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Characterizing the Influence of Topology on Graph Learning Tasks

Kailong Wu, Yule Xie, Jiaxin Ding, Yuxiang Ren, Luoyi Fu, Xinbing Wang, Chenghu Zhou

TL;DR

This work addresses how graph topology influences GNN performance by introducing TopoInf, an edge-level metric that quantifies the compatibility between topology and downstream tasks through a graph-filter perspective. By modeling topology via $\boldsymbol{f}(\mathbf{A})$ and the ideal signal $\mathbf{L}$, the authors define a global compatibility score $\mathcal{C}(\mathbf{A})=\mathcal{I}(\mathbf{A})-\lambda\mathcal{R}(\mathbf{A})$ and an edge-wise influence measure $\nabla\mathcal{C}_{\mathbf{A}}(e_{ij})$, enabling local topology analysis and optimization. Theoretical grounding is provided via contextual SBMs, showing how the bias term $\|\boldsymbol{f}(\mathbf{A})\mathbf{L}-\mathbf{L}\|$ and the regularization term $\|\boldsymbol{f}(\mathbf{A})\|$ jointly shape learning, complemented by motivating experiments and extensive real-data validation. Empirically, TopoInf-guided topology modification and DropEdge strategies improve performance across nine GNNs and multiple datasets, demonstrating practical utility for topology-aware graph learning and interpretability.

Abstract

Graph neural networks (GNN) have achieved remarkable success in a wide range of tasks by encoding features combined with topology to create effective representations. However, the fundamental problem of understanding and analyzing how graph topology influences the performance of learning models on downstream tasks has not yet been well understood. In this paper, we propose a metric, TopoInf, which characterizes the influence of graph topology by measuring the level of compatibility between the topological information of graph data and downstream task objectives. We provide analysis based on the decoupled GNNs on the contextual stochastic block model to demonstrate the effectiveness of the metric. Through extensive experiments, we demonstrate that TopoInf is an effective metric for measuring topological influence on corresponding tasks and can be further leveraged to enhance graph learning.

Characterizing the Influence of Topology on Graph Learning Tasks

TL;DR

This work addresses how graph topology influences GNN performance by introducing TopoInf, an edge-level metric that quantifies the compatibility between topology and downstream tasks through a graph-filter perspective. By modeling topology via and the ideal signal , the authors define a global compatibility score and an edge-wise influence measure , enabling local topology analysis and optimization. Theoretical grounding is provided via contextual SBMs, showing how the bias term and the regularization term jointly shape learning, complemented by motivating experiments and extensive real-data validation. Empirically, TopoInf-guided topology modification and DropEdge strategies improve performance across nine GNNs and multiple datasets, demonstrating practical utility for topology-aware graph learning and interpretability.

Abstract

Graph neural networks (GNN) have achieved remarkable success in a wide range of tasks by encoding features combined with topology to create effective representations. However, the fundamental problem of understanding and analyzing how graph topology influences the performance of learning models on downstream tasks has not yet been well understood. In this paper, we propose a metric, TopoInf, which characterizes the influence of graph topology by measuring the level of compatibility between the topological information of graph data and downstream task objectives. We provide analysis based on the decoupled GNNs on the contextual stochastic block model to demonstrate the effectiveness of the metric. Through extensive experiments, we demonstrate that TopoInf is an effective metric for measuring topological influence on corresponding tasks and can be further leveraged to enhance graph learning.
Paper Structure (13 sections, 2 theorems, 8 equations, 3 figures, 1 table)

This paper contains 13 sections, 2 theorems, 8 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Methodology
  4. Matching graph topology and tasks
  5. TopoInf: measuring the influence of graph topology on tasks
  6. Motivation
  7. Theoretical Analysis
  8. Motivating Example
  9. Experiments
  10. Validating TopoInf on Graphs
  11. Validating Estimated TopoInf
  12. Related Works
  13. Conclusion

Key Result

theorem thmcountertheorem

For $0 < \delta < 1$, with probability at least $1 - \delta$, we have where $c_1 = O\left( \left\| \boldsymbol{\mu} \right\| \right)$, $c_2 = O\left( \mathbb{E} \left\{ \left\| \textbf{N} \right\| \right\} / \delta \right)$.

Figures (3)

  • Figure 1: The performance of GCN and MLP on the synthetic graph datasets generated by cSBM with different $\sigma, p, q$, corresponding to the noise variance of features, intra-community and inter-community connection probability. The synthetic datasets share the same statistics with Cora CoraAndCiteSeer, such as the number of nodes and edges and the dimension of features. $p$ and $q$ used to generate SBMs are $(0.9, 0.1)$, $(0.8, 0.2)$, and $(0.7, 0.3)$ respectively from left to right.
  • Figure 2: Performance change while deleting edges by TopoInf. Horizontal axis's meaning: zero point corresponds to the original graph without deleting any edges. The absolute value of the coordinate denotes the ratio of deleted edges to all edges. The negative/positive coordinate corresponds to the case that the graph is obtained by deleting edges with negative/positive TopoInf values of the corresponding proportion, in the descendant order of the absolute value of TopoInf. The vertical axis is the accuracy of the node classification in the GNN models. The TopoInf values are calculated with $\mathcal{V}_t$ in Equation \ref{['eq:I_A_node_wise']} set as the test set. For each step, we remove 10% of edges in the negative/positive set.
  • Figure 3: GCN performance with different edge deletion strategies on Cora. The left figure shows the results of removing positive edges to increase performance (higher is better), and the right figure shows the results of removing negative edges to decrease performance (lower is better). The horizontal axis denotes the ratio of deleted edges to all edges. The vertical axis is the accuracy of the node classification in GCN. The TopoInf values are calculated with $\mathcal{V}_t$ in the Equation \ref{['eq:I_A_node_wise']} set as a test set.

Theorems & Definitions (5)

  • definition thmcounterdefinition
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof