MPGE: A Multi-Perspective Graph Explainer for Molecular Classification Explanation

Mahtab Sarvmaili

cs.LGOct 8, 2026

MPGE: A Multi-Perspective Graph Explainer for Molecular Classification Explanation

Authors: Mahtab Sarvmaili

Organizations: Dalhousie University Halifax, Nova Scotia, Canada

Abstract

Graph neural networks (GNNs) predict molecular properties from chemical graph data, but predictive accuracy does not explain how graph information supports an individual decision. A compact prediction-preserving rationale does not necessarily reveal which changes reverse the decision or which modifications the model tolerates. We propose the Multi-Perspective Graph Explainer (MPGE), unifying factual support, counterfactual sensitivity, and exemplar tolerance for a frozen classifier. The factual view, originally termed prototype (PT), seeks a compact retained edge set with the same label and required confidence. Counterfactual (CF) explanations seek bounded prediction-changing deletions; exemplar (EXE) explanations seek non-trivial bounded deletions that preserve the label and confidence. A shared constrained formulation connects prediction behavior, compactness, and edit cost, while separate objectives generate the three views. Our graph-classification extension of CF-GNNExplainer learns symmetric edge rankings and verifies discrete candidates, recording unsuccessful searches. A separate BBBP fragment backend returns RDKit-sanitized molecules. We evaluate the primary GCN implementation on MUTAG, Mutagenicity, AIDS, COX2_MD, and BBBP using semantic coverage, conditional quality, stability, and runtime. Successful factual masks retained 8.6%--15.5% of input edges on average across datasets; bounded counterfactual coverage was 4.8%--67.6%, and exemplar preservation coverage was 98.9%--100.0%. Exploratory controls reveal the influence of hard projection and retained node information. Quantitative comparisons and molecular visualizations characterize model support, sensitivity, and tolerance without treating them as validated chemical mechanisms.

Figures & tables

Dataset	$N$	Pos.	$\|V\|$	$\|E\|$	Features	Split	GT
MUTAG	188	0.66	$17.9\pm 4.6$	$19.8\pm 5.7$	N:L7; E:L4	Stratified 10-Fold	no
Mutagenicity	4337	0.45	$30.3\pm 20.1$	$30.8\pm 16.8$	N:L14; E:L3	Stratified 10-Fold	no
AIDS	2000	0.80	$15.7\pm 13.7$	$16.2\pm 15.0$	N:L38; N:A4; E:L3	Stratified 10-Fold	no
COX2_MD	303	0.49	$26.3\pm 2.5$	$335.1\pm 65.3$	N:L7; E:L5; E:A1	Stratified 10-Fold	no
BBBP	2039	0.77	$24.1\pm 10.6$	$26.0\pm 11.7$	Atom/Bond Categorical	Scaffold 80/10/10	no

TABLE I: Datasets and locked evaluation protocol.

|E|

counts each undirected edge once. N:L and E:L give node/edge categorical-label cardinalities; N:A and E:A give continuous-attribute dimensions. BBBP uses MoleculeNet atom/bond categorical features. GT denotes a documented edge-level explanation mask; none of these five datasets provides one. Features describe stored data, not all inputs used by GCN. AIDS explanation reporting uses folds 0–7; predictor reporting uses all ten folds.

Dataset	Architecture	Balanced accuracy	AUROC	AUPRC	F1
AIDS	GCN	$0.971\pm 0.014$	$0.985\pm 0.015$	$0.991\pm 0.012$	$0.989\pm 0.005$
BBBP	GCN	$0.698\pm 0.014$	$0.797\pm 0.010$	$0.942\pm 0.008$	$0.907\pm 0.008$
COX2_MD †	GCN	$0.519\pm 0.069$	$0.561\pm 0.133$	$0.586\pm 0.099$	$0.522\pm 0.181$
MUTAG	GCN	$0.688\pm 0.160$	$0.836\pm 0.102$	$0.929\pm 0.043$	$0.835\pm 0.062$
Mutagenicity	GCN	$0.781\pm 0.024$	$0.856\pm 0.022$	$0.819\pm 0.029$	$0.757\pm 0.027$

TABLE II: Primary GCN out-of-fold or scaffold-held-out predictive performance. Values are mean

\pm

standard deviation over outer-fold/model-seed runs.

Dataset	Architecture	Brier	ECE	Log loss	Best epoch
AIDS	GCN	$0.015\pm 0.006$	$0.023\pm 0.007$	$0.077\pm 0.050$	37.8
BBBP	GCN	$0.122\pm 0.005$	$0.094\pm 0.010$	$0.466\pm 0.035$	10.3
COX2_MD	GCN	$0.285\pm 0.065$	$0.204\pm 0.127$	$0.865\pm 0.382$	6.9
MUTAG	GCN	$0.162\pm 0.045$	$0.204\pm 0.046$	$0.494\pm 0.114$	51.2
Mutagenicity	GCN	$0.153\pm 0.012$	$0.052\pm 0.012$	$0.494\pm 0.053$	86.8

TABLE III: Predictive calibration and selected training epoch; mean

\pm

standard deviation over the same outer-fold/model-seed runs.

	Factual			Counterfactual			Exemplar
Dataset	Success	Retained	Comp.	Success	Edits	$\Delta p_{y}$	Success	Edited	Retained $p_{y}$ ratio
MUTAG	100.0%	$0.105\pm 0.131$	$0.120\pm 0.220$	67.6%	$2.59\pm 1.57$	$0.436\pm 0.192$	98.9%	$0.279\pm 0.048$	$1.041\pm 0.161$
Mutagenicity	100.0%	$0.155\pm 0.203$	$0.142\pm 0.241$	63.7%	$2.85\pm 1.91$	$0.315\pm 0.155$	99.0%	$0.246\pm 0.068$	$1.046\pm 0.129$
AIDS	100.0%	$0.103\pm 0.079$	$0.019\pm 0.113$	4.8%	$3.48\pm 2.10$	$0.419\pm 0.165$	100.0%	$0.285\pm 0.059$	$1.002\pm 0.040$
COX2_MD	100.0%	$0.092\pm 0.218$	$0.048\pm 0.173$	5.5%	$4.68\pm 2.33$	$0.099\pm 0.114$	100.0%	$0.025\pm 0.005$	$0.998\pm 0.028$
BBBP	100.0%	$0.086\pm 0.093$	$0.090\pm 0.163$	54.6%	$3.65\pm 2.24$	$0.382\pm 0.146$	99.7%	$0.277\pm 0.052$	$1.008\pm 0.094$

TABLE IV: Primary GCN multi-perspective explanations on all five datasets. Success is the percentage of all attempts; continuous outcomes are mean

\pm

standard deviation after averaging explainer repetitions within each graph–model pair. Comp. denotes comprehensiveness and

\Delta p_{y}

is the drop in the original-class probability.

Dataset	Method	Success	Retained	Comp.
MUTAG	MPGX-CF	100.0%	$0.105\pm 0.131$	$0.120\pm 0.220$
MUTAG	GNNExplainer	100.0%	$0.160\pm 0.210$	$0.103\pm 0.213$
MUTAG	PGExplainer	100.0%	$0.219\pm 0.280$	$0.112\pm 0.243$
MUTAG	CF 2 reimplementation	96.8%	$0.445\pm 0.154$	$0.519\pm 0.355$
MUTAG	Random	100.0%	$0.183\pm 0.222$	$0.114\pm 0.220$
Mutagenicity	MPGX-CF	100.0%	$0.155\pm 0.203$	$0.142\pm 0.241$

TABLE V: Factual native-output comparisons. Success includes failures; quality is conditional on success and uses graph–model means

\pm

standard deviation after averaging explainer repetitions.

Dataset	Method	Success	Edits	$\Delta p_{y}$
MUTAG	MPGX-CF	67.6%	$2.594\pm 1.569$	$0.436\pm 0.192$
MUTAG	CF-GNNExplainer	3.7%	$1.190\pm 0.402$	$0.398\pm 0.118$
MUTAG	CF 2 reimplementation	7.2%	$5.854\pm 1.663$	$0.756\pm 0.202$
MUTAG	Random	31.3%	$3.452\pm 1.910$	$0.396\pm 0.187$
Mutagenicity	MPGX-CF	63.7%	$2.851\pm 1.912$	$0.315\pm 0.155$
Mutagenicity	CF-GNNExplainer	2.2%	$1.286\pm 0.590$	$0.335\pm 0.127$

TABLE VI: Counterfactual native-output comparisons under the locked budget. Success includes failures; quality is conditional on success. Native discretization and optimization procedures differ across methods.

Dataset	Method	Success	Edited	Retained $p_{y}$ ratio
MUTAG	MPGX-CF	98.9%	$0.279\pm 0.048$	$1.041\pm 0.161$
MUTAG	Greedy control	100.0%	$0.305\pm 0.026$	$1.169\pm 0.232$
MUTAG	Random	94.6%	$0.268\pm 0.060$	$1.023\pm 0.145$
Mutagenicity	MPGX-CF	99.0%	$0.246\pm 0.068$	$1.046\pm 0.129$
Mutagenicity	Greedy control	99.9%	$0.257\pm 0.069$	$1.191\pm 0.211$
Mutagenicity	Random	96.5%	$0.231\pm 0.074$	$1.027\pm 0.127$

TABLE VII: Exemplar comparisons. Coverage requires a non-trivial prediction-preserving edit; conditional quality uses the same aggregation as Table IV .

Fig. 1: Hard-constraint coverage for compatible methods on all five datasets. Bars are means and error bars are standard deviations over graph–model analysis units after averaging explainer repetitions.

Fig. 2: Input, factual, counterfactual, and exemplar panels for MUTAG, Mutagenicity, AIDS, COX2_MD, and BBBP. A common graph and fixed layout are used within each row; panels print class-1 probability, label, and deleted unique-edge count. TU graphs are categorical renderings, not reconstructed or sanitized molecules. Unavailable rows remain visible rather than being silently omitted; the primary chemistry-aware BBBP case is shown separately in Fig. 3 . Selections are MUTAG: mutag-000074, fold 5, model seed 29, explainer seed 101; Mutagenicity: mutagenicity-003801, fold 7, model seed 47, explainer seed 211; AIDS: aids-001512, fold 0, model seed 11, explainer seed 101; COX2_MD: cox2_md-000133, fold 3, model seed 29, explainer seed 307; BBBP: bbbp-001663, fold 0, model seed 47, explainer seed 307.

Fig. 3: Primary-test chemistry-aware BBBP fragment explanations. The factual and exemplar fragments preserve the query decision, whereas the counterfactual fragment reverses it. Probabilities printed in the panels are the frozen GCN’s class-1 probabilities. Graph bbbp-001204 , scaffold-held-out BBBP test set, GCN predictor seed 11 and fragment-search seed 0; selected by median counterfactual Tanimoto similarity among complete successful fragment triplets in the primary run.

Fig. 4: Conditional quality of successful factual, counterfactual, and exemplar explanations. Structural size and prediction behavior are shown together because neither alone establishes semantic validity. Diamonds share the fractional axis except in the counterfactual panel, where they use the right-hand probability-drop axis.

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

View	Success	Steps	Atoms removed	Tanimoto	Retained $p_{y}$ ratio	Candidates
Factual	97.7%	$1.71\pm 0.70$	$19.91\pm 8.79$	$0.10\pm 0.11$	$0.92\pm 0.12$	$56.98\pm 59.75$
Counterfactual	27.4%	$1.40\pm 0.61$	$3.22\pm 2.66$	$0.62\pm 0.14$	$0.56\pm 0.15$	$73.96\pm 66.97$
Exemplar	96.4%	$2.19\pm 0.78$	$6.63\pm 2.60$	$0.35\pm 0.12$	$1.00\pm 0.14$	$57.54\pm 59.95$

Appendix

TABLE VIII: GCN chemistry-aware fragment search on the scaffold-held-out BBBP test set. Success is percentage-only over all attempts, including failures; other values are conditional means

\pm

standard deviation over graph–model units. Fragment factual size is retained heavy atoms, not retained edges.

Fig. 5: Macro-averaged hard-constraint success over the complete GCN and GINE validation grids. Failed searches remain in every denominator.

Fig. 6: Validation coverage as the permitted edit fraction changes. Error bars show variation across dataset-level means.

Dataset	Ranking	Units	Native	Common projection
MUTAG	MPGX-CF	72	54.2%	54.2%
MUTAG	CF-GNNExplainer	72	5.6%	41.7%
MUTAG	CF 2	72	4.2%	51.4%
MUTAG	Random	72	26.4%	26.4%
Mutagenicity	MPGX-CF	72	63.9%	63.9%
Mutagenicity	CF-GNNExplainer	72	1.4%	51.4%

Appendix

TABLE IX: Exploratory counterfactual control on a class-balanced held-out subset: native hard output versus identical ranked-prefix projection of each method’s stored scores. Percentages include failures; 72 graph–model units represent 24 graph IDs and three predictor seeds per dataset.

Dataset	Units	Proposed 0.5 threshold	Proposed ranked prefix
MUTAG	72	16.7%	54.2%
Mutagenicity	72	9.7%	63.9%
AIDS	72	1.4%	6.9%
COX2_MD	72	0.0%	9.7%
BBBP	72	15.3%	65.3%

Appendix

TABLE X: Proposed counterfactual ranking: fixed threshold versus hard-prefix search on the same selected held-out units.

Dataset	Graphs	Units	No-edge feasible	Induced factual feasible
MUTAG	24	72	73.6%	37.5%
Mutagenicity	24	72	56.9%	55.6%
AIDS	24	72	91.7%	61.1%
COX2_MD	24	72	69.4%	56.9%
BBBP	24	72	48.6%	38.9%

Appendix

TABLE XI: Factual null and induced-node controls. The no-edge input retains all query nodes, features, and predictor self-loops; the induced input retains only nodes incident to the proposed factual edges. Percentages use all 72 graph–model units per dataset.

Dataset	Ranking	Units	Cap 8	Cap 32
AIDS	MPGX-CF	72	6.9%	6.9%
AIDS	CF-GNNExplainer	72	5.6%	5.6%
AIDS	CF 2	72	6.9%	6.9%
AIDS	Random	72	1.4%	1.4%
COX2_MD	MPGX-CF	72	9.7%	23.6%
COX2_MD	CF-GNNExplainer	72	8.3%	22.2%

Appendix

TABLE XII: Exploratory hard-budget sensitivity under a common projection on the same class-balanced held-out subset.

Dataset	View	Runtime (s)	Runtime 95% CI	Components	Isolated nodes	Units
MUTAG	Factual	$0.54\pm 0.04$	$[0.53,0.56]$	$15.99\pm 4.95$	$14.64\pm 5.12$	564
MUTAG	Counterfactual	$0.55\pm 0.04$	$[0.54,0.55]$	$2.00\pm 1.00$	$0.52\pm 0.79$	564
MUTAG	Exemplar	$0.57\pm 0.04$	$[0.56,0.58]$	$4.02\pm 1.08$	$1.46\pm 0.78$	564
Mutagenicity	Factual	$0.58\pm 0.22$	$[0.51,0.68]$	$26.02\pm 19.56$	$23.52\pm 19.58$	13011
Mutagenicity	Counterfactual	$0.56\pm 0.16$	$[0.51,0.64]$	$3.57\pm 6.14$	$1.79\pm 6.04$	13011
Mutagenicity	Exemplar	$0.58\pm 0.16$	$[0.52,0.65]$	$6.79\pm 9.10$	$3.92\pm 8.99$	13011

Appendix

TABLE XIII: Proposed GCN per-view runtime and connectivity. Runtime is unconditional wall time in seconds, including failed searches; components and isolated nodes are conditional on hard success, with predictor self-loops excluded. Values are mean

\pm

standard deviation after averaging explainer seeds within graph–model units. Units counts timing units, not independent training replications. Runtime intervals are descriptive hierarchical 95% bootstrap intervals; hardware and implementation limit timing comparisons.

Dataset	View	Present	Removed	Removed frac.	$\|\Delta p\|$	$\Delta$ betw.	$\Delta$ harmonic	Stability
MUTAG	Factual	$1.98\pm 2.71$	$17.82\pm 6.05$	$0.90\pm 0.13$	$0.08\pm 0.09$	$0.22\pm 0.04$	$6.27\pm 1.25$	$0.25\pm 0.35$
MUTAG	Counterfactual	$17.33\pm 5.22$	$2.59\pm 1.57$	$0.13\pm 0.07$	$0.44\pm 0.19$	$0.12\pm 0.05$	$1.68\pm 0.97$	$0.74\pm 0.29$
MUTAG	Exemplar	$14.27\pm 4.33$	$5.52\pm 1.75$	$0.28\pm 0.05$	$0.08\pm 0.08$	$0.16\pm 0.04$	$3.33\pm 0.78$	$0.19\pm 0.13$
Mutagenicity	Factual	$4.33\pm 6.97$	$26.44\pm 16.25$	$0.84\pm 0.20$	$0.12\pm 0.10$	$0.21\pm 0.05$	$7.92\pm 2.50$	$0.42\pm 0.38$
Mutagenicity	Counterfactual	$26.13\pm 15.08$	$2.85\pm 1.91$	$0.11\pm 0.07$	$0.31\pm 0.16$	$0.10\pm 0.06$	$2.44\pm 1.28$	$0.74\pm 0.30$
Mutagenicity	Exemplar	$24.11\pm 15.79$	$6.69\pm 1.83$	$0.25\pm 0.07$	$0.08\pm 0.06$	$0.16\pm 0.06$	$4.33\pm 0.98$	$0.17\pm 0.13$

Appendix

TABLE XIV: Structural and probability diagnostics for the proposed method. Continuous metrics are conditional on hard success; stability is computed across explainer seeds within the same graph–model unit.

Dataset	Factual–CF overlap	Factual–exemplar overlap	Contrast
MUTAG	$0.095\pm 0.187$	$0.017\pm 0.048$	$0.078\pm 0.177$
Mutagenicity	$0.150\pm 0.212$	$0.036\pm 0.065$	$0.114\pm 0.214$
AIDS	$0.259\pm 0.211$	$0.075\pm 0.083$	$0.184\pm 0.204$
COX2_MD	$0.014\pm 0.029$	$0.005\pm 0.008$	$0.009\pm 0.030$
BBBP	$0.177\pm 0.210$	$0.026\pm 0.048$	$0.151\pm 0.199$

Appendix

TABLE XV: Cross-perspective overlap for the proposed method. Overlap is the Jaccard similarity between factual retained edges and the edges deleted by the counterfactual (CF) or exemplar view. Contrast is CF minus exemplar overlap. Values are mean

\pm

standard deviation after averaging explainer seeds within graph–model units on complete successful triplets. Counts and intervals are in the supplementary metric records.

Dataset	Baseline	Coverage difference	Holm $p$	Comprehensiveness difference	Holm $p$
MUTAG	GNNExplainer	$0.000$ [0.000, 0.000]; $n=188$	1	$0.018$ [0.006, 0.029]; $n=188$	0.00729
MUTAG	PGExplainer	$0.000$ [0.000, 0.000]; $n=188$	1	$0.008$ [-0.008, 0.024]; $n=188$	1
MUTAG	CF 2 reimplementation	$0.032$ [0.018, 0.048]; $n=188$	0.000139	$-0.400$ [-0.445, -0.357]; $n=188$	8.57e-31
MUTAG	Random	$0.000$ [0.000, 0.000]; $n=188$	1	$0.006$ [-0.005, 0.018]; $n=188$	1
Mutagenicity	GNNExplainer	$0.000$ [0.000, 0.000]; $n=4337$	1	$0.024$ [0.022, 0.026]; $n=4337$	4.79e-69
Mutagenicity	PGExplainer	$0.000$ [0.000, 0.000]; $n=4337$	1	$0.002$ [-0.001, 0.005]; $n=4337$	0.456

Appendix

TABLE XVI: Paired factual effects against compatible baselines.

Dataset	Baseline	Coverage difference	Holm $p$	Removed-edge difference	Holm $p$
MUTAG	CF-GNNExplainer	$0.638$ [0.580, 0.694]; $n=188$	2.08e-26	$-0.141$ [-0.392, 0.088]; $n=17$	0.256
MUTAG	CF 2 reimplementation	$0.603$ [0.546, 0.660]; $n=188$	8.94e-26	$-3.370$ [-3.851, -2.876]; $n=47$	1.39e-08
MUTAG	Random	$0.363$ [0.320, 0.406]; $n=188$	2.79e-24	$-1.299$ [-1.530, -1.073]; $n=121$	2.45e-16
Mutagenicity	CF-GNNExplainer	$0.615$ [0.603, 0.628]; $n=4337$	$<10^{-300}$	$-0.130$ [-0.191, -0.070]; $n=239$	6.06e-06
Mutagenicity	CF 2 reimplementation	$0.624$ [0.612, 0.637]; $n=4337$	$<10^{-300}$	$-3.581$ [-3.889, -3.275]; $n=184$	6.76e-29
Mutagenicity	Random	$0.412$ [0.402, 0.423]; $n=4337$	$<10^{-300}$	$-1.694$ [-1.761, -1.626]; $n=1984$	8.45e-263

Appendix

TABLE XVII: Paired counterfactual effects against compatible baselines.

Dataset	Baseline	Coverage difference	Holm $p$	Edit-fraction difference	Holm $p$
MUTAG	Greedy control	$-0.011$ [-0.017, -0.005]; $n=188$	0.000941	$-0.026$ [-0.030, -0.021]; $n=188$	1.6e-19
MUTAG	Random	$0.044$ [0.030, 0.059]; $n=188$	1.35e-07	$0.011$ [0.005, 0.017]; $n=188$	0.000736
Mutagenicity	Greedy control	$-0.010$ [-0.011, -0.008]; $n=4337$	1.15e-42	$-0.011$ [-0.012, -0.010]; $n=4337$	9.02e-236
Mutagenicity	Random	$0.025$ [0.022, 0.027]; $n=4337$	8.28e-74	$0.015$ [0.014, 0.016]; $n=4335$	1.99e-157
AIDS	Greedy control	$-0.000$ [-0.001, 0.000]; $n=1600$	0.102	$-0.002$ [-0.002, -0.001]; $n=1600$	7.09e-13
AIDS	Random	$0.003$ [0.002, 0.004]; $n=1600$	0.000112	$0.001$ [0.000, 0.001]; $n=1600$	0.0013

Appendix

TABLE XVIII: Paired exemplar effects against compatible baselines.

Fig. 7: Expanded MUTAG query, factual, counterfactual, and exemplar case.

Fig. 8: Expanded Mutagenicity query, factual, counterfactual, and exemplar case.

Fig. 9: Expanded AIDS query, factual, counterfactual, and exemplar case.

Fig. 10: Expanded COX2_MD query, factual, counterfactual, and exemplar case. The light complete-graph context and red dashed edits prevent the dense input from obscuring the selected changes.

Fig. 11: Expanded BBBP query, factual, counterfactual, and exemplar edge-mask case using RDKit coordinates.

Dataset	View	Failure rate	Most common reason	Count
MUTAG	Factual	0.0%	–	–
MUTAG	Counterfactual	32.4%	No Flip Within 8 Edits	173
MUTAG	Exemplar	1.1%	No Nontrivial Preserving Edit Within 3 Edits	9
Mutagenicity	Factual	0.0%	–	–
Mutagenicity	Counterfactual	36.3%	No Flip Within 8 Edits	10406
Mutagenicity	Exemplar	1.0%	No Nontrivial Preserving Edit Within 8 Edits	95

Appendix

TABLE XIX: Failure audit for the primary GCN and proposed method. Failure rate is the percentage of all attempts; the final column counts the most common machine-readable reason.

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