[Paper Review] Modelling Graph Errors: Towards Robust Graph Signal Processing

J.K. Miettinen, Sergiy A. Vorobyov+1 more|arXiv (Cornell University)|Mar 20, 2019

Advanced Graph Neural Networks47 references4 citations

TL;DR

This paper introduces practical graph error models to study the impact of adjacency matrix estimation errors on graph signal processing (GSP) tasks. It proposes a framework based on Erdös-Rényi random graphs and graphon theory to model perturbations in the adjacency matrix, analytically and numerically evaluating their effects on GSP methods like graph filtering and independent component analysis, with key results showing that error effects depend on graph topology and signal structure.

ABSTRACT

The first step for any graph signal processing (GSP) procedure is to learn the graph signal representation, i.e., to capture the dependence structure of the data into an adjacency matrix. Indeed, the adjacency matrix is typically not known a priori and has to be learned. However, it is learned with errors. A little attention has been paid to modelling such errors in the adjacency matrix, and studying their effects on GSP methods. However, modelling errors in the adjacency matrix will enable both to study the graph error effects in GSP and to develop robust GSP algorithms. In this paper, we therefore introduce practically justifiable graph error models. We also study, both analytically when possible and numerically, the graph error effect on the performance of GSP methods in different types of problems such as filtering of graph signals and independent component analysis of graph signals (graph decorrelation).

Motivation & Objective

• To address the lack of systematic modeling of errors in learned adjacency matrices in graph signal processing (GSP).
• To study how errors in the graph structure—specifically in the adjacency matrix—affect the performance of GSP methods.
• To develop theoretically grounded and practically justifiable models for graph errors to enable robust GSP algorithm design.
• To analyze the impact of adjacency matrix uncertainty on key GSP tasks such as graph filtering and independent component analysis (ICA) of graph signals.
• To provide analytical and numerical tools for assessing the sensitivity of GSP methods to topological perturbations in the graph.

Proposed method

• Uses Erdös-Rényi random graph models as a baseline for modeling adjacency matrix uncertainty.
• Introduces a perturbation model where the true adjacency matrix is subject to random edge additions or deletions with known probabilities.
• Applies graphon theory to model large-scale graph structures and derive asymptotic behavior of eigenvalues and eigenfunctions.
• Employs the minimum distance index to quantify the discrepancy between true and estimated graphs.
• Derives analytical expressions for expected signal energy and autocorrelation under graph error models using probabilistic and matrix algebra techniques.
• Uses the law of large numbers and symmetry properties of random graph models to approximate the expected behavior of graph signal processing operations.

Experimental results

Research questions

• RQ1 How do errors in the learned adjacency matrix affect the performance of graph filtering and graph signal decorrelation?
• RQ2 What are the theoretical and practical implications of modeling graph errors using Erdös-Rényi and graphon-based models?
• RQ3 How does the structure of the graph (e.g., degree distribution, eigenvalue spectrum) influence the sensitivity of GSP methods to adjacency matrix errors?
• RQ4 To what extent can the effects of graph errors be analytically quantified using random matrix theory and graphon models?
• RQ5 How do different error models (e.g., edge deletion, edge addition) impact the stability of GSP algorithms like graph ICA and filtering?

Key findings

• The expected squared norm of the filtered signal is approximated as $ \frac{1}{N} \mathbb{E}\{\|\mathbf{z}\|^2\} \approx \sigma_y^2 \left( (\alpha - \alpha^2)\theta^2 a^2 N - 2\alpha\theta a + 1 \right) $, showing dependence on graph density $\alpha$, filter gain $\theta$, and signal variance.
• Graph error effects are analytically tractable under the Erdös-Rényi model, with the expected value of the adjacency matrix perturbation converging to $ c\epsilon $ via the law of large numbers.
• The eigenvalues and eigenfunctions of the adjacency matrix can be approximated using the graphon kernel, enabling asymptotic analysis of large graphs.
• The minimum distance index provides a metric to quantify the difference between true and estimated graphs, supporting error modeling and robustness analysis.
• The analysis reveals that graph error effects are not equivalent to signal measurement errors and require distinct modeling approaches in GSP.
• Numerical results confirm that graph error effects are non-trivial and depend on the interplay between signal structure and graph topology, especially in filtering and ICA tasks.