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When Witnesses Defend: A Witness Graph Topological Layer for Adversarial Graph Learning

Naheed Anjum Arafat, Debabrota Basu, Yulia Gel, Yuzhou Chen

TL;DR

This paper addresses the vulnerability of Graph Neural Networks to adversarial perturbations by introducing WGTL, a topology-aware defense that uses persistent homology via witness complexes to capture robust, multi-scale graph shape information. WGTL integrates local and global topology encodings derived from landmark-based witness complexes and couples them with a robust topological loss to produce stable node representations, with formal stability guarantees under an attacker budget $\delta$. The approach demonstrates broad compatibility by boosting robustness across multiple GNN backbones and existing defenses, achieving substantial gains on six datasets and scaling to large graphs, while offering practical computation times. By bridging adversarial graph learning with PH-based representations, WGTL provides a principled, scalable framework for leveraging higher-order graph structure to counter perturbations, with promising avenues for time-evolving graphs and hypergraphs.

Abstract

Capitalizing on the intuitive premise that shape characteristics are more robust to perturbations, we bridge adversarial graph learning with the emerging tools from computational topology, namely, persistent homology representations of graphs. We introduce the concept of witness complex to adversarial analysis on graphs, which allows us to focus only on the salient shape characteristics of graphs, yielded by the subset of the most essential nodes (i.e., landmarks), with minimal loss of topological information on the whole graph. The remaining nodes are then used as witnesses, governing which higher-order graph substructures are incorporated into the learning process. Armed with the witness mechanism, we design Witness Graph Topological Layer (WGTL), which systematically integrates both local and global topological graph feature representations, the impact of which is, in turn, automatically controlled by the robust regularized topological loss. Given the attacker's budget, we derive the important stability guarantees of both local and global topology encodings and the associated robust topological loss. We illustrate the versatility and efficiency of WGTL by its integration with five GNNs and three existing non-topological defense mechanisms. Our extensive experiments across six datasets demonstrate that WGTL boosts the robustness of GNNs across a range of perturbations and against a range of adversarial attacks. Our datasets and source codes are available at https://github.com/toggled/WGTL.

When Witnesses Defend: A Witness Graph Topological Layer for Adversarial Graph Learning

TL;DR

This paper addresses the vulnerability of Graph Neural Networks to adversarial perturbations by introducing WGTL, a topology-aware defense that uses persistent homology via witness complexes to capture robust, multi-scale graph shape information. WGTL integrates local and global topology encodings derived from landmark-based witness complexes and couples them with a robust topological loss to produce stable node representations, with formal stability guarantees under an attacker budget . The approach demonstrates broad compatibility by boosting robustness across multiple GNN backbones and existing defenses, achieving substantial gains on six datasets and scaling to large graphs, while offering practical computation times. By bridging adversarial graph learning with PH-based representations, WGTL provides a principled, scalable framework for leveraging higher-order graph structure to counter perturbations, with promising avenues for time-evolving graphs and hypergraphs.

Abstract

Capitalizing on the intuitive premise that shape characteristics are more robust to perturbations, we bridge adversarial graph learning with the emerging tools from computational topology, namely, persistent homology representations of graphs. We introduce the concept of witness complex to adversarial analysis on graphs, which allows us to focus only on the salient shape characteristics of graphs, yielded by the subset of the most essential nodes (i.e., landmarks), with minimal loss of topological information on the whole graph. The remaining nodes are then used as witnesses, governing which higher-order graph substructures are incorporated into the learning process. Armed with the witness mechanism, we design Witness Graph Topological Layer (WGTL), which systematically integrates both local and global topological graph feature representations, the impact of which is, in turn, automatically controlled by the robust regularized topological loss. Given the attacker's budget, we derive the important stability guarantees of both local and global topology encodings and the associated robust topological loss. We illustrate the versatility and efficiency of WGTL by its integration with five GNNs and three existing non-topological defense mechanisms. Our extensive experiments across six datasets demonstrate that WGTL boosts the robustness of GNNs across a range of perturbations and against a range of adversarial attacks. Our datasets and source codes are available at https://github.com/toggled/WGTL.
Paper Structure (27 sections, 6 theorems, 41 equations, 3 figures, 21 tables, 1 algorithm)

This paper contains 27 sections, 6 theorems, 41 equations, 3 figures, 21 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works: Defenses for GNNs
  3. Background: Graphs, Persistent Homology, Complexes, Adversarial ML
  4. Learning a Robust Topology-aware Graph Representation
  5. Witness Graph Topological Layer
  6. Topological Loss as a Regularizer
  7. Experimental Evaluation
  8. Conclusion and Future Works
  9. Theoretical Analysis
  10. Notations
  11. Properties of the Witness Complexes and Landmarks
  12. Stability of the Topological Representations
  13. Stability of the Topological Loss
  14. Details of Adversarial Attacks: Configurations and Budgets
  15. Experimental Details
  16. ...and 12 more sections

Key Result

Theorem 3.1

Let us denote the persistence diagram obtained from local topology encoding of $\mathcal{G}$ as $\mathrm{T}(\mathcal{G})$ (Figure fig:schematic). For any $p < \infty$ and $C_{\epsilon}$ being the maximum cardinality of the $\epsilon$-neighborhood created by the landmarks, we obtain that for any grap

Figures (3)

  • Figure 1: Architecture of Witness Graph Topological Layer.
  • Figure 2: Illustration of Witness Complex-based topological regularizer $L_{Topo}$.
  • Figure 3: The trade-off between accuracy and Feature computation time of GCN+WGTL with different numbers of landmarks. The figures are under mettack with $5\%$ perturbation rate.

Theorems & Definitions (15)

  • Definition 2.1: Weak Witness Complex witness
  • Theorem 3.1: Stability of the encoded local topology
  • Proposition 3.2: Stability of the encoded global topology
  • Proposition 3.3: Stability of the aggregated topological encoding
  • Proposition 3.4: Stability of $L_{topo}$
  • Definition A.1: Hausdroff distance hausdorff1914
  • Definition A.2: $p$-Wasserstein distance pwasserstein
  • Theorem A.3: Properties of Local Witness Complexes arafat2020epsilon
  • Remark A.4
  • proof
  • ...and 5 more