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SCNode: Spatial and Contextual Coordinates for Graph Representation Learning

Md Joshem Uddin, Astrit Tola, Varin Sikand, Cuneyt Gurcan Akcora, Baris Coskunuzer

TL;DR

SCNode tackles the limitations of traditional message-passing GNNs by introducing spatial and contextual coordinates that encode local neighborhood structure and global class-relational signals. It defines class landmarks in feature space and computes distances to these landmarks, pairing them with k-hop label distributions to form a concatenated embedding gamma(u) that expands the model's receptive capabilities. The approach is backbone-agnostic and demonstrated to achieve state-of-the-art results on node classification and competitive or leading performance on link prediction across both homophilic and heterophilic graphs, with efficient runtime characteristics. Overall, SCNode provides a practical, plug-and-play augmentation that mitigates underreaching and enhances expressivity by blending local structural cues with global semantic context, with promising directions for future temporal extensions.

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 \textit{SCNode}, a \textit{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.

SCNode: Spatial and Contextual Coordinates for Graph Representation Learning

TL;DR

SCNode tackles the limitations of traditional message-passing GNNs by introducing spatial and contextual coordinates that encode local neighborhood structure and global class-relational signals. It defines class landmarks in feature space and computes distances to these landmarks, pairing them with k-hop label distributions to form a concatenated embedding gamma(u) that expands the model's receptive capabilities. The approach is backbone-agnostic and demonstrated to achieve state-of-the-art results on node classification and competitive or leading performance on link prediction across both homophilic and heterophilic graphs, with efficient runtime characteristics. Overall, SCNode provides a practical, plug-and-play augmentation that mitigates underreaching and enhances expressivity by blending local structural cues with global semantic context, with promising directions for future temporal extensions.

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 \textit{SCNode}, a \textit{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.
Paper Structure (33 sections, 3 theorems, 9 equations, 5 figures, 13 tables)

This paper contains 33 sections, 3 theorems, 9 equations, 5 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. SCNode Framework
  4. Spatial Node Embeddings
  5. Contextual Node Embeddings
  6. Class Landmarks.
  7. SCNode Embedding
  8. Relation to label-structure methods.
  9. Homophily and SCNode
  10. Experiments
  11. Datasets.
  12. Experimental Setup.
  13. Runtime.
  14. Node Classification Results
  15. Baselines.
  16. ...and 18 more sections

Key Result

Theorem 3.7

For a given $\mathcal{G}=(\mathcal{V},\mathbb{E})$, let $\vec{\alpha}_1(v)$ be SCN spatial-1 vector. Let $\widehat{\alpha}_1(v)$ be the vector where the entry corresponding to the class of $v$ is set to $0$. Then, $1-\mathcal{H}_{\hbox{node}}(\mathcal{G})=\frac{1}{|\mathcal{V}|}\sum_{v\in\mathcal{V}

Figures (5)

  • Figure 1: SCNode framework. Spatial and contextual embeddings are incorporated in class-aware contrast-based learning.
  • Figure 2: Computing spatial (a) and contextual (b) embeddings for node $u$ on the graph.
  • 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.
  • Figure 4: 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 5: Training set vs Accuracy

Theorems & Definitions (13)

  • Definition 3.1: Class Landmark
  • Definition 3.2: Distances to Class Landmarks
  • Definition 3.3: Contextual Embedding
  • Definition 3.4: SCNode Embedding
  • Definition 3.5: SCNode Homophily Matrices
  • Definition 3.6: $\alpha$ Homophily Ratio
  • Theorem 3.7
  • Theorem A.1
  • proof
  • Definition A.2: Higher Homophily
  • ...and 3 more