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arxiv: 2605.23673 · v1 · pith:FXAOP5PUnew · submitted 2026-05-22 · 💻 cs.LG

Relevant Walk Search for Explaining Graph Neural Networks

Ping Xiong , Thomas Schnake , Michael Gastegger , Gr\'egoire Montavon , Klaus-Robert M\"uller , Shinichi Nakajima This is my paper

Pith reviewed 2026-05-25 05:09 UTC · model grok-4.3

classification 💻 cs.LG
keywords graph neural networksexplainabilityGNN-LRPrelevant walksmax-product algorithmpolynomial timelayer-wise relevance propagation
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The pith

Polynomial-time max-product algorithms identify top-K relevant walks for GNN explanations exactly at the neuron level.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops algorithms to find the most relevant walks in graph neural networks for explanation purposes using GNN-LRP. It replaces the exponential time requirement of enumerating all possible walks with polynomial time methods based on the max-product algorithm. The approach works exactly when tracking relevance at the neuron level and approximately at the node level. Readers would care because this makes higher-order explanations practical for large graphs in domains like molecular science and disease spread modeling.

Core claim

The central claim is that algorithms derived from the max-product method can locate the top-K relevant walks in polynomial time, exactly at the neuron level and approximately at the node level, which allows GNN-LRP to scale to large problems while maintaining its higher-order explanatory power.

What carries the argument

The max-product algorithm, which finds maximum likelihood configurations in graphical models, adapted here to propagate and select the highest relevance scores along walks in the GNN.

If this is right

  • Scales GNN-LRP explanations to graphs too large for exhaustive walk search.
  • Preserves exact top-K walks at neuron level without losing higher-order information.
  • Provides practical approximations at node level for broader applicability.
  • Validated on benchmarks from epidemiology, molecular chemistry, and natural language processing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may generalize to other path-based explanation techniques in neural networks.
  • Could support interactive explanation tools for GNNs in production settings.
  • Opens the way for combining with sampling methods for even larger networks.

Load-bearing premise

The max-product algorithm can be directly adapted to GNN-LRP relevance scores to find the exact top-K walks at the neuron level.

What would settle it

Compare the top-K walks found by the new algorithm against those from full enumeration on a small-depth GNN where exponential computation is still feasible.

Figures

Figures reproduced from arXiv: 2605.23673 by Gr\'egoire Montavon, Klaus-Robert M\"uller, Michael Gastegger, Ping Xiong, Shinichi Nakajima, Thomas Schnake.

Figure 1
Figure 1. Figure 1: We aim to find the most important information flows for a GNN prediction in terms of walk relevance. Naively one can apply a brute force search, where the relevances of all possible walks are evaluated and the most relevant ones are chosen. Our proposed methods are based on the max-product algorithm to find the most relevant walks by local message passing, which reduces the computational complexity drastic… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of neuron-level and node-level walks. The node-level path of the top neuron-level walk may differ from the top node-level walk, because many weakly relevant neuron-level walks may sum up to a strongly relevant node-level walk. network (FFNN), Xiong et al. (2022) defined the neuron￾level walk (see [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Precision-recall curves of AMP-ave in top-K∗ node-level walk search on BA-2motif (top) and Mutagenicity (bottom) [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The histograms of the cosine similarity (16) between the column vectors and their average of the propagation matrices for γ = [3, . . . , 0]. Each panel corresponds to each dataset. poor performance in general (Schnake et al., 2022). In the subsequent experiments, we focus on the recommended set￾ting γ = [3, · · · , 0]. Column-similarity Assumption: Here, we investigate to what extent the assumption requir… view at source ↗
Figure 5
Figure 5. Figure 5: Visual explanation by AMP-ave (ours), Edge-IG, and Node-IG on Infection dataset. The deeper the red color is, the higher the relevance is. The star node at the top is the target node for which the prediction is explained, while the square nodes at the bottom are initial carriers. For clarity, we only plot the nodes within 4-hops from the target node, and the nodes involved in the 3 possible infection chain… view at source ↗
Figure 7
Figure 7. Figure 7: Recall of infection chain detection on Infection dataset [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Recall of motif’s edge detection on BA-2motif dataset. Positive samples and negative samples are plotted separately. AMP-ave to popular edge-level explanability baseline meth￾ods with comparable computational complexity,4 including Edge-IG, GNNExplainer, edge-level GNN-LRP (relevance propagated to edges in the input layer) and simple Gradient￾based heatmap for edges. Here the edge scoring by AMP￾ave is sim… view at source ↗
Figure 9
Figure 9. Figure 9: Computation time dependence on the network depth L (left) and on the graph size M (right). Note the different vertical scales in the top and bottom parts, and that y-axis in (a) is in log-scale. For the depth larger than L > 4, the computation time of exhaustive search is estimated from partial computation, since the whole computation is infeasible. (a) Top-1 walk search with GIN-L for L = 2, . . . , 7 on … view at source ↗
Figure 10
Figure 10. Figure 10: The proportion K Ke of positive relevant neuron-level walks in the top-Ke absolute relevant walks. Dataset γ + − 0 BA-2motif 0 3.70% 4.12% 92.10% 0.2 6.38% 2.78% 90.84% [3, . . . , 0] 5.34% 2.66% 92.01% +∞ 6.70% 2.67% 90.63% MUTAG 0 11.82% 2.48% 85.70% 0.2 12.39% 2.23% 85.38% [3, . . . , 0] 12.86% 1.95% 85.19% +∞ 14.28% 1.12% 84.60% Graph-SST2 0 39.17% 3.30% 57.53% 0.2 41.45% 2.41% 56.14% [3, . . . , 0] 4… view at source ↗
Figure 11
Figure 11. Figure 11: Precision-recall curves of AMP-ave for the top-K∗ node-level walks on MUTAG (top) and Graph-SST2 (bottom). 20 [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Heatmaps of different explanation methods of 4 molecules from the Mutagenicity dataset (+: mutagenic, -: non-mutagenic). 22 [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Explanations of parse trees from Graph-SST2 (+: positive sentiment, -: negative sentiment). 24 [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
read the original abstract

Graph Neural Networks (GNNs) have become important machine learning tools for graph analysis, and its explainability is crucial for safety, fairness, and robustness. Layer-wise relevance propagation for GNNs (GNN-LRP) evaluates the relevance of \emph{walks} to reveal important information flows in the network, and provides higher-order explanations, which have been shown to be superior to the lower-order, i.e., node-/edge-level, explanations. However, identifying relevant walks by GNN-LRP requires {\em exponential} computational complexity with respect to the network depth, which we will remedy in this paper. Specifically, we propose {\em polynomial-time} algorithms for finding top-$K$ relevant walks, which drastically reduces the computation and thus increases the applicability of GNN-LRP to large-scale problems. Our proposed algorithms are based on the \emph{max-product} algorithm -- a common tool for finding the maximum likelihood configurations in probabilistic graphical models -- and can find the most relevant walks exactly at the neuron level and approximately at the node level. Our experiments demonstrate the performance of our algorithms at scale and their utility across application domains, i.e., on epidemiology, molecular, and natural language benchmarks. We provide our codes under \href{https://github.com/xiong-ping/rel_walk_gnnlrp}{github.com/xiong-ping/rel\_walk\_gnnlrp}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes polynomial-time algorithms, based on the max-product algorithm, for identifying the top-K most relevant walks under GNN-LRP. These algorithms are claimed to recover the walks exactly at the neuron level and approximately at the node level, thereby replacing the exponential enumeration required by standard GNN-LRP with scalable dynamic programming. Experiments on epidemiology, molecular, and NLP benchmarks are presented to demonstrate runtime and utility, and code is released.

Significance. If the exactness claim at the neuron level holds, the work would materially increase the applicability of higher-order walk-based explanations to deeper and larger GNNs, a recognized bottleneck in current GNN explainability. The public code release is a concrete strength that supports reproducibility and follow-up work.

major comments (2)
  1. [§3 (algorithm derivation)] The central claim of exact top-K recovery at the neuron level (abstract and §3) rests on repurposing max-product without introducing unstated approximation errors. The manuscript must explicitly show that GNN-LRP relevance scores satisfy the non-negativity and independence conditions required for the dynamic program to match exhaustive enumeration; otherwise the polynomial algorithm may return a different set than the exponential baseline.
  2. [Table 2 / experimental verification] Table 2 or the corresponding runtime table: the reported speed-ups are only meaningful if the returned walks are verified to be identical (neuron level) or within a stated bound (node level) to the exhaustive top-K; without such a verification column the empirical results do not yet confirm the exactness property.
minor comments (2)
  1. [§2 / §3] Notation for neuron-level versus node-level walks is introduced without a clear running example; a small illustrative graph with explicit relevance scores would clarify the distinction.
  2. [Abstract / §3] The abstract states the algorithms are 'based on the max-product algorithm' but does not cite the specific PGM reference or the precise variant (e.g., with or without beam pruning) used for the top-K extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the exactness and empirical verification of our algorithms. We address each major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3 (algorithm derivation)] The central claim of exact top-K recovery at the neuron level (abstract and §3) rests on repurposing max-product without introducing unstated approximation errors. The manuscript must explicitly show that GNN-LRP relevance scores satisfy the non-negativity and independence conditions required for the dynamic program to match exhaustive enumeration; otherwise the polynomial algorithm may return a different set than the exponential baseline.

    Authors: We agree that an explicit derivation of the required conditions is needed for full rigor. In the revised version we will insert a short subsection in §3 that proves GNN-LRP relevance scores are non-negative (by construction of the standard LRP rules) and that the additive decomposition of walk relevance satisfies the independence property needed for the max-product dynamic program to recover exactly the same top-K set as exhaustive enumeration at the neuron level. revision: yes

  2. Referee: [Table 2 / experimental verification] Table 2 or the corresponding runtime table: the reported speed-ups are only meaningful if the returned walks are verified to be identical (neuron level) or within a stated bound (node level) to the exhaustive top-K; without such a verification column the empirical results do not yet confirm the exactness property.

    Authors: We accept this observation. We will augment Table 2 with an additional verification column that reports (i) exact match rate with exhaustive enumeration on all neuron-level instances small enough for brute-force comparison, and (ii) the empirical deviation from the stated approximation bound on the node-level instances. This will directly confirm the claimed exactness and approximation guarantees. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic reduction is independent of inputs

full rationale

The paper's core contribution is a new polynomial-time algorithm adapting the max-product method from PGMs to compute top-K relevant walks for GNN-LRP, claimed to be exact at the neuron level. This is presented as an independent algorithmic engineering step that reduces exponential enumeration to polynomial time without any fitted parameters, self-definitional mappings, or load-bearing self-citations in the abstract or claimed derivation. No equations or results reduce by construction to the input relevance scores; the exactness claim is an asserted property of the DP adaptation rather than a renaming or re-expression of existing quantities. The derivation chain is therefore self-contained against external algorithmic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5796 in / 1054 out tokens · 19706 ms · 2026-05-25T05:09:48.992577+00:00 · methodology

discussion (0)

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