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Higher Order Structures For Graph Explanations

Akshit Sinha, Sreeram Vennam, Charu Sharma, Ponnurangam Kumaraguru

TL;DR

FORGE addresses the limitation of graph explainers in capturing higher-order, multi-node interactions by lifting graphs to cell complexes and applying higher-order message passing during explanation. It introduces a lifting algorithm to obtain a restricted cell complex X(C,Σ), explains on this higher-order structure, and then propagates the resulting importance back to the original graph via multiple propagation strategies. Empirical evaluation on GraphXAI real-world datasets and synthetic motifs shows FORGE consistently improves explanation accuracy and faithfulness across diverse explainers, with notable gains and scalable performance. The work establishes higher-order structures as a practical path to more faithful, interpretable GNN explanations and suggests future work on adaptive propagation and broader task applicability.

Abstract

Graph Neural Networks (GNNs) have emerged as powerful tools for learning representations of graph-structured data, demonstrating remarkable performance across various tasks. Recognising their importance, there has been extensive research focused on explaining GNN predictions, aiming to enhance their interpretability and trustworthiness. However, GNNs and their explainers face a notable challenge: graphs are primarily designed to model pair-wise relationships between nodes, which can make it tough to capture higher-order, multi-node interactions. This characteristic can pose difficulties for existing explainers in fully representing multi-node relationships. To address this gap, we present Framework For Higher-Order Representations In Graph Explanations (FORGE), a framework that enables graph explainers to capture such interactions by incorporating higher-order structures, resulting in more accurate and faithful explanations. Extensive evaluation shows that on average real-world datasets from the GraphXAI benchmark and synthetic datasets across various graph explainers, FORGE improves average explanation accuracy by 1.9x and 2.25x, respectively. We perform ablation studies to confirm the importance of higher-order relations in improving explanations, while our scalability analysis demonstrates FORGE's efficacy on large graphs.

Higher Order Structures For Graph Explanations

TL;DR

FORGE addresses the limitation of graph explainers in capturing higher-order, multi-node interactions by lifting graphs to cell complexes and applying higher-order message passing during explanation. It introduces a lifting algorithm to obtain a restricted cell complex X(C,Σ), explains on this higher-order structure, and then propagates the resulting importance back to the original graph via multiple propagation strategies. Empirical evaluation on GraphXAI real-world datasets and synthetic motifs shows FORGE consistently improves explanation accuracy and faithfulness across diverse explainers, with notable gains and scalable performance. The work establishes higher-order structures as a practical path to more faithful, interpretable GNN explanations and suggests future work on adaptive propagation and broader task applicability.

Abstract

Graph Neural Networks (GNNs) have emerged as powerful tools for learning representations of graph-structured data, demonstrating remarkable performance across various tasks. Recognising their importance, there has been extensive research focused on explaining GNN predictions, aiming to enhance their interpretability and trustworthiness. However, GNNs and their explainers face a notable challenge: graphs are primarily designed to model pair-wise relationships between nodes, which can make it tough to capture higher-order, multi-node interactions. This characteristic can pose difficulties for existing explainers in fully representing multi-node relationships. To address this gap, we present Framework For Higher-Order Representations In Graph Explanations (FORGE), a framework that enables graph explainers to capture such interactions by incorporating higher-order structures, resulting in more accurate and faithful explanations. Extensive evaluation shows that on average real-world datasets from the GraphXAI benchmark and synthetic datasets across various graph explainers, FORGE improves average explanation accuracy by 1.9x and 2.25x, respectively. We perform ablation studies to confirm the importance of higher-order relations in improving explanations, while our scalability analysis demonstrates FORGE's efficacy on large graphs.
Paper Structure (18 sections, 1 theorem, 7 equations, 7 figures, 3 tables, 2 algorithms)

This paper contains 18 sections, 1 theorem, 7 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Higher-Order Representations of Graphs
  4. Explainability in
  5. Cell Complexes
  6. Proposed Approach
  7. Lifting Graphs to Cell Complexes
  8. Generating Explanations for Cell Complexes
  9. Information Propagation
  10. Experimental Settings
  11. Datasets
  12. Evaluation Criteria
  13. Baselines
  14. Results
  15. Ablation Studies
  16. ...and 3 more sections

Key Result

Theorem 3.1

For a graph $G(V, E)$ with adjacency matrix $A$ having cycles of length at most $K$, let $W_k$ represent the number of closed walks of length $k$ which are not $k$-cycles. Let $deg(v)$ represent the degree of a node $v$ in the graph. The corresponding cell complex $X$ will have cells $C$ and boundar

Figures (7)

  • Figure 1: (a) Ground Truth for an example from BENZENE graphxai (b) Explanation generated by GNNExplainer gnnexplainer (c) Explanation generated by using FORGE. Green nodes and edges signify the subgraph considered important for GNN prediction. By incorporating FORGE, we can capture important multi-node interactions, resulting in more accurate explanations.
  • Figure 2: Visual representation of FORGE. The input graph is lifted to a cell complex, which is then given as input to (i) a GNN to train on, as well as to (ii) a graph explainer. Propagation is then done on the output cell complex explanation to map it to an explanation for the original graph. The green color on cells, nodes, and edges signify the substructure considered important for GNN prediction (the explanation).
  • Figure 3: The variation of $|\Sigma|$ for a conventional cell complex and our proposed reduced cell complex with increasing $|E|$ (Theorem \ref{['theorem']}) (left) and increasing $|V|$ (right), showing that our proposed variation is more space efficient.
  • Figure 4: Example of a computation cell complex. The figure on the left shows $2$-hop horizontal message passing, while the figure on the right represents $2$-hop vertical message passing, introduced by FORGE.
  • Figure 5: Different motifs used to generate synthetic graphs: (left to right) Bull, Wheel, Cube, and House.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Definition 1: $p$-cell
  • Definition 2: boundary/coboundary
  • Definition 3: $p$-chain
  • Definition 4: $p$-skeleton
  • Definition 5: boundary relation
  • Theorem 3.1