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Are GNNs doomed by the topology of their input graph?

Amine Mohamed Aboussalah, Abdessalam Ed-dib

TL;DR

This work examines how the topology of the input graph constrains GNN learning, introducing $k$-hop similarity to connect local neighborhood structure with global learning dynamics. It proposes a weight-consistency conjecture: for a $k$-layer GNN, training on $k$-hop similar graphs yields approximately equivalent learned functions, and it develops efficient methods to generate such graphs. Through SBMs-based experiments, it provides strong evidence of function-level consistency across $k$-hop similar graphs and leverages this to offer a topological explanation for oversmoothing: deep message passing effectively turns large components into complete graphs, causing indistinguishable node representations within components. The findings illuminate the role of local topology in GNN expressiveness and suggest scalable perturbation strategies and broader validations on real data.

Abstract

Graph Neural Networks (GNNs) have demonstrated remarkable success in learning from graph-structured data. However, the influence of the input graph's topology on GNN behavior remains poorly understood. In this work, we explore whether GNNs are inherently limited by the structure of their input graphs, focusing on how local topological features interact with the message-passing scheme to produce global phenomena such as oversmoothing or expressive representations. We introduce the concept of $k$-hop similarity and investigate whether locally similar neighborhoods lead to consistent node representations. This interaction can result in either effective learning or inevitable oversmoothing, depending on the inherent properties of the graph. Our empirical experiments validate these insights, highlighting the practical implications of graph topology on GNN performance.

Are GNNs doomed by the topology of their input graph?

TL;DR

This work examines how the topology of the input graph constrains GNN learning, introducing -hop similarity to connect local neighborhood structure with global learning dynamics. It proposes a weight-consistency conjecture: for a -layer GNN, training on -hop similar graphs yields approximately equivalent learned functions, and it develops efficient methods to generate such graphs. Through SBMs-based experiments, it provides strong evidence of function-level consistency across -hop similar graphs and leverages this to offer a topological explanation for oversmoothing: deep message passing effectively turns large components into complete graphs, causing indistinguishable node representations within components. The findings illuminate the role of local topology in GNN expressiveness and suggest scalable perturbation strategies and broader validations on real data.

Abstract

Graph Neural Networks (GNNs) have demonstrated remarkable success in learning from graph-structured data. However, the influence of the input graph's topology on GNN behavior remains poorly understood. In this work, we explore whether GNNs are inherently limited by the structure of their input graphs, focusing on how local topological features interact with the message-passing scheme to produce global phenomena such as oversmoothing or expressive representations. We introduce the concept of -hop similarity and investigate whether locally similar neighborhoods lead to consistent node representations. This interaction can result in either effective learning or inevitable oversmoothing, depending on the inherent properties of the graph. Our empirical experiments validate these insights, highlighting the practical implications of graph topology on GNN performance.

Paper Structure

This paper contains 15 sections, 7 theorems, 6 equations, 4 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exploring $k$-Hop Similarity in GNNs
  4. Definition and Properties
  5. Constructing $k$-Hop Similar Graphs
  6. Conjecture Validation
  7. Synthetic Dataset Generation
  8. Experimental Setting
  9. Numerical Results
  10. Number of nodes.
  11. Intracluster Probability.
  12. Intercluster Probability.
  13. Number of classes.
  14. Oversmoothing through Graph Topology
  15. Conclusion and Future Work

Key Result

Theorem 2.2

Given two graphs $G_1$ and $G_2$ that are $k$-hop isomorphic, a GNN with $k$ layers cannot distinguish between them. Formally, for any GNN $f_{\theta}$ with $k$ layers, we have: where $f_{\theta}(G)$ represents the learned representation of graph $G$.

Figures (4)

  • Figure 1: Illustration of two $2$-hop similar graphs with distinct edge arrangements. Although the $2$-hop neighborhoods are identical, the graphs are not $2$-hop isomorphic.
  • Figure 2: Example of two graphs that are $2$-hop similar but not $2$-hop isomorphic. The left graph shows a complete connection between nodes $a$, $b$, and $c$, while the right graph has the same nodes connected in a path configuration.
  • Figure 3: Average predicted probability distribution across ten classes for a GNN model trained on the original graph.
  • Figure 4: Average predicted probability distribution across ten classes for a GNN model trained on the $2$-hop similar graph.

Theorems & Definitions (12)

  • Conjecture 1.1: Weight Consistency under $k$-Hop Similarity in GNNs
  • Definition 2.1: $k$-Hop Isomorphism
  • Theorem 2.2: Weight Consistency under $k$-Hop Isomorphism in GNNs
  • Definition 2.3: $k$-Hop Similarity
  • Proposition 3.1: Isomorphism Implies Similarity
  • Definition 3.2: $k$-th Power of a Graph
  • Proposition 3.3: Binary $k$-Hop Reachability
  • Theorem 3.4: NP-Hardness
  • Definition 3.5: Level of Disagreement
  • Proposition 4.1: $k$-th Power of a Connected Graph
  • ...and 2 more