et al., 2022aa
14.64
73.19
87.86
74.60 18.37
77.84....1
75.49.1572
68.42 1.58
10.6
NIGAT (Lān al., 2021)
86.50 1.40
1..20
88,20
62. 10
54.70.7.00
56.9017.
65.70
17.5
GloGNN++ (Li et al., 2022)
85.33
77.22:1.76
89.24
84. 000
85, 10
38.04
71.21
GGCN (Yan et al., 2022)
87.95
17.14
80. $0.15.19.23
84.86.1455
85.68
86.8611 2m
71.14_1.8
4.7
Gen-NSD (Bodnar et al., 2022;
17.30
16.32
80.33
82. 82.97
85.68
89.21
07.93.1..5
5.3
ACM-GCN (Lman et al., 2022)
87.91
77.32=1.70
90.00=0.62
87.84=4.40
85. 14
88. 13-33+322 22
69.14:21 14
4.3
LRGNN (Liung al., 2023)
85.3019.0
1..50
90,20
90.30.440
86.501570
88.2013350
29.16
2.2
Ordered-GNN (8ong at al., 2023)
85.37
17.31
90. 15.
86.22.1412
67.03
68.0413.63
72.28
3.7
TEDGON (Yan et al., 2024)
77.81=1.7
84.84=1.21
91.41+a.22
86. 96.53:33.0
91 22
67.33
4.1
FGSAM-SAGE (Luo et al., 2021s)
88.36+1151
77.13=00.0
89. 79.
$1.35e6.10
82. 82.133=3,88
86. 86.174441
51.31=2.06
8.4
SCNode
88.65.11..
77.83.1.m
90.53lu.e1
94.59..a.as
88.09
92.01 1..8
84.08.11.aa
RevGNN-Deey requires 13.5 days and ReyGNN- Wide takes 17.1 days to train for 2000 epochs on a single NVIDIA V100 for the OGB datasets (Li et al., 2021).
4.1 Node Classification Results
Baselines. We compare SCNode against a range of state-of-the-art models( Tables 2 to 4), including three classical approaches: GCN (Kipf & Welling, 2017), GraphSAGE (Hamilton al., 2017), and GAT (Veličković et al., 2018a), well graph transformer model. All baselines in Tables 2 to 4 are evaluated in the transductive setting, except for GraphSAGE, which follows the inductive setting.
Performance. We present the node classification results for benchmark homophilic and heterophilic datasets in Tables 2 and 3 and for OGB datasets in Table 4. On the two homophilic benchmarks, SCNODE achioves stato-of-the-art performance, consistently outperforming prior methods. Even more strikingly, on the four heterophilic graphs, SCNODE delivers dramatic gains, surpassing the previous best by 4 points. For the remaining five heterophilic datasets, SCNODE achieves the best results on two and demonstrates competitive performance on the others, shown in Table 3. This uniform superiority across low- and high- homophily settings underscores SCNODE's ability to adaptively integrate both local feature similarity and global topological cues, yielding a single architecture that excels regardless of underlying graph structure. Moreover, on the OGB benchmarks, SCNODE achieves results within 3-4 % of the state-of-the-art, despite relying on a far more compact architecture than competing deep-learning models.
SCN-GNNs: GNNs with SCNode Embeddings. To evaluate the integration of SCNode with GNNs as plug-and-play components, we replaced the initial node embeddings in GNNs with SCN embeddings and assessed their effectiveness. We tested three classical GNN models (GCN, GraphSAGE, and GAT), along with the recent LINKX model. A two-layer GNN framework was implemented using the Adam optimizer with an initial lcarning rate of 0.001, dropout rate of 0.5, a weight decay of 5c-4, and 32 hidden channels. The results, shown in Table and Figures 3 and 4, indicate significant performance improvements when using SCN embeddings. These embeddings accelerate convergence and maintain a consistent accuracy advantage over the vanilla models. Notably, combining SCNode with the LINKX model yields the best results across all baselines for both datasets, showcasing the potential of SCNode to enhance GNN performance. This improvement is attributed to the effective integration of spatial and contextual information provided by
8

---

Published in Transactions on Machine Learning Research (11/2025)
Table 3: More Heterophilic Datasets, Node classification results our models compared to GNN and graph transformer baselines on heterophilic benchmarks. The last column reports the average gap to the best- best-performing model in each dataset. Baseline models are sourced from (Platonov et al., 2023b; Yadati, 2025; Luo et al., 2024b)
Model
Squirrel
Chameleon"
Amazon
Tolokers
Questions
Avg. Gap
GCN (Kipf & Welling, 2017)
34.50
53.53
47 04
83.64
63.04.1.01
8.58
GruphSAGE (Hamilton et al. 2017)
35. 25.19
76.001277
43.4110.4
82 4310.4
75.27
5.64
GAT (Veličkovrí et al., 2018x)
34.41
38.7923.07
49.2522.73
44.9821.90
73.42
13.94
H2GCN (Zhu et al., 2020)
35.10=1.16
26.75+a.64
20.4
13.35
63.59
15.05
GPRGNN (Chien et al., 2020)
38.95
39.93+3.30
44.88+0.22
72.94
55. 56.48+0.91
11.67
CPGNN (Zhu et sl., 2021)
30.04:2.00
33 33.00
79.7
73.36:1101
65.96&1.96
13. 68
ESGNN (Maurya et al., 2022)
35.92+1.32
35 25.60
52.74
82.76a0.61
8..66.
4.93
GloGNN (Li al., 2022)
35.1141:24
25.9011.80
36.8020
73.39
65.7421.18
LRGNN (Liang ct al., 2023)
39.5142211
41.24
42:231
66.41a1.75
9.08
NodeFormer (Wu al., 2022)
38.52#1.87
34.78+4.14
43.86
78.10=108
74.27=1.14
8.20
GraphGPS (Rampášek al., 2022)
35.49:2.00
41.04
74.9420.77
83.71+0.18
72.15 =1.18
SGFormer (Wu al., 2023)
35.81+2.02
42.51+a.58
48.18
83.333a0.0
73.05#1116
5.52
Polynormer (Deng et al., 2024)
40.87
41.8213.45
54.8110.49
84.83.0.72
78.92:00.9
1.86
SCNode
45.85
46.12s9.21
52.57:0.42
79.00
76.01a1.02
Table 5: Vanilla-GNN va. SCNode+ (SCN-GNN) accu-
Table 4: Classification accuracy of baselines and our SCNode model on OGBN datasets. Model
racy results for node classification.
Dataset
Model
GNN
SCN-GNN
1mp-(1)
ARXIV
MAG
GCN
86.14+1.1
88, 13ana2
2.29
GCN (Kipf & Welling, 2017)
71.74
34.87
SAGE
86.26+1.64
88.98=1a7
2.72
CORA
GSAGE (Hamilton et al., 2017)
71.49
37 04
GAT
85.03+1.41
87. 87.18=2:2
2.15
GAT (Veličković et al., 2018a)
73.91
37.67
LINKX
81.6311.57
63.8011443
DeepGCN (Li et al., 2019)
72.32
GCN
75.39
77.18:1.
DAGNN (Liu al., 2020)
72.09
SAGE
7:165+1.6
77. 17.3
2.71
UniMP-v2 (Shi al., 2020)
73.92
CITESEER
GAT
74.85+1.45
77.47
2.62
RevGAT (Li al., 2021)
74.26
LINKX
70.51+2.45
73.01=1.76
2.53
RGCN (Yu al., 2022a)
47.96 96
GCN
56.22ss.81
HGT (Yu et al., 2022a)
R-HGNN (Yu et al., 2022a)
52.04
49.21
TEXAS
SAGE
GAT
51.32=6.30
75.951a.01
87.84
122.66i5.70
11.89
7.84
LEGNN (Yu et al., 2022b)
73.71
53.78
LINKX
74.60as.a7
93.78s4.04
19.18
SaGCL (Wan et al., 2024)
71.36
GCN
17
63.53
LMSPS (Liu et al., 2024)
54.83
SAGE
81.181a.a8
84.90a
3.72
WISCONSIN
SCNode
71.56
50.03
GAT
49.4114.0
55.201se4
5.88
LINKX
75.49ss.72
91.18=1.39
13.69
SCNode embeddings. The accuracy gap remains stable, further demonstrating their robustness. In future work, we aim to explore this integration more comprehensively to further enhance GNN performance.
4.2 Link Prediction Results
In this part, we show the utilization of SCNode Embeddings in the link prediction task. We utilize the common setting described in (Zhou et al., 2022) where the datasets were partitioned into training, validation, and test sets with a ratio of 85%, 5%, and 10%, respectively.
The model architecture for the link prediction problem consists of three MLP layers. In this framework, for each considered node pair (u, v), the normalized SCNode encodings are term-wise multiplied, h., ho and feed into MLP. We configure the model with hidden neuron dimension of 16, learning rate of 0.001, and train it over 100 epochs with batch size of 128.
Table 6 reports the prediction results. We report the baselines from (Fu et al., 2023) and (Zhou et al., 2022), which use our experiment settings. Out of three homophilic and three heterophilic datasets, SCNode outperforms existing models in three datasets and gets the second best result in one. On the six datasets, SCNode reaches the highest mean AUC of 0.905.
Ablation Study. To evaluate the efficacy of our feature vectors, we conducted ablation studies for the node classification task. We employed three submodels: utilizing (1) spatial embeddings, (2) contectual embeddings, and (3) both (SCNode embeddings). As given in Table 7, we observe that in the CORA,
9

---

Published in Transactions on Machine Learning Research (11/2025)
Table 6: Link Prediction Performances. AUC results for Link Prediction. Baselines are reported from (Fu et al., 2023; Zhou et al., 2022; Wu et al., 2024). In the Overall column, we report mean AUC results across all datasets.
Dataset
CORA
CITESEER
PUBMED
WISC.
CORNELL
TEXAS
Overall
Node2vee (Grover et al., 2016)
0.856
0.891+0.014
0.919
GAE [Kipl & Welling, 2016)
0.89520.165
0.887+
0,957
a.680
L.730
0.753
0.820
VGAE (Kipf & Welling, 2016)
0. 0.85220.09
0.510+0.339
0.929
8.660
0.783
0.767
0.802
ARVGE (Pan et al., 2018)
0.913=0.079
0.878+0.177
0.965
0.7.1
0. 0.789e0.001
0.765
0.837
DGI (Veličkovié et al., 2018h)
0. 198888..00 uso
0.956 100
0.912
G-VAE (Grover et sl., 2019)
0. 9.917700013
0.9 n.903
0.904
GNAB (Ahn & Kim, 2021)
0.941:20.063
0.969 1.9699 0.0002
0.954:10.019
0.78220..00
0.729 088
0.751 1:007
0.854
VGNAE (Ahn Kim, 2021)
89220 067
0.955.10.000
0.897.0.010
0.703 100
0.733 57.5
0.789
0.528
GIC (Maveomatiss
2021)
0.935=0..00
0.970=0.050
0.937
LINKX (Lim al., 2021a)
0.934=00000
0.035
0,801
0. 0,758
0.837
DisenLink (Zhou et al., 2022)
0.971 1711:0.00
0.983=0.080
0.811100.10
0,807
0.901
DGAE (Fu et al., 2023)
0.958g0.0st
0.9722.0004
0.978a9002
0.757 s0.566
0.681 1007
0.683
0.838
VDGAE (Fu et al., 2023)
0. 0.959200042
0.978
0.970=0.012
0.850
0. 0.76140.47
0.813:0.541
0. 1.889
HAGNN (Wu et al., 2024)
0.93620.00s
0.924:20.06
0.967 00.003
0.858:0.048
0.7 20.7000
0.795220002
0.875
SCNode
0. 0.967
0.952a0.000
0.980=0.012
0. 0.796=0.506
0.814
0.831 20.39
0.905
1.0
1.0
1.0
0.8
0.8
0.8
Accuracy
Accuracy
0.6
Accuracy
0.4
0.2
GCN
0.2
GraphSAGE
0.2
GAT
SCN_GON
SCN GraphhssE
SON_GAT
0
28
50
75
125 150 Epoch
100
175
200
0
25
50
75
100
125
150
175
200
25
50
75
100 Epoch
125
15o
175
200
Epoch
(a)
(b)
(c)
Figure 3: Performance comparison of three GNN models (GCN, GSAGE, GAT) starting with the original node embeddings (blue) and SCNode node embeddings (orange) on the CORA dataset.
CHAMELEON, and PUBMED datasets, using only spatial or domain feature embeddings individually yields satisfactory performance. However, their combination significantly enhances performance in most cases. CHAMELEON (+5.27) experiences significant increase in accuracy in the combined setting. We note that when the ablation study has large accuracy gap between the spatial and domain-only models for a dataset (e.g., in TEXAS), the accuracies of the SOTA models in Table show huge accuracy deviations for the dataset well (e.g., TEXAS accuracies range from 54.32 to 90.30). A possible explanation is that models might individually capture either spatial or contextual information. Hence, they may be unable to combine these two sets of features to counterbalance the insufficient information present in one of them, leading to diminished accuracy scores. In contrast, the ablation study offers evidence that the SCNode approach is resilient to this limitation and experiences accuracy gain (e.g., 91.35 + 94.59 for TEXAS in Table 7). We present further ablation studics on the effect of neighborhood size choice on spatial embeddings, and similarity metric choice contextual embeddings in Appendix B.4.
Table 7: Spatial vs. Contextual. Accuracy results of our model considering different feature subsets.
Features
CORA
CITESEER PUBMED
TEXAS
CORNELL
WISC.
CHAM.
Spatial Only
84.61:1.20
71.82419
85.8610.32
67.20
49.45:1422
57.8416.14
63.70_3.05
Context Only 73.98+2.56
67.42+oos
89.70++050
91.35
92.71+4.28
94.71+2..2
78.81-1.38
Both (SCNode)
+..65
77.83+H1.00
90.53 to.e1
94.59 +5.25
88.09+4..1
92.01++406
84.08+1.58
Effectiveness in smaller training Settings. Our model leverages both spatial and contextual en- codings, where contextual embeddings are derived from class cluster centroids as landmarks, and spa-
tial encodings capture class distributions within k-hop neighborhoods. For these embeddings to be ef-
10

---

Published in Transactions on Machine Learning Research (11/2025)
1.0
1.0
M
1.0
0.8
Accuracy
Accuracy
0.6
Accuracy
0.4
0.4
0.4
0.2-
0.2
GraphSAGE
0.2
SCN_GON
SCN, GrrphhhsSEE
SCN_GAI
50
75
100
125
150
175
200
25
50
75
100
125
150
175
200
25
50
100
125
175
200
Epoch
Epech
Epoch
(a)
(b)
(c)
Performance comparison of three GNN models (GCN, GraphSAGE, GAT) starting with the original node embeddings (blue) and SCNode node embeddings (orange) on the TEXAS dataset.
Figure 4:
fective, two conditions are particularly important: reliable landmarks and balanced class distribution. To evaluate the robustness of SCNode under limited supervision, we fix 20% of the nodes as the test set and vary the size of the training set from 5% to 75% across three benchmark datasets: Cora, Cite- seer, and PubMed (see Figure 5). The results show that SCNode achieves stable performance across all datasets once at least 50% of the nodes are included in the training set. Tt also worth noting that on PubMed, our model performs noticeably well even under low training ratios. This because, de- spite only using 5% of nodes for training, the dataset still provides nearly 1,000 labeled nodes across only three classes. This ensures sulficient number of representatives per class, enabling SCNode to learn effective contextual embeddings. Consequently, SCNode produces stronger node representations when enough class representatives are available, making the embeddings more effective for supervised learning.
Training Data Size vs Accuracy
Limitations. A limitation of SCNode lies in its reliance on the quality and relevance of the landmarks used to define the contextual embeddings. If these landmarks fail to adcquately represent the underlying data distributions, the model's perfor- mance may suffer. However, this issue could be mitigated by leveraging multiple landmarks employing contrastive learn- ing in unsupervised setting to optimize landmark selection. We aim to address this limitation in future work.
85
Accuracy (%)
80
75
70
65
Cora
Citesear
60
Pubmed
5 Conclusion
20
30
40
50
60
70
Training Data (%)
In this work, we introduced SCNode, a novel method that in- tegrates spatial and contextual information from graphs. Our results show that SCNode overcomes the predictive limitations relying solely on either spatial or contextual features, achieving significant performance gains when both are informative. The model is computationally efficient. and consistently outperforms matches SOTA GNNs across diverse graph tasks, including small and large graphs as well homophilic and heterophilic settings. Furthermore, the plug-and-play design of SCNode vectors highlights their flcxibility, providing powerful enhancement to existing GNN architectures. In future work, we plan to refine SCNode for deeper integration with GNNs and expand its application to temporal graphs by incorporating temporal dynamics alongside node attributes.
Figure 5: Training set vs Accuracy
Broader Impact Statement
Our SCNode framework advances graph representation learning by addressing the fundamental limitations of GNNs, particularly their struggles with heterophilic graphs, underreaching, and oversquashing. By inte- grating spatial and contextual information through a novel landmark-based relative positioning approach, SCNode enhances the adaptability of GNNs across diverse datasets. This breakthrough enables more accu-
11

---

Published in Transactions on Machine Learning Research (11/2025)
rate node classification and link prediction, improving applications in bioinformatics, social networks, and financial modeling. Additionally, SCNode embeddings serve as plug-and-play features, significantly boosting existing GNN models' performance, thereby bridging the gap between homophilic and heterophilic settings. This research contributes to the broader goal of more generalizable and interpretable graph learning models, expanding their impact across scientific and industrial domains.
Acknowledgments
This research was partially supported by the National Science Foundation under grants DMS-2220613 and The authors acknowledge the Texas Advanced Computing Center (TACC) at UT Austin for providing computational resources that. have contributed this paper.
References
Sami Abu-El-Haija, Bryan Perozzi, Amol Kapoor, Nazanin Alipourfard, Kristina Lerman, Hrayr Harutyun- yan, Greg Ver Steeg, and Aram Galstyan. Mixhop: Higher-order graph convolutional architectures via sparsified neighborhood mixing. In international conference on machine learning, pp. 21-29. PMLR, 2019.
Seong Jin Ahn and MyoungHo Kim. Variational graph normalized autoencoders. In Proceedings of the 30th
ACM international conference information & knowledge management, pp. 2827 2831, 2021.
Andrew Arnold, Ramosh Nallapati, and William W Cohen. A comparative study of methods for transductive transfer learning. In Seventh IEEE internalional conference on dala mining workshops (ICDMW 2007), pp. 77 82. IEEE, 2007.
Pablo Barceló, Egor V Kostylev, Mikael Monet, Jorge Pérez, Juan Reutter, and Juan-Pablo Silva. The logical expressiveness of graph neural networks. In 8th International Conference Learning Representations (ICLR 2020), 2020.
Cristian Bodnar, Francesco Di Giovanni, Benjamin Chamberlain, Pietro Lio, and Michael Bronstein. Neural sheaf diffusion: A topological perspective on heterophily and oversmoothing in gnns. Advances in Neural Information Processing Systems, 35:18527-18541, 2022..
Karsten M. Borgwardt and Hans-Peter Kriegel. Shortest-path kernels graphs In Proceedings of the 5th IEEE International Conference on Data Mining (ICDM). IEEE, 2005.
Jie Chen, Tengfei Ma, and Cao Xiao. Fastgen: Fast learning with graph convolutional networks via impor- tance sampling, In International Conference Learning Represendations, 2018.
Ming Chen, Zhewei Wei, Zengfeng Huang, Bolin Ding, and Yaliang Li. Simple and deep graph convolutional networks. In International conference on machine learning, pp. 1725 1735. PMLR, 2020.
Eli Chien, Jianhao Peng, Pan Li, and Olgica Milenkovic. Adaptive universal generalized pagerank graph neural network. In International Conference Learning Representations, 2020.
Giorgio Ciano, Alberto Rossi, Monica Bianchini, and Franco Scarselli. On inductive transductive learning with graph neural networks. IEFE Transactions Pattern Analysis and Machine Intelligence, 44(2): 758-769, 2021.
Gabriele Corso, Hannes Stark, Stefanie Jegelka, Tommi Jaakkola, and Regina Barzilay. Graph neural networks. Nature Reviews Methods Primers, 4(1):17, 2024.
Chenhui Deng, Zichao Yue, and Zhiru Zhang. Polynormer: Polynomial-expressive graph transformer in linear time. arXiv preprint arXiv:2403. xrXt::2403.01233, 2024.
Jun Fu, Xiaojuan Zhang, Shuang Li, and Dali Chen. Variational disentangled graph auto-encoders for link prediction. arXiv preprint ar arXiv:2306.1 11315, 2023.
Zhengyang Geng, Xin-V Yu Zhang, Shaojie Bai, Yisen Wang, and Zhouchen Lin. On training implicit models. NewrIPS, 34:24247 24224247-44260, 2021.
12

---

Published in Transactions on Machine Learning Research (11/2025)
Aditya Grover et al. node2vee: Scalable feature learning for networks. In Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 855-864, 2016.
Aditya Grover et Graphite: Iterative generative modeling of graphs. In International conference on machine learning, pp. 2434 2444. PMLR, 2019.
Will Hamilton et al. Inductive representation learning on large graphs. NeurIPS, 30, 2017.
Mingcheng Hou and Issei Sato. A closer look prototype classifier for few-shot image classification. Advances in Newral Information Processing Systems, 2022.
Weihua Hu, Matthias Fey, Marinka Zitnik, Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta, and Jure Leskovec. Open graph benchmark: Datasets for machine learning on graphs. NewrIPS, 53:22118-22138 2020.
Wenbing Huang, Tong Zhang, Yu Rong, and Junzhou Huang. Adaplive sampling towards fast graph repre- sentation learning. NeurIPS, 31, 2018.
Di Jin, Rui Wang, Meng Ge, Dongxiao He, Xiang Li, Wei Lin, and Weixiong Zhang, Raw-gun: Random walk aggregation based graph neural network. In 31st International Joint Conference on Artificial Intelligence, IJCAI 2022, pp. 2108 2114. International Joint Conferences Artificial Intelligence, 2022.
Bharti Khemani, Shruti Patil, Ketan Kotecha, and Sudeep Tanwar. A review graph neural networks: concepts, architectures, techniques, challenges, datasets, applications, and future directions. Journal of Big Data, 11(1):18, 2024.
Thomas N Kipf and Max Welling, Variational graph auto-encoders. arXiv preprint arXie:1611.07900 2016.
Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. ICLR, 2017.
Guohao Li, Matthias Muller, Ali Thabet, and Bernard Ghanem. Deepgens: Can gens go as deep as enns?
In Proceedings of the LEEB/CVE international conference on computer vision, pp. 9267 9276, 2019.
Guohao Li, Matthias Müller, Bernard Ghanem, and Vladlen Koltun. Training graph neural networks with 1000 layers. In International conference on machine learning, pp. 6437 6449. PMLR, 2021.
Xiang Li, Renyu Zhu, Yao Cheng, Caihua Shan, Siqiang Luo, Dongsheng Li, and Weining Qian. Finding global homophily in graph neural networks when meeting heterophily. In International Conference on Machine Learning, pp. 13242-13256, PMLR, 2022.
Langzhang Liang, Xiangjing Hu, Zenglin Xu, Zixing Song, and Trwin King. Predicting global label rela- tionship matrix for graph neural networks under heterophily. Advances in Newral Information Processing Systems, 36, 2023.
Derek Lim, Felix Hohne, Xiuyu Li, Sijia Linda Huang, Vaishnavi Gupta, Omkar Bhalerao, and Ser Nam Lim. Large scale learning on non non-homophilous graphs: New benchmarks and strong simple methods, NeurIPS, 34:20887- 20902, 2021a.
Derek Lim, Xiuyu Li, Felix Hohne, and Ser-Nam Lim. New benchmarks for learning on non-homophilous graphs. Xi preprint arXiv:2104.0140, 2021b.
Cheng Liu, Chenhuan Yu, Ning Gui, Zhiwu Yu, and Songgaojun Deng. Simgel: graph contrastive learning by finding homophily heterophily. Knowledge and Information Systems, 66(3))0089-2) 2214, 2024.
Juncheng Liu, Bryan Hooi, Kenji Kawaguchi, and Xiaokui Xiao. Mgnni: Multiscale graph neural networks with implicit layers. NeurIPS, 35:21358 -117588-217,,--111 2022.
Meng Liu, Hongyang Gao, and Shuiwang Ji. Towards deeper graph neural networks. KDD, pp. 338 338-348, 2020.
13

---

Published in Transactions on Machine Learning Research (11/2025)
Meng Liu, Zhengyang Wang, and Shuiwang Ji. Non-local graph neural networks. IEEE transactions pollern analysis and machine intelligence, 41(12):10270-102776 2021.
Sitao Luan, Chenqing Hua, Qincheng Lu, Jiaqi Zhu, Mingde Zhao, Shuyuan Zhang, Xiao- Wen Chang, and Doina Precup. Revisiting heterophily for graph neural networks. NewrIPS, 35:1362- 1111221377,, 2022.
Sitao Luan, Chenqing Hua, Minkai Xu, Qincheng Lu, Jiaqi Zhu, Xiao-Wen Chang, Jie Fu, Jure Leskovec, and Doina Precup. When do graph neural networks help with node classification: Investigating the homophily principle on node distinguishability. NewrIPS, 2023.
Sitao Luan, Chenqing Hua, Qincheng Tu, Liheng Ma, Lirong Wu, Xinyu Wang, Minkai Xu, Xiao- Wen Chang, Doina Precup, Rex Ying, et al. The heterophilic graph learning handbook: Benchmarks, models, theoretical analysis, applications and challenges. arXiv preprint ar Xii:2247.006188, 2024.
Yihong Luo, Yuhan Chen, Siya Qiu, Yiwei Wang, Chen Zhang, Yan Zhou, Xiaochun Cao, and Jing Tang. Fast graph sharpness-awar minimization for enhancing and accelerating lew-shot node classification. Advances in Neural Information Processing Systems, 37:132366 1T:112364-132387, 2024a.
Yuankai Luo, Lei Shi, and Xiao-Ming Wu. Classic gnns strong baselines: Reassessing gnns for node classification. Advances in Neural Information Processing Systems, 11497600-7666) 2024b.
Yao Ma, Xiaorui Liu, Neil Shah, and Jiliang Tang. Is homophily necessity for graph neural networks?
arXiv preprint ar Xix:2 1eXi:2106.000 202221060004444 2021.
Sunil Kumar Maurya, Xin Liu, and Tsuyoshi Murata. Simplifying approach to node classification in graph neural networks. Journal of Computational Science, 62:101695, 2022.
Costas Mavromatis et al. Graph infoclust: Maximizing coarse-grain mutual information in graphs. In Pacific-Asia Conference on Knowledge Discovery and Data Mining, pp. 541-553. Springer, 2021.
Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. NewrIPS, 26, 2013.
Christopher Morris, Martin Ritzert, Matthias Fey, William Hamilton, Jan Eric Lenssen, Gaurav Rattan, and Martin Grohe. Weisfeiler and leman go neural: Higher-order graph neural networks. In Proceedings of the AAAI conference artificial intelligence, volume 33-01, pp. 4602-4609, 2019.
Suphakit Niwattanakul, Jatsada Singthongchai, Ekkachai Nacmudorn, and Supachamun Wanapu. Using of jaccard coefficient for keywords similarity. In Proceedings of the international multiconference of engineers and computer scientists, volume pp. 380-384, 2013.
Kenta Oono and Taiji Suzuki. Graph neural networks exponentially lose expressive power for node classifi- cation. In International Conference on Learning Represenlations, 2020.
Shirui Pan, Ruiqi Hu, Guodong Long, Jing Jiang, Lina Yao, and Chengqi Zhang. Adversarially regularizod graph autoencoder for graph embedding. arXiv preprint w=Xiv:1802.04000 2018.
Hongbin Pei, Bingzhe Wei, Kevin Chen-Chuan Chang, Yu Lei, and Bo Yang. Geom-gen: Geometric graph convolutional networks. In International Conference on Learning Representations, 2019.
Oleg Platonov, Denis Kuznedelev, Artem Babenko, and Liudmila Prokhorenkova. Characterizing graph datasets for node classification: Homophily-heterophhi dichotomy and beyond. Advances in Neural In- formation Processing Systems, 36, 2023a.
Oleg Platonov, Denis Kuznedelev, Michael Diskin, Artem Babenko, and Liudmila Prokhorenkova. A crit- ical look at the evaluation of gnns under heterophily: Are we really making progress? arXiv preprint arXiv:2302. 116400 2023b.
Ladislav Rampášck, Michael Galkin, Vijay Prakash Dwivedi, Anh Tuan Luu, Guy WOlf, and Dominique Beaini. Recipe for a general, powerful, scalable graph transformer. Advances in Neural Information Processing Systems, 35:14501-1455 2022.
14

---

Published in Transactions on Machine Learning Research (11/2025)
Prithviraj Sen, Galilco Namata, Mustafa Bilgic, Lise Getoor, Brian Galligher, and Tina Eliassi-Rad Collec- tive classification in network data. AT Magazine, 29(3):93 93, 2008.
Nino Shervashidze, S. V. N. Vishwanathan, Tobias Petri, Kurt Mehlhorn, and Karsten M. Borgwardt. Efficient graphlet kernels for large graph comparison. In Proceedings of the 12th International Conference on Artificial Intelligence and Statistics (AISTATS), 2009.
Nino Shervashidze, Pascal Schweitzer, Erik Jan Van Leeuwen, Kurt Mehlhorn, and Karsten M Borgwardt. Weisfeller-lehman graph kernels. JMLR, 12(9), 2011.
Yunsheng Shi, Zhengjie Huang, Shikun Feng, Hui Zhong, Wenjin Wang, and Yu Sun. Masked label prediction: Unified message passing model for semi-supervised classification. arXiv preprint arXiv:2009.0S500 2020.
Jake Snell, Kevin Swersky, and Richard S. Zemel. Protolypical networks for few-shot learning. In Advances in Neural Information Processing Systems, 2017.
Yunchong Song, Chenghu Zhou, Xinbing Wang, and Zhouhan Lin. Ordered gnn: Ordering message passing to deal with hetcrophily and over-smoothing. arXiv preprint wrXiv:2902.01524 2023.
Susheel Suresh, Vinith Budde, Jennifer Neville, Pan Li, and Jianzhu Ma. Breaking the limit of graph neural networks by improving the assortativity of graphs with local mixing patterns. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pp. 1541-1551, 2021.
Jake Topping, Francesco Di Giovanni, Benjamin Paul Chamberlain, Xiaowen Dong, and Michael M Bronstein. Underslanding over-squashing and bottlenecks on graphs via curvature. arXiv preprinl arXiv:2111.14522 2021.
Petar Veličkovié, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua Bengio. Graph attention networks. ICLR, 2018a.
Petar Veličković, William Fedus, William L Hamilton, Pietro Liò, Yoshua Bengio, and R Devon Hjelm. Deep graph infomax. arXiv preprint MXXi:11099.1344 2018b.
S. V. N. Vishwanathan, Nicol N. Schraudolph, Risi Kondor, and Karsten M. Borgwardt. Graph kernels. Journal of Machine Learning Research, 11:1201-1242, 2010.
Guancheng Wan, Yijun Tian, Wenke Huang, Nitesh V Chawla, and Mang Ye. S3gel: Spectral, swift, spatial graph contrastive learning. In ICML, 2024.
Kuansan Wang, Zhihong Shen, Chiyuan Huang, Chich-Han Wu, Yuxiao Dong, and Anshul Kanakia. Mi- crosoft academic graph: When experts are not enough. Quantitative Science Studies, 1(1):396-413, 413, 2020.
Dengsheng Wu, Huidong Wu, and Jianping Li. Hierarchy-aware adaptive graph neural network. IEEE Transactions on Knowledge and Data Engineering, 2024.
Qitian Wu, Wentao Zhao, Zenan Li, David P Wipf, and Junchi Yan. Nodeformer: scalable graph structure learning transformer for node classification. Advances in Neural Information Processing Systems, 35:27387- 27401, 2022.
Qitian WI, Wentao Zhao, Chenxiao Yang, Hengrui Zhang, Fan Nic, Haitian Jiang, Yatao Bian, and Junchi Yan. Sgformer: Simplifying and empowering transformers for aage-graph representations. Advances in Newral Information Processing Systems, 36:64753-6445 2023.
Shunxin Xiao, Shiping Wang, Yuanfei Dai, and Wenzhong Guo. Graph neural networks in node classification: survey and evaluation. Machine Vision and Applications, 33:1-19, 2022.
Keyulu Xu, Weihua Hu, Jure Leskovee, and Stefanie Jegelka. How powerful are graph neural networks? In International Conference on Learning Representations, 2019.
15

---

Published in Transactions on Machine Learning Research (11/2025)
Zhe Xu, Yuzhong Chen, Qinghai Zhou, Yuhang Wu, Menghai Pan, Hao Yang, and Hanghang Tong. Node classification beyond homophily: Towards general solution. In Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 2862 2873, 2023.
Naganand Yadati. Localformer: Mitigating over-globalising in transformers graphs with localised training. Transactions on Machine Learning Research, 2025.
Yuchen Yan, Yuzhong Chen, Huiyuan Chen, Minghua Xu, Mahashweta Das, Hao Yang, and Hanghang Tong, From trainable negative depth to edge heterophily in graphs. Advonces in Neural Information Processing Systems, 36, 2024.
Yujun Yan, Milad Hashemi, Kevin Swersky, Yaoqing Yang, and Danai Koutra. Two sides of the same coin: Heterophily and oversmoothing in graph convolutional neural networks. In 2022 IEEE International Conference on Data Mining (ICDM), pp. 1287-1292. IEEE, 2022.
Le Yu, Leilei Sun, Bowen Du, Chuanren Liu, Weifeng Lv, and Hui Xiong, Heterogeneous graph representation learning with relation awareness. IEEE Transactions on Knowledge and Data Engineering, 2022a.
Le Yu, Leilei Sun, Bowen Du, Tongyu Zhu, and Weifeng Lv. Label-cnhanced graph neural network for semi-supervised node classification. IEEE Transactions on Knowledge and Dala Engineering, 2022b.
Li Zhang, Heda Song, Nikolaos Aletras, and Haiping Lu. Node-feature convolution for graph convolutional networks. Pattern Recognition, 128:108661, 2022.
Jianan Zhao, Hesham Mostafa, Mikhail Galkin, Michael Bronstein, Zhaocheng Zhu, and Jian Tang. Graphany: A foundation model for node classification on any graph. arXiv preprint orXiv:2405.20444 2024.
Xin Zheng, Yixin Liu, Shirui Pan, Miao Zhang, Di Jin, and Philip S Yu. Graph neural networks for graphs with heterophily: A survey. arXiv preprint arXiv:2202.0700 07082, 2022.
Shijic Zhou, Zhimeng Guo, Charn Aggarwal, Xiang Zhang, and Suhang Wang. Link prediction on hetcrophilic graphs via disentangled representation learning. arXiv preprint arXiv:2208.018 01180, 2022.
Jiong Zhu, Yujun Yan, Lingxiao Zhao, Mark Heimann, Leman Akoglu, and Danai Koutra. Beyond homophily in graph neural networks: Current limitations and effective designs. NewrIPS, 33:7793-768 7804, 2020.
Jiong Zhu, Ryan A Rossi, Amup Rao, Tung Mai, Nedim Lipka, Nesreen K Ahmed, and Danai Koutra. Graph neural networks with heterophily. In Proceedings of the AAAT conference on artificial intelligence, volume 35, pp. 11168 11176, 2021.
Jiong Zhu, Yujun Yan, Mark Heimann, Lingxiao Zhao, Leman Akoglu, and Danai Koutra. Heterophily and graph neural networks: Past, present and future. IEEE Data Engineering Bulletin, 2023.
16

---

Published in Transactions on Machine Learning Research (11/2025)
Appendix
A Homophily and SCNode
A.1 Recent Homophily Metrics
Until recently, the prevailing homophily metrics were node homophily Pei et. al. (2019) and edge ho- mophily Abu-El-Haija et al. (2019); Zhu et al. (2020). Node homophily simply computes, for cach node, the proportion of its neighbors that belong to the same class, and averages across all nodes, while edge homophily measures the proportion of edges connecting nodes of the same class compared to all edges in the network. In the past few years, to study heterophily phenomena in graph representation learning, several new homophily metries were introduced, e.g., class homophily Lim et al. (2021b), generalized edge homophily Jin al. (2022) and aggregation homophily Luan et al. (2022), adjusted homophily Platonov al. (2023a) and label informativeness Platonov et al. (2023a). In Table 8, we give these metrics for our datasets. The details of these metrics can be found in Luan et al. (2023).
Hunde (9)
1
Σ Hiode
Σ
{u a E No, Zu: Zo.i
[V]
1
dp
VEV
Hedge (g) |fEwel | euv E E, Lu: = Ze.i3
[3]
1
{8l
Z = =1}
Evev, Hu No, Zu. Zu Eve(viZen=11) do
Helass C-1
h
where h4
N
k=
Σ cos(xIt,
HGE(9) (3.f)EE
[3
Hapz( (9)= |2| x {u)
Mcan, (IS(A, 2) =2.;))) 2 Mean, (1S(A, 2) #2.
Hodge DE
-c=1 Pe
Hadj
1 Σe
Σ
Eanog Pes,oz log
Perice ded Pep
Eeg,rr Pe log Pe
LI
E.Po log
E. Pe log Pe
where Hhode is the local homophily value for node v; [a]+ max(0, a); h2 is the class-wise homophily metric Lim et al. (2021b); Mcan, (f-H) takes the average and S S(A,2) ÁZ(ÁZ) is the post-aggregation node similarity matrix; Dr Ev:r,=e do. Dr. 227 1(z. =0r. =e) C, (1....,C).
of given multiset of values or variables
Dr
Per.02 D(u,)e
2
Table 8: Homophily metrics for our datasets.
Metric
CORA
CITES.
PUBMED
TEXAS
CORNELL
WISC.
CHAM.
MssN-spaat11
0.8129
0.6861
0.7766
0.1079
0.1844
0.2125
0.2549
Hscc--context
0.1702
0.1949
0.3245
0.2352
0.2409
0.2449
0.2564
Hando
0.8252
0.7175
0.7924
0.3855
0.1498
0.0968
0.2470
Hedge
0.8100
0.7362
0.8024
0.5669
0.4480
0.4106
0.2795
Hales
0.7657
0.6270
0.66-1
0.0468
0.0941
0.0013
0.0620
Hass
0.9904
0.9826
0.9432
0.8032
0.7768
0.6940
0.6100
HGB
0.1700
0.1900
0,2700
0.3100
0.3400
0.3500
0.0152
Halij
0.8178
0.7588
0.7431
0.1889
0.0826
0.0258
0.0663
LI
0.5904
0.4508
U.4093
0.0169
0.1311
0.1923
0.0480
17

---

Published in Transactions on Machine Learning Research (11/2025)
Table 9: Homophily matrices for CORA dataset. Every row represents the corresponding homophily ratio of the row's class. In every row, the highest homophily ratio is marked bold.
SCNode Spatisl-1 Homophily Matrix
SCNode Context-S Homophily Matrix
SCNode Context-1 Homophiiyy Matrix
C2
C2
C2
C,
022
C2
Co
C,
C:
C2
C2
C,
C:
0.000
0.001
0.137
0.023
0.151
0.124
Table 10: Homophily matrices for the WISCONSIN dataset. Every row represents the corresponding homophily ratio of the row's class. In every row, the highest homophily ratio marked bold.
SCNode Spatial-1 Homophily Matrix
SCNode Spatial-2 Homophily Matrix
SCNode Domain Homophily Matrix
C
C2
C2
C,
C2
G
C2
C2
C,
C2
C
C2
C2
C:
C2
C2
0.323
0.000
0.000
0.323
0.323
C1
0.444
0.278
0.111
0.111
0.056
C1
0.221
0.198
0.207
0.192
0.181
0.219
0.595
0.185
0.001
C2
0.001
0.439
0.106
0.016
C2
0.148
0.247
0.223
0.201
Cs
0.000
0.547
0.225
0.145
0.084
Ca
0.011
0.351
0.481
0.111
0.046
Cs
0.174
0.205
0.228
0.204
0.189
C₁ 0.044
0.268
0.468
0.071
0.150
C4
0.029
0.238
0.485
0.110
0.138
C2
0.145
0.203
0.217
0.237
0.199
C, 0.149
0.039
0.397
0.2001
0.215
C
0.149
0.084
0.374
0.136
0.258
C,
0.149
0.204
0,209
0.209
0.230
In Table 9 below, we give SCNode Homophily matrices which provides detailed insights on the class in- teractions in CORA dataset. In particular, Spatial-1 matrix interprets as for any class, the nodes likely to form link with same class node in their one neighborhood. In the following Contextual-SSlective and Contextual-Inchen Homophily matrices, independent of graph distance, we see the positions of attribute vectors similarity to the chosen class landmarks.
A.2 SCNode Homophily Matrices
We provide further homophily matrices for one homophilic and one heterophilic graphs. In Table 9, for CORA dataset, we observe in SCN Spatial-1, Context-S and Context-I Homophily matrices show the strong homophily behavior in both spatial and contextual aspects. In Spatial-1 matrix the entry hy represents how likely the nodes in C, to connect nodes in C, among their 1-neighborhood. The very high numbers in the diagonal shows that most classes likely to connect with their own class, as CORA's node homophily ratio (0.83) suggest. From the matrix, we have finer information that the class C3 is very highly homophilie. The domain matrices represents the average similarity/ clossness to class landmarks. We observe that in both domain matrices, the node fcature vectors likely to land close to their own class landmark, and SCNode vectors captures this crucial information.
In Table 10, for WISCONSIN dataset, spatial-1 1 and Spatial-2 matrices show the irregular behavior as the WISCONSIN node homophily ratio (0.09) suggest. The nodes are unlikely to connect their adjacent nodes in the same class. Even in their two neighborhoods, they don't have many of their fellow classmate for C₁ and C3. However, while the other three classes are not connecting their common neighbors. Finally, the domain homophily matrix again shows that while they are not very close in the graph, they all share common interests, every node's feature vector lands close to their classmates!.
classmate, they have several
While different homophily ratios are summarizing crucial information about node tendencies, we observe that our matrices are giving much finer and easily intepretable information no class behaviors.
A.3 Proof of the Theorem and New Homophily Metrics
In Soction 3.4, we have defined homophily and discussed the relation between SCNode vectors and the homophily notion. Here we give details how to generalize this idea give finer homophily notions for graphs. First, we give the proof of Theorem 3.7.
18

---

Published in Transactions on Machine Learning Research (11/2025)
Theorem A.1. For undirected graph G EV, let (I()) be the spotial vector defined in Appendir A. Let (i(1) be the vector where the entry corresponding to class of is set to O. Then,
Σ [ NE
1- 1-
| (u)I
Proof. For an undirected graph 9 (V, R) with node class assignment function C V - (1,2...., N); ã1 (v) = (v) 02(v) ... (x)] where as (v( = #(u E N(()) C(u) = i). Let C(v) fv. Then, by setting (v) O, we get a new vector ãi (v) = (v)... (z.-1(v) 0 (1.+1(e) ...]
L'-norm of ãi (o) is [ = (v). Similarly, [ (v)]l Eizj. a,(e). Hence, we have ||ā (v)ll= [ā](v)] aj. (v). Notice that by definition, (z.(v) (0)), the number of neighbors in the same class with v. Similarly, [0)(1) = deg(v). Hence,
| (v)ih
(+)( - [+)(1)]
T(v) deg(v)
(1)
[
[ Evey TTv)
1
As ((
we have - (9) as
deg(v))
(e() 1 EvEy deg(v) MV Evev(1
n(v) 1
a (
EvEy
[V]
deg(v)
= M
(vi(l)
where the last equality follows by Equation (1). The proof follows.
This perspective inspires different ways to generalize the homophily concept by using SCNode vectors. Notice that in SCNode matrices above, employed classwise grouping to measure class interactions. If we do not use any grouping for the nodes, we get natural generalizations of existing homophily ratios. First, we define higher homophily by using the ratio of the number of nodes in the 2-neighborhood N2 with the same class to [N2(.)].
Definition A.2 (Higher Homophily). Given g = (V, with C V + (1,2,..., N} representing node classes. Let (2))) be the number of nodes in (2 in the same class with v. Then, homophily ratio of g is defined as
1
me(e) (N2(v))
(g)
[V]
VEV
By applying similar ideas to the proof of Theorem 3.7, obtain the following result.
Theorem A.3. For undirected graph G (V,R), let ōz(u) be the spotial vector defined above. Let (z( be the vector where the entry corresponding to the class of is set to O. Then,
1 (2( =
Σ DEV
V
While the above notions represent structural homophily, we introduce another homophily by comparing the contextual vectors of neighboring nodes with the central node's.
(1, representing node classes. Let A(v) be the contextual vector as defined in Section 3.2. Let 3(v) be the vector where the entry corresponding to the class of o is set to 0. o.Then, define the contextual homophily of G as
Definition A.4 (Contextual Homophily). Given G = (V, with C: (1,
1 V
(G) =
Similar homophily metrics, as introduced in Luan al. (2023); Lim et al. (2021b); Zhu al. (2020); Jin et al. (2022), were employed to investigate the concept of heterophily in graph representation learning. Tn
19

---

Published in Transactions on Machine Learning Research (11/2025)
particular, our SCNode features inherently encompass this information, enabling the ML classifier to adapt accordingly. This explains the outstanding performance of SCNode in both homophilic and heterophilic settings for node classification and link prediction.
Remark A.5 (SCNode and Underreaching). Underreaching arises because a depth-K message-passing GNN is K-local: its output at node u depends only on the K-hop ego-network and features within it, 80 nodes that are K-hop isomorphic remain indistinguishable. SCNNde breaks this locality at the input level before message passing.
From a speciral perspective, spatial coordinates on (u) are class-distribution statistics over K-hop neighbor- hoods and can be expressed as 0g Pe)A),, where Pk is degree- K polynomial in the normalized adjacency A and Y encodes class indicators on labeled nodes. Thus, ag lies within the span of polynomial filters in A. In contrast, contextual coordinates 3(u) [d(Xu.)1)..... ,(X,,, Ex]] are distances to class landmarks (G) in attribute space and are independent of A. Concatenating [ox(u) II [(w) therefore injects nonlocal, zero-hop channel that cannot be reproduced by any finite-degree polynomial in A, strictly enlarging the representable function class.
From an expressicity perspective, two nodes that are K-hop isomorphic receive identical inputs under any K-local MPGNN. With contextual embeddings B(u), such nodes can still be distinguished if they occupy different positions relative to class landmarks in attribute space, allowing even a shallow classifier on [ax || 3] to separate them.
Empirically, Table 7 supports this analysis: contextual-only embeddings succeod in hetcrophilic graphs where local neighborhoods are misleading, while spatial-only embeddings are strongest on homophilic graphs. Their combination consistently achieves the best results, aligning with the theoretical claim that supplies global class information inaccessible to local filters, while a exploits reliable neighborhood structure.
B Generalizations of SCNode
B.1 Inductive and Transductive Settings
Inductive Setting. In the inductive learning framework, model trained by using the training data Dtrasn while the test data, Diest is completely hidden during the training time. This means that no information about test nodes (c.g., edges between training and test nodes) is provided during the training stage. The learning procedure aims to minimize suitable loss function to capture the statistical distribution of the training data. Once an inductive model has undergone training, it can be utilized to make predictions on new (unscen) data, thereby determining the labels for unlabeled nodes.
Transductive Setting. In the transductive learning framework, which closely aligns with semi-supervised learning, both the training data Dtrain and the test data Prest can be simultaneously leveraged capitalize their interconnectedness. This interrelationship can be employed cither during the training phase, the prediction phase, or both. Specifically, in the training stage, the information about Usest and its position in the graph is known, while its label Utest remains concealed. Consequently, the model is trained with explicit awareness of the nodes it will be evaluated on after the training process. This can serve as a valuable asset for the model, enabling it to establish a sound decision function by exploiting the characteristics observed in Vtest-
To clarify the differences between the inductive and transductive settings in graph- learning, consider given graph G, with datasets Drrain and Dust. In the inductive setting, all test nodes and their connected edges are removed to create the training subgraph Graan The model is trained exclusively on Gtaai and only gains access to the complete graph G during the testing stage. Conversely, in the transductive learning approach, the model has access lo the entire graph G during training; however, the labels of the test nodes remain hidden throughout this process. It is noteworthy that any dataset configured for transductive learning can be adapted to the inductive sctting by excluding test nodes and their connecting odges during the training phase. However, converting from the inductive to the transductive setting is not generally feasible. For further details, see the references Ciano et al. (2021); Arnold et al. (2007).
20

---

Published in Transactions on Machine Learning Research (11/2025)
B.2 SCNode for Transductive Setting and Iterated Predictions
So far, outlined our SCNode vectors for simplicity in inductive setting, To adapt to the transductive setting, we make adjustments without altering contextual vectors. In transductive setting, test node labels are hidden during training, but connection information to training nodes is available. We introduce new 'unknown' class Cu for test nodes with unknown labels, considering them as neighbors to training nodes. div], where
Each node u E V is represented by a N + dimensional vector of = [4% af 4% %f the count of neighboring test nodes (unknown labels) of node u, and al is the count of neighboring training nodes in class Cj for j N. Similar representations are defined for āii(u), dio(u), and (z( The superscript 0 indicates no iterations have occurred yet.
Iterated Predictions:
Recall that the ultimate goal in the node classification problem is to predict the labels of new (Lest) nodes. Afler we oblain all SCNode vectors for training and lest nodes above, we let our ML model make a prediction for each test node U E Vieet. Let Po : Vieet + CC......CN) be our predictions. Hence, we have a label for each node in our graph G.
While the original spotial veclor (u) connol use any class information for the Lest nodes, in the nexl slep, we remedy this by using our class predictions for test nodes. In particular, by using the predictions Po, we define a new (improved) spatial vector ()) I al2 als ... divl where we use predictions Po for neighboring test nodes. Notice that this is no longer (N + 1)-dimensional vector as there is no unknown class anymore. Similarly, we update all spatial vectors, train our MI. model with these new node labels, and make a new prediction for test nodes. Then, we get new label predictions Pi Veest + C.... By using predictions PI, we define the next iteration ai(u) = lof of aix] and train our model with these updated vectors. Again, we get new predictions P2 Viest + CCy]. In our experiments, we observe that 1 or 2 itcrations (Pi or P2) improve the performance significantly, but further iterations do not, in the transductive setting.
B.3 Spatial Embeddings on Directed, Weighted Graphs
Directed Graphs. When G directed, to obtain finer information about node neighborhoods, we produce two different embeddings of size N, áua(u) and (u) for the k-hop neighborhood:
(u) = jaki agp2 dndl (go(u) = [a 2x
axxl
where dy is the count of k-hop neighbors incoming to belonging to class Cj while obj is the count of neighbors outgoing from u belonging to class C, (See Figure 2(a)).
Weighted Graphs: In weighted graphs, the counts incorporate edge weight information. Specifically, we define the weighted feature vector oz (u) for node u as: o. = [a "K. .., [EN] where og is the sum of the weights of the edges connecting u to the k-hop neighbors belonging to class Cj. If the weight of an edge is inversely proportional to the similarity between nodes,
use the sum of the reciprocals of the weights instead: "t = EvEN (w)nC, weight(u.r) This approach allows for adjustments in how edge weights inffuence the calculation of proximity and class association. Additionally, in the context of directed graphs within weighted graph setting, different vectors can be defined for incoming and outgoing neighborhood connections, such as (u) for incoming edges and (u) for outgoing edges, providing more detailed representation of node relationships based on directionality and weight.
Table 11: The dimension of the SCNode embedding used for each dataset in our model
CORA CITESEER PUBMED OCEN-ARXIV
OGBN-MAG TEXAS CORNELL WISCONSIN CHAMELEON
Dimeusion
36
120
1,524
B.4 Further Ablation Studies
In this part, we present further ablation studies to evaluate the sensitivity of SCNode to neighborhood size in spatial embeddings and to similarity metrics in contextual embeddings. Recall that the two embeddings
21

---

Published in Transactions on Machine Learning Research (11/2025)
serve distinct roles: (1) Spatial embeddings are derived from neighborhood label distributions up fixed hop size and do not involve landmarks; (2) Contextual embeddings are constructed using class landmarks in attribute space, with similarity metrics selected according to feature type (e.g., Jaccard for sparse binary features such as Cora and Citeseer, Euclidean for real-valued features such as PubMod).
Spatial Embeddings:: Effect of Neighborhood Size. We compared spatial embeddings derived from 1- hop and 2-hop neighborhoods. Results show that 2-hop neighborhoods yield modest accuracy improvements across datasets, though at higher computational cost. For example, on PubMed the accuracy increases from 77.59 + 0.61 with 1-hop to 82.28 = 0.09 with 2-hop, while runtime grows from 63s to 232s. Based on this trade-off, we adopt 2-hop neighborhoods the main experiments as a balanced choice. The detailed results are reported in Table 12.
Contextual Embeddings: Effect of Similarity Metrics. We further evaluated contextual embeddings using Jaccard, cosine, and Euclidean metrics. On binary datasets (Cora, Citeseer), Jaccard provides the best performance, while cosine achieves similar results but at higher computational cost. Euclidean performs substantially worse in this setting, as expected for sparse binary vectors. On real-valued datasets (PubMed), both Euclidean and cosine produce strong results. These findings confirm that the effectivencss of contextual embeddings depends on aligning the similarity metric with the feature type. Full results are provided Table 13.
These ablations demonstrate that SCNode is robust to landmark quantity and neighborhood size, and that contextual similarity metrics should be selected according to feature type to achieve the best balance between accuracy and efficiency.
Table 12: Effect of Neighborhood Size. Performances of spatial-only vectors used by different k-hop neighborhoods.
Accuracy
Ngbd Size
Cora
Citeseer
PubMed
Cora
Citeseer
PubMed
I-hop
83.96 # 1.35
71.28 # 1.84
77.59 = 0.61
18
18
63 8
2-hop
84.61 + 1.20
71.82 + 1.95
82.28 0.00
36
3
232
Table 13: Effect of Metric Choice. similarity metrics.
Performances of contextual-only vectors obtained by different
Accurary
Time
Metric
Cora
Citeseer
PubMed
Cora
Citeseer
PubMed
Jaceard
73.98 2.56
67.42
n/a 85.62 + 0.41
2
n/a
Casine
73.50 2.48
67.09 + 1.80
5s
15.
17
Enclidean
62.00 + 2.65
60.89 + 1.75
85.92 + 0.52
2s
6
1
C Details of SCNode Embeddings
In all datasets, we basically used the same method obtain our vectors, however, when the graph type (directed, undirected), or node altribute vector format varies, our methodology naturally adapts the corre- sponding setting detailed below. Note that details of OGBN datasets can be found at Hu et al. (2020) and at Stanford's Open Graph Benchmark site
CORA: The CORA dataset a directed graph of scientific publications classified into one of the 7 classes. Each node u represents a paper and comes with a binary (0/1) vector X,, of length 1433 indicating the presenee/absence of the corresponding word in the paper from a dictionary of 1433 unique words. The
htttps://obbetaane stanford. wer.aordddee///bccc//odeeoooop/
22

---

Published in Transactions on Machine Learning Research (11/2025)
task is to predict the subject area of the paper, e.g. Neural Networks, Theory, Case-Based. The directed graph approach is used to extract the attribute vector, 7!)), from the CORA dataset. Recall that this is transductive setting, and the first row is an 8 dimensional spatial vector ( (Section 3.1) where the first 7 entrics represent the count of citing (incoming) papers from the corresponding class and the 8th entry is the count. of unknown citing paper (v, E Ptest) (Appendix B.2). The second row (f)(w) is defined similarly using the count of cited (outgoing) papers of the corresponding class. The third and fourth row of the spatial feature vector is obtained similarly using the second neighborhood information of each node for citing and cited paper respectively. For the first iteration 7 (u), the same setup is followed ignoring the 8 entry of each row because there is no unknown class now.
For the contextual vector (w), we follow the landmark approach described in Section 3.2. We define the first landmark vector E] which is a binary word vector of length 1433 such that the entry is if the corresponding word is present in any of binary vectors X₂ belonging to class Cj, and the entry 0 otherwise. Then, each entry رıj of (u) is the count of common words between N₂ and E) for each class, which produces a 7 dimensional contextual vector. Similarly, for the contextual vector Bg(u), we use a more selective landmark vector E3 is defined as a binary word vector of length 1433 indicating the presence/absene corresponding word in at least 10% nodes in the class C,. Hence, the initial vector 7 is 46 dimensional (32 spatial, 14 contextual), in the next iterations, (1),), ((())) are both 42-dimensional (28 spatial, 14 contextual).
of the
CITESEER: The CITESEER is also a directed graph of scientific publication classified into one of the 6 classes. Each node represents paper and comes with a binary vector like CORA from dictionary of 3703 unique words. Here, the aim of the node classification task is to make a prediction about the subject area of the paper. Since the graph properties/strucuure and node representing word vector is similar to the CORA dataset, the same feature-extractingg techniques for both spatial and contextual vectors is followed here. Hence, the initial vector (() is 40 dimensional (28 spatial, 12 contextual), in the next iterations, y M()) (u)) are both 36-dimensional (24 spatial, 12 contextual).
PUBMED:
The PUBMED dataset is a directed graph of 19717 scientific publications from the PubMed database pertaining to diabetes classified into one of three classes. Each node represents publication and is described by a TF /IDF weighted word vector from dictionary which consists of 500 unique words. Since the graph structure ure is quite similar to CORA and CITESEER, a similar method is followed to extract the spatial features. So the initial spatial vector is 16 dimensional and it is 12 dimensional for the second iteration. For the contextual vector 3(u), Principal component analysis (PCA) is used to reduce the dimension of the given weighted word vector from 500 to 100. Hence, the initial vector, ?(()) is 16 dimensional (spatial only) and the vector is 112 dimensional (12 spatial, 100 contextual) in the next iteration.
The OGBN-ARXIV dataset is a directed graph, representing the citation network between all Computer Science (CS) arXiv papers indexed by MAG Wang et al. (2020). Each node is an arXiv paper and each directed edge indicates that one paper cites another one. Each paper comes with a 128-dimensional vector obtained by averaging the emboddings of words in its title and abstract. The emboddings of individual words are computed by running the skip-gram model Mikolov el al. (2013) over the MAG corpus. We also provide the mapping from MAG paper IDs into the raw texts of titles and abstracts here. In addition, all papers are also associated with the year that the corresponding paper was published. The task is to predict the 40 subject areas of arXiv CS papers, e.g., AI, cs. and CS.
OGBN-ARXIV:
The vector y(u) for OGBN-ARXIV is obtained by using our directed graph approach (Section 3.1). The first row of spatial vector (u) is 40-dimensional, and each entry is the count of citing (incoming) papers from the corresponding class. The second row is (1)(u) and is defined similarly, where each entry is the count of cited (outgoing) papers from the corresponding class. For contextual vector ((u), follow the landmark approach described in Section 3.2, employing just one landmark. Since the vectors are weighted vector, we use Euclidean distance to determine the distance between the landmark and given node. Hence, for each node 7u) has 80-dimensional spatial, and 40-dimensional contextual vector, which totals 120-dimensional SCNode vector.
23

---

Published in Transactions on Machine Learning Research (11/2025)
The OGBN-MAG dataset is a heterogeneous network composed of subset of the Microsoft Academic Graph (MAG) Wang et al. (2020). It contains four types of entities entities-papers papers (736,389 nodes); authors (1,134,649 nodes), institutions (8,740 nodes), and fields of study (59,963 nodes) as well as four types of directed relations connecting two types of entitics an author is "affiliatd with" an institution, an author "writes" " a paper, a paper "ciles" a paper, and a paper "has a topic of" a field of study, Similar to OGBN-ARXIV, each paper associated with a 128-dimensional word2vee vector, and all the other types of entities are not associated with input node features. Given the heterogeneous OGBN-MAG data, the task is to predict one of 349 venues (conference or journal) of each paper.
OGBN-MAG:
The vector 7(u) for OGBN-MAG is bit different than OGEN-ARXIV, as OGBN-MAG is heterogencous network. We first collapse the network to a homogeneous network for papers. Similar to OGBN-ARXIV, we obtain -49edimnnll spatial vectors, ie., (1.(u) (citing papers), and a1o(u) (cited papers). As another spatial vector from a different level of the heterogeneous network, we use author information as follows. Each author has a natural 349-dimensional vector where cach entry is the number of papers the author publishod in the corresponding venue. For each paper, we consider the author with the most publications and assign their attribute vector to the paper's attribute vector. We call it author vector aguthor (u). We construct similar set of vectors for field of study another type of node information. Each paper belongs to 1 or more fields of studies (or topics), and for each unique topic, we construct a attribute vector T = {ti, t,,.... fnum d such that & denotes the number of papers assigned to venue for the given topic. We then aggregate these topic attribute vectors for each paper as follows: for a given paper with assigned topics topic let @topic(u) Σx= Tj and append this final aggregate vector to the paper's attribute vector. For contextual vector 3(n), we directly use 128-dimensional vector for each node as it is. Hence, 7()) is concatenation of spatial vectors (1:(u), a1o(u); Gauthor (u), Oropic(u)), and ((u) which totals 4 4-349 + 128 1524 dimensional vector.
WebKB (TEXAS, CORNELL and WISCONSIN): Carnegie Mellon University collected the WebKB dataset from computer science departments of various universitios. Three subsets of this dataset are TEXAS, CORNELL and WISCONSIN. The dalaset contains links between web pages, indicating relationships like "is located in" or "is a student of forming a directed graph structure. Node features are represented as bag-of-words, creating a binary vector for each web page. The classification task involves categorizing nodes into five types: student, project, course, staff, and faculty. Similar to the CORA dataset, these datasets share a directed graph structure and binary vector representation for node features, leading to the use of comparable feature extraction methods for spatial and contextual vectors. Therefore, the initial attribute vector (u) is 34 dimensional, comprising 24 spatial and 10 contextual dimensions. In subsequent iterations, it is 30-dimensional, with 20 spatial and 10 contextual dimensions each.
Wikipedia Network (CHAMELEON): The dataset depict page-page networks focused on specific topics such as chameleons. In these networks, nodes represent articles, and edges denote mutual links between them. Node features are derived from several informative nouns found in the corresponding Wikipedia pages. If feature is present in the feature list, it signifies the occurrence of an informative noun in the text of the Wikipedia article. The objective is to classify the nodes into five categories based on the average monthly traffic of the respective web pages. In the context of the Wikipedia network, each link between articles does not imply a one way relationship; instead, it signifies a mutual connection between the two articles, making it an undirected graph. Therefore, undirocted feature extraction approaches are employed for 7 (()).
Regarding spatial features, the first row consists of 6-dimensional vector. The first five entries represent the count of five classes, while the sixth entry represents the count of the unknown class in the 1-hop neighborhood. In the second row, the same procedure is applied for the 2-hop neighborhood. Subsequent iterations follow a similar process, but the 6th entries are ignored because there is no unknown class at this point in the analysis. For contextual features in the Wikipedia network, a similar approach as used for the CORA dataset is employed due to their analogous nature. This approach results in a 10-dimensional vector. Consequently, the initial vector is 22-dimensional, and subsequent iterations reduce it to 20-eimens..nal.
24


Reviews:
[{"review_id": "GkO69h2HBO", "reviewer": "Reviewer_7Xoe", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The spatial and contextual embeddings can capture information from both local and global scopes of a graph.\n2. The SCNode vectors can be viewed as plug-and-play components to enhance existing GNN methods.\n3. The contextual embedding is interesting while measuring the distance between a node and all class landmarks.\n\nWeaknesses:\n1. The SCNode are highly rely on the training set label since the both spatial and contextual node embeddings need label information. It may work badly in real-world low label rate setting. In addition, it seems the calculation of class landmark need all labels rather that training set according to the description, which can leads to data leakage.\n2. The datasets and baseline methods are insufficient to evaluate the proposed method. For datasets, the used heterophilic datasets have some drawbacks [1]. For baseline methods, some representative methods [2,3] are not included. In addition, the performance of SCNode on OGB datasets are not satisfactory.\n3. As a plug-and-play component, SCNode vectors may bring more computing consumption. How much the performance gain in Figure 4 is attribute to the additional dimensions of node embedding.\n4. The claim about “no prior work has proposed a systematic approach to analyze class-wise relations” is incorrect. Here are some works on investigating class-wise relations [4,5,6].\n\nReference:\n\n[1] A critical look at the evaluation of GNNs under heterophily: Are we really making progress? ICLR 2023.\n\n[2] Revisiting Heterophily For Graph Neural Networks. NeurIPS 2022.\n\n[3] Ordered GNN: Ordering Message Passing to Deal with Heterophily and Over-smoothing. ICLR 2023.\n\n[4] Is Homophily a Necessity for Graph Neural Networks? ICLR 2022.\n\n[5] Heterophily and Graph Neural Networks: Past, Present and Future. IEEE Data Eng. Bull. 2023.\n\n[6] Graph Neural Networks with Heterophily. AAAI 2021.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposed a Spatial-Contextual Node Embedding framework (SCNode), which  integrates both spatial and contextual information from local and global scope. It can be easily integrated into any GNN model as plug-and-play components to improve the performance in both homophilic and heterophilic settings. In addition, a new homophily matrices is proposed to understand class interactions and tendencies.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. A case study on low-label rate setting is needed to evaluate the ability of SCNode with tiny number of labels. And it is necessary to investigate whether the effect gain comes from data leakage\n2. More strong datasets and baseline methods are needed to show the effectiveness of SCNode.\n3. More ablation studies on the plug-and-play component are needed.\n4. It it necessary to discuss the connections and difference between other works that analyze the class-wise relations.", "broader_impact_concerns": "No", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "47mEbWUc6i", "reviewer": "Reviewer_y9rn", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\n1. Clear Addressing of GNN Pathologies: By explicitly modeling spatial and contextual cues, SCNode tackles oversmoothing/undersquashing and heterophily in a unified way.\n\n2. Strong Empirical Gains: Outperforms prior methods on homophilic benchmarks and heterophilic ones, with consistent improvements when plugged into standard GNNs.\n\n\n**Weaknesses**\n\n1. Dependence on Label Quality: Contextual embeddings require representative class centroids; poor or imbalanced labels could degrade performance.\n\n2. Limited Ablation on Landmark Choices: Only two landmark types (Euclidean/Jaccard) are explored; impact of number and selection method of landmarks is not fully studied.\n\n3. Scalability to Massive Heterogeneous Graphs: While OGBN-MAG is handled, the approach to collapse heterogeneous relations into paper-only views may lose information and the dimension may be prohibitive in larger knowledge graphs.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this work, the authors identify key limitations of message-passing GNNs—namely oversquashing, oversmoothing, underreaching, and reliance on homophily—and propose SCNode, a hybrid embedding framework that augments each node with (1) spatial coordinates (counts of labeled neighbors up to two hops) and (2) contextual coordinates (distances to learned class landmarks in feature space). They introduce class-aware homophily matrices to analyze class interactions, demonstrate that SCNode yields state-of-the-art (SOTA) results on both homophilic and heterophilic benchmarks, and show that SCNode embeddings drop in as plug-and-play features to improve existing GNNs—with negligible extra training cost compared to multi-day baselines.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Ablation on Landmark Quantity & Metrics: Evaluate SCNode with 1–k landmarks and additional similarity measures (e.g., cosine, learned prototypes), to gauge sensitivity.\n\n2. Heterogeneous Graph Treatment: Provide more principled handling (e.g., relational landmarks) rather than flattening authors/topics into vote vectors for OGBN-MAG.\n\n3. Runtime Breakdown: Detail memory and parallelization overhead for spatial/contextual computations, especially on graphs with large amounts of nodes.\n\n4. Benchmark Against Newer Models: Include comparisons to the latest heterophily-focused GNNs (e.g., H2GCN 2024, GraphAny 2024) to ensure novelty.\n\n5. Theoretical Insights: Expand on why the concatenation of spatial/contextual coordinates provably overcomes underreaching, possibly via spectral or expressivity analysis.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "jbqtDqZevB", "reviewer": "Reviewer_nKnE", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n- SCNode consistently outperforms or matches the best models in both types of graph structures, where many GNNs struggle.\n- The proposed method is evaluated in several datasets and settings.\n\n\n**Weakness**\n- While SCNode introduces a novel and effective framework for generating spatial and contextual node embeddings, the methodology bears a strong conceptual resemblance to traditional graph kernel approaches, particularly those that rely on structural similarity and relative positioning—such as Weisfeiler-Lehman kernels, shortest-path kernels, or graphlet kernels. Like SCNode, these methods transform node neighborhoods and relational structures into high-dimensional representations to facilitate downstream tasks.\nHowever, the paper does not include any discussion or empirical comparison with kernel-based methods, nor are such approaches mentioned in the related work. This omission creates a gap in the theoretical framing and historical positioning of the work.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces SCNode, a framework that enhances node representations in graphs by integrating both spatial coordinates (from local neighborhoods) and contextual coordinates (based on distances to class-specific landmarks). This dual representation allows SCNode to perform robustly in both homophilic and heterophilic graphs, overcoming limitations of traditional GNNs such as oversquashing, oversmoothing, and underreaching. SCNode achieves state-of-the-art results in node classification and link prediction across diverse benchmark datasets.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Could the authors discuss the similarity of the proposed method with feature augmentation techniques and kernel-based methods?", "broader_impact_concerns": "I do not have any concern.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)