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SAGMAN: Stability Analysis of Graph Neural Networks on the Manifolds

Wuxinlin Cheng, Chenhui Deng, Ali Aghdaei, Zhiru Zhang, Zhuo Feng

TL;DR

A distance-preserving graph dimension reduction (GDR) approach that utilizes spectral graph embedding and probabilistic graphical models to create low-dimensional input/output graph-based manifolds for meaningful stability analysis and illustrates its utility in downstream tasks, notably in enhancing GNN stability and facilitating adversarial targeted attacks.

Abstract

Modern graph neural networks (GNNs) can be sensitive to changes in the input graph structure and node features, potentially resulting in unpredictable behavior and degraded performance. In this work, we introduce a spectral framework known as SAGMAN for examining the stability of GNNs. This framework assesses the distance distortions that arise from the nonlinear mappings of GNNs between the input and output manifolds: when two nearby nodes on the input manifold are mapped (through a GNN model) to two distant ones on the output manifold, it implies a large distance distortion and thus a poor GNN stability. We propose a distance-preserving graph dimension reduction (GDR) approach that utilizes spectral graph embedding and probabilistic graphical models (PGMs) to create low-dimensional input/output graph-based manifolds for meaningful stability analysis. Our empirical evaluations show that SAGMAN effectively assesses the stability of each node when subjected to various edge or feature perturbations, offering a scalable approach for evaluating the stability of GNNs, extending to applications within recommendation systems. Furthermore, we illustrate its utility in downstream tasks, notably in enhancing GNN stability and facilitating adversarial targeted attacks.

SAGMAN: Stability Analysis of Graph Neural Networks on the Manifolds

TL;DR

A distance-preserving graph dimension reduction (GDR) approach that utilizes spectral graph embedding and probabilistic graphical models to create low-dimensional input/output graph-based manifolds for meaningful stability analysis and illustrates its utility in downstream tasks, notably in enhancing GNN stability and facilitating adversarial targeted attacks.

Abstract

Modern graph neural networks (GNNs) can be sensitive to changes in the input graph structure and node features, potentially resulting in unpredictable behavior and degraded performance. In this work, we introduce a spectral framework known as SAGMAN for examining the stability of GNNs. This framework assesses the distance distortions that arise from the nonlinear mappings of GNNs between the input and output manifolds: when two nearby nodes on the input manifold are mapped (through a GNN model) to two distant ones on the output manifold, it implies a large distance distortion and thus a poor GNN stability. We propose a distance-preserving graph dimension reduction (GDR) approach that utilizes spectral graph embedding and probabilistic graphical models (PGMs) to create low-dimensional input/output graph-based manifolds for meaningful stability analysis. Our empirical evaluations show that SAGMAN effectively assesses the stability of each node when subjected to various edge or feature perturbations, offering a scalable approach for evaluating the stability of GNNs, extending to applications within recommendation systems. Furthermore, we illustrate its utility in downstream tasks, notably in enhancing GNN stability and facilitating adversarial targeted attacks.
Paper Structure (54 sections, 4 theorems, 38 equations, 11 figures, 10 tables, 1 algorithm)

This paper contains 54 sections, 4 theorems, 38 equations, 11 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Stability Analysis of ML Models on the Manifolds
  4. Previous Investigations on the Stability of GNNs
  5. The SAGMAN Framework for Stability Analysis of GNNs
  6. Challenges in Applying DMD Metrics to GNN Stability Analysis.
  7. Overview of SAGMAN
  8. Phase 1: Embedding Matrix Construction for GDR
  9. Graph Dimensionality Reduction via Laplacian Eigenmaps.
  10. Spectral Embedding with Eigengaps.
  11. Using Eigengaps for Graph Dimension Estimation.
  12. Phase 2: Manifold Construction via PGMs
  13. Probabilistic Graphical Models (PGMs) for Graph-based Manifold Learning.
  14. Scalable PGM via Spectral Sparsification.
  15. Spectral Sparsification via Graph Decomposition.
  16. ...and 39 more sections

Key Result

Theorem 3.2

Maximizing the objective function in Equation opt2 can be achieved in nearly-linear time via the following edge pruning strategy equivalent to spectral sparsification of the initial dense nearest-neighbor graph. Specifically, edges with small distance ratios are pruned, where $d^{\mathrm{eff}}(p, q)$ is the effective resistance distance between nodes $p$ and $q$, $d^{\mathrm{dat}}(p, q) = \| X_p

Figures (11)

  • Figure 1: The proposed SAGMAN framework for stability analysis of GNNs on the manifolds.
  • Figure 2: The proposed spectral sparsification algorithm. (a) The initial graph. (b) LRD decomposition for graph clustering. (c) LSSTs for pruning non-critical edges within clusters. (d) The final graph-based manifold with two inter-cluster edges.
  • Figure 3: The horizontal axes, denoted by $X$, represent the perturbation applied. 'Random EdgesPT' refers to the DICE attack. 'Random FeaturePT' indicates the application of Gaussian noise perturbation, expressed as $X \eta$ perturbation, where $\eta$ represents Gaussian noise. SAGMAN Stable/Unstable denotes the samples classified as stable or unstable by SAGMAN, respectively.
  • Figure 4: Examples of three different graph structures with nodes $A$ and $B$. Despite having the same geodesic distance (shortest path length), the effective resistance distances $\Omega_{AB}$ vary due to the different global structures of the graphs.
  • Figure 5: The horizontal axes, denoted by $X$, represent the magnitude of perturbation applied. 'Random EdgesPT' refers to the DICE adversarial attack scenario, in which pairs of nodes with different labels are connected and pairs with the same label are disconnected, with the number of pairs being equal to $X$. 'Random FeaturePT' indicates the application of Gaussian noise perturbation, expressed as $FM + X \eta$, where $FM$ denotes the feature matrix and $\eta$ represents Gaussian noise. The upper and lower subfigures illustrate the cosine similarity and the Kullback–Leibler Divergence (KLD). 'Stable/Unstable' denotes the samples that are classified as stable or unstable without GDR, respectively.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Lemma A.1
  • Lemma A.2