Propagate, Then Sharpen: Post-Hoc Refinement of Frozen Node Classifiers

Preben Johnsen Bentdal; Nello Blaser; Xue-Cheng Tai

cs.LGSep 28, 2026

Propagate, Then Sharpen: Post-Hoc Refinement of Frozen Node Classifiers

Authors: Preben Johnsen Bentdal, Nello Blaser, Xue-Cheng Tai

Organizations: Department of Informatics, University of Bergen · Norwegian Research Centre (NORCE)

Abstract

We study post-hoc refinement of frozen node classifiers: given only the graph $G$ and class distributions $Q$ predicted by a frozen model, can we improve accuracy without access to node features, model parameters, or gradients? APPNP answers this by propagating logits with a restart towards the initial predictions, minimizing the anchored Dirichlet energy. Instead, we consider the Potts energy, and decompose it into a Dirichlet term, which penalizes disagreement between neighbouring nodes, and a Gini term, which penalizes indecision within each node. This decomposition motivates Propagate, Then Sharpen (PtS), which alternates between propagation of class probabilities and node-wise, mass-preserving sharpening, with only one additional hyperparameter selected using labelled validation nodes. Across nine homophilic graphs, with a frozen MLP backbone, PtS improves mean test accuracy over independently tuned APPNP by $1.71$ percentage points on clean inputs and $3.90$ under severe Gaussian feature corruption. Gains over APPNP become smaller, but remain positive with frozen GCN and GraphSAGE backbones. Sharpening also removes most of the accuracy loss of deep propagation: on clean inputs without restart, accuracy falls by $2.2$ points between $2$ and $100$ propagation steps under PtS, compared with $33.8$ for APPNP.

Figures & tables

Panel of Figure 1 — Figure 1 : Propagate, Then Sharpen. Node colours show two-class proportions in an illustrative first iteration ( $\alpha=0.1$ , $\eta=3.5$ ), initialized at $Q$ . Propagation and sharpening update all nodes, and the sharpened state becomes the current state for the next iteration. Propagation includes self-loops and a restart towards the fixed $Q$ . Pie charts show normalised class proportions.

	$\sigma=0$				$\sigma=2$
Dataset	$Q$ (MLP)	APPNP	PtS	PtS $-$ APPNP	$Q$ (MLP)	APPNP	PtS	PtS $-$ APPNP
WikiCS	72.67	77.78	78.42	+0.64 $\pm$ 0.50	46.36	67.97	72.15	+4.18 $\pm$ 1.75
Cora-TAPE	60.51	76.16	79.74	+3.58 $\pm$ 1.23	43.82	68.16	72.30	+4.13 $\pm$ 1.99
PubMed-TAPE	81.59	84.35	85.18	+0.83 $\pm$ 0.64	64.56	78.87	80.73	+1.86 $\pm$ 1.44
TAPE-Arxiv23	67.59	69.53	69.72	+0.19 $\pm$ 0.10	28.56	35.63	38.91	+3.29 $\pm$ 0.26
ogbn-arxiv	55.95	66.10	66.56	+0.46 $\pm$ 0.51	22.65	33.86	40.62	+6.76 $\pm$ 1.73

Table 1: Test accuracy (%) with a frozen MLP backbone. Each method is tuned independently at the evaluation severity. The mean excludes the heterophilic controls.

Method	Sharpening	Accuracy
PPR-Prob	$\eta=0$	59.15
PtS	Selected $\eta$	60.24
Reaction OFF	$\eta=0$	54.93

Table 2 : Mean accuracy (%) over the nine main graphs at

\sigma=2

with the frozen MLP. PPR-Prob is independently tuned, Reaction OFF reuses the PtS-selected propagation settings without retuning.

Figure 2 : Mean accuracy over the nine main graphs at fixed

(\alpha,\eta)

. Columns vary the restart, rows vary the corruption severity.

Frozen backbone	$\sigma=0$	$\sigma=2$
MLP	+1.71	+3.90
GCN	+0.41	+0.88
GraphSAGE	+0.47	+1.23

Table 3: Mean test-accuracy gain of PtS over independently tuned APPNP across the nine main datasets, in percentage points.

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Nodes	Edges	Features	Classes	$h$
WikiCS	11,701	215,603	300	10	0.65
Cora-TAPE	2,708	5,278	768	7	0.81
PubMed-TAPE	19,717	44,324	768	3	0.80
TAPE-Arxiv23	46,198	38,863	300	40	0.64
ogbn-arxiv	169,343	1,157,799	128	40	0.65
ogbn-products	2,449,029	61,859,012	100	47	0.81

Appendix

Table 4 : Dataset statistics. Edges are the reported undirected-pair counts, and

h

denotes edge homophily.

Dataset	$D$	$C$	MLP	GCN	GraphSAGE	Conv. layers $\times$ width
WikiCS	300	10	146,442	80,138	159,498	$2\times 256$
Cora-TAPE	768	7	265,479	199,175	397,575	$2\times 256$
PubMed-TAPE	768	3	264,451	198,147	395,523	$2\times 256$
TAPE-Arxiv23	300	40	154,152	87,848	174,888	$2\times 256$
ogbn-arxiv	128	40	110,120	110,120	218,664	$3\times 256$
ogbn-products	100	47	104,751	36,015	71,215	$3\times 128$

Appendix

Table 5: Number of parameters of each backbone, with the feature dimension

D

and the number of classes

C

. The last column gives the depth and width of the graph backbones.

Method	Search	Budget
APPNP / PPR	$\alpha\in[0,1]$ , $K\in\{1,\ldots,100\}$	250
PtS	Same, $\eta=0$ or $\log_{10}\eta\in[-2,2.408]$	250
LAME-Graph	$T\in\{1,\ldots,100\}$ , $\gamma=0$ or $\log_{10}\gamma\in[-2,2.408]$	250
Graph-TV	$\lambda=0$ or $\log_{10}\lambda\in\{-3,\ldots,3\}$ (25 log-spaced values), $\varepsilon=1$ fixed	$\leq 26$
C&S	$\alpha_{c},\alpha_{s}\in[0,1]$	250
C&S-PtS	Same, plus the PtS reaction search	250

Appendix

Table 6 : Search spaces and maximum numbers of hyperparameter evaluations per unit and severity. All methods use Optuna TPE except Graph-TV, which uses a grid search. The PtS search includes the exact sharpening off setting

\eta=0

alongside the logarithmic range. For hyperparameter tuning on ogbn-products, the upper limits are reduced to

K=50

and

T=50

Dataset	$\sigma$	$K_{\mathrm{APPNP}}$	$K_{\mathrm{PPR}}$	$K_{\mathrm{PtS}}$	$\alpha_{\mathrm{APPNP}}$	$\alpha_{\mathrm{PtS}}$	$\eta_{\mathrm{PtS}}$	off (%)
WikiCS	0	2	2	3	0.11	0.09	4.11	7
	0.5	2	2	4	0.08	0.06	6.40	8
	1	2	3	4	0.04	0.04	6.96	3
	1.5	2	2	6	0.03	0.03	12.86	0
	2	2	3	8	0.02	0.00	34.68	0
Cora-TAPE	0	3	6	54	0.06	0.13	2.24	0

Appendix

Table 7 : Median selected values over the units of each (graph, severity), where

K

is the number of propagation steps,

\alpha

the restart probability,

\eta

the sharpening strength, and “off” the percentage of units for which the search selected the exact sharpening off setting

\eta=0

Dataset	$Q$	APPNP	PPR-Prob	LAME-Graph	Graph-TV	PtS
WikiCS	46.36	67.97	68.98	67.80	65.10	72.15
Cora-TAPE	43.82	68.16	70.47	63.94	68.72	72.30
PubMed-TAPE	64.56	78.87	80.22	73.87	79.64	80.73
TAPE-Arxiv23	28.56	35.63	38.26	35.26	36.54	38.91
ogbn-arxiv	22.65	33.86	40.43	40.38	35.25	40.62
ogbn-products	22.46	28.54	27.24	27.47	–	27.33

Appendix

Table 8 : Comparison of post-hoc graph refinement methods at

\sigma=2

, including the LAME-Graph and the Graph-TV methods.

Dataset	$Q$	APPNP	PPR-Prob	LAME-Graph	Graph-TV	PtS
WikiCS	72.67	77.78	78.24	78.18	78.18	78.42
Cora-TAPE	60.51	76.16	78.42	76.78	76.82	79.74
PubMed-TAPE	81.59	84.35	84.96	85.05	85.11	85.18
TAPE-Arxiv23	67.59	69.53	69.68	69.82	69.81	69.72
ogbn-arxiv	55.95	66.10	65.96	67.53	65.80	66.56
ogbn-products	59.58	70.58	71.32	71.40	–	72.17

Appendix

Table 9: External

Q{+}G

baselines on clean features (

\sigma=0

), including the LAME-Graph and the Graph-TV methods.

Dataset	APPNP	PPR-Prob	PtS	Reaction OFF	$\Delta_{\mathrm{space}}$ PPR $-$ APPNP	$\Delta_{\mathrm{PtS}}$ PtS $-$ PPR
WikiCS	67.97	68.98	72.15	53.81	+1.01 $\pm$ 1.59	+3.17 $\pm$ 0.91
Cora-TAPE	68.16	70.47	72.30	64.83	+2.31 $\pm$ 1.90	+1.83 $\pm$ 0.65
PubMed-TAPE	78.87	80.22	80.73	76.80	+1.35 $\pm$ 1.25	+0.51 $\pm$ 0.27
TAPE-Arxiv23	35.63	38.26	38.91	36.90	+2.64 $\pm$ 0.28	+0.65 $\pm$ 0.37
ogbn-arxiv	33.86	40.43	40.62	39.95	+6.57 $\pm$ 1.71	+0.18 $\pm$ 0.22
ogbn-products	28.54	27.24	27.33	27.33	-1.30 $\pm$ 0.22	+0.09 $\pm$ 0.15

Appendix

Table 10: Decomposition of PtS

-

APPNP at

\sigma=2

into the change from moving propagation into probability space,

\Delta_{\mathrm{space}}

and the change from adding the reaction,

\Delta_{\mathrm{PtS}}

. APPNP, PPR-Prob, and PtS are tuned independently. Reaction OFF sets

\eta=0

at the

(\alpha,K)

selected for PtS, without retuning

	Clean ( $\sigma=0$ )				Noisy ( $\sigma=2$ )
Dataset	$Q(GCN)$	APPNP	PtS	$\Delta$	$Q(GCN)$	APPNP	PtS	$\Delta$
WikiCS	79.08	79.05	79.40	+0.35 $\pm$ 0.21	76.73	76.91	77.68	+0.77 $\pm$ 0.30
Cora-TAPE	81.96	82.37	83.10	+0.73 $\pm$ 0.49	76.20	78.34	79.94	+1.60 $\pm$ 0.61
PubMed-TAPE	85.77	85.74	85.81	+0.06 $\pm$ 0.05	80.41	82.27	82.73	+0.46 $\pm$ 0.17
TAPE-Arxiv23	69.28	69.26	69.26	-0.00 $\pm$ 0.02	43.43	44.04	44.33	+0.29 $\pm$ 0.11
ogbn-arxiv	71.83	71.90	72.48	+0.58 $\pm$ 0.28	62.73	64.66	64.78	+0.12 $\pm$ 0.33

Appendix

Table 11 : Accuracy (%) with the frozen GCN backbone.

\Delta

is PtS minus APPNP in percentage points. Bold marks the highest displayed accuracy in each condition.

	Clean ( $\sigma=0$ )				Noisy ( $\sigma=2$ )
Dataset	$Q(SAGE)$	APPNP	PtS	$\Delta$	$Q(SAGE)$	APPNP	PtS	$\Delta$
WikiCS	78.72	79.08	79.41	+0.33 $\pm$ 0.18	71.06	75.12	76.38	+1.27 $\pm$ 0.43
Cora-TAPE	73.54	74.76	75.96	+1.20 $\pm$ 1.17	68.18	70.89	72.69	+1.80 $\pm$ 0.95
PubMed-TAPE	84.10	84.47	84.80	+0.33 $\pm$ 0.18	72.26	78.63	80.04	+1.41 $\pm$ 0.77
TAPE-Arxiv23	70.04	70.20	70.35	+0.15 $\pm$ 0.09	39.67	41.59	43.09	+1.50 $\pm$ 0.44
ogbn-arxiv	71.85	72.17	72.43	+0.26 $\pm$ 0.06	50.00	55.30	57.44	+2.14 $\pm$ 0.94

Appendix

Table 12 : Accuracy (%) with the frozen GraphSAGE backbone.

\Delta

is PtS minus APPNP in percentage points. Bold marks the highest displayed accuracy in each condition.

Dataset	C&S 0	C&S-PtS 0	C&S 2	C&S-PtS 2	Gain (pp)
WikiCS	76.90	78.82	70.06	75.43	+5.38 $\pm$ 0.85
Cora-TAPE	87.38	87.23	86.18	86.00	-0.17 $\pm$ 0.20
PubMed-TAPE	86.97	86.98	84.44	84.44	0.00 $\pm$ 0.05
TAPE-Arxiv23	70.32	70.31	45.54	45.55	+0.02 $\pm$ 0.04
ogbn-arxiv	70.91	71.00	68.35	68.61	+0.26 $\pm$ 0.30
ogbn-products	77.97	78.67	75.08	76.73	+1.65 $\pm$ 0.53

Appendix

Table 13: Comparison of Correct and Smooth (C&S) with C&S-PtS, where the categorical sharpening step is applied after each smoothing step. All methods use an MLP backbone. Subscripts give the corruption severity, and Gain is C&S-PtS minus C&S at

\sigma=2

in percentage points.

Dataset	Method	$\sigma=0$	$\sigma=0.5$	$\sigma=1$	$\sigma=1.5$	$\sigma=2$
WikiCS	APPNP	77.78	77.12	75.33	72.19	67.97
	PPR-Prob	78.24	77.48	75.69	73.09	68.98
	PtS	78.42	77.61	76.14	74.38	72.15
Cora-TAPE	APPNP	76.16	75.60	73.87	71.43	68.16
	PPR-Prob	78.42	77.92	76.28	73.52	70.47
	PtS	79.74	78.95	77.32	75.21	72.30

Appendix

Table 14 : Accuracy across the full Gaussian corruption ladder. APPNP propagates logits, PPR-Prob isolates probability-space propagation, and PtS additionally applies the categorical sharpening step.

Dataset	Method	$\sigma=0$	$\sigma=0.5$	$\sigma=1$	$\sigma=1.5$	$\sigma=2$
WikiCS	C&S	76.90	75.64	73.14	71.19	70.06
	C&S-PtS	78.82	78.27	77.32	76.31	75.43
Cora-TAPE	C&S	87.38	87.14	86.81	86.54	86.18
	C&S-PtS	87.23	87.05	86.66	86.42	86.00
PubMed-TAPE	C&S	86.97	86.39	85.48	84.89	84.44
	C&S-PtS	86.98	86.38	85.48	84.88	84.44

Appendix

Table 15 : Correct & Smooth across the full Gaussian corruption ladder. C&S-PtS applies the same categorical sharpening step during the smoothing stage.

Dataset	PtS-rn	PtS	$\Delta_{\mathrm{PtS}}$	$\Delta_{\mathrm{PPR}}$
Clean ( $\sigma=0$ )
WikiCS	78.24	78.42	$+0.18\pm 0.27$	$+0.15\pm 0.27$
Cora-TAPE	79.15	79.74	$+0.59\pm 0.66$	$+0.17\pm 0.58$
PubMed-TAPE	85.12	85.18	$+0.07\pm 0.16$	$+0.03\pm 0.08$
TAPE-Arxiv23	69.75	69.72	$-0.03\pm 0.06$	$-0.08\pm 0.04$
ogbn-arxiv	67.45	66.56	$-0.89\pm 0.34$	$-1.11\pm 0.32$

Appendix

Table 16: Effect of preserving the propagated row mass.

\Delta_{\mathrm{PtS}}

is PtS minus PtS-rn and

\Delta_{\mathrm{PPR}}

is PPR-Prob minus PPR-rn, both computed within (split, seed, draw). Bold marks the higher of the two accuracies in each row.

$\sigma=0$ , $\alpha=0$ (no restart)
Curve	$K{=}1$	$K{=}2$	$K{=}3$	$K{=}5$	$K{=}10$	$K{=}20$	$K{=}40$	$K{=}100$
frozen $Q$	67.5	67.5	67.5	67.5	67.5	67.5	67.5	67.5
APPNP	74.5	74.3	73.4	70.7	64.8	58.2	50.2	40.5
Logit-Sharp $\eta{=}16$	74.5	74.9	74.7	74.3	72.7	68.0	60.7	52.9
PPR-Prob	75.1	75.8	75.2	73.3	68.5	61.4	53.7	44.6
PtS $\eta{=}16$	75.1	75.8	75.8	75.3	74.4	73.5	71.9	70.6

Appendix

Table 17 : Depth sensitivity under clean and corrupted inputs.

Figure 6 : Evolution of the Dirichlet energy

E_{\mathrm{D}}

(left), Gini energy

E_{\mathrm{G}}

(middle), and validation accuracy (right) under gaussian feature corruption at

\sigma=2

. Gold and red markers show values after propagation and sharpening. Black markers denote the frozen predictions

Q

. Propagation reduces disagreement between nodes, but increases indecision within nodes, while sharpening has the opposite effect

Node stratum	Bucket	PtS $-$ APPNP ( $\sigma{=}0$ )	PtS $-$ APPNP ( $\sigma{=}2$ )
Local homophily	0-.2	-1.29	-1.75
	.2-.4	-0.51	-0.35
	.4-.6	+1.88	+4.83
	.6-.8	+3.69	+9.12
	.8-1	+1.90	+5.76
	isolated	+0.00	+0.00

Appendix

Table 18 : PtS

-

APPNP accuracy difference in percentage points, with nodes stratified by local homophily, by degree quintile, and by the confidence of the frozen prediction.

		MLP	Per step (ms)			$K=100$ (ms)			PtS/APPNP
Dataset	Device	fwd (ms)	APPNP	PPR-Prob	PtS	APPNP	PPR-Prob	PtS	per step
WikiCS	CPU	23.0	1.00	2.02	4.72	100	202	472	4.7
Cora-TAPE	CPU	5.4	0.09	0.44	1.31	8.8	43.6	131	14.9
PubMed-TAPE	CPU	59.0	0.18	0.90	3.08	17.7	89.8	308	17.4
TAPE-Arxiv23	CPU	87.6	5.45	12.1	30.7	545	1,211	3,068	5.6
ogbn-arxiv	A100	9.9	0.90	1.69	3.85	90.0	169	385	4.3

Appendix

Table 19: Runtime and overhead: MLP forward-pass and post-hoc call times in milliseconds at

\sigma=2

$\sigma$	Method	$T$	raw NLL	scaled NLL	raw ECE	scaled ECE
0	Frozen $Q$	1.34	1.226	1.132	0.088	0.040
	APPNP	0.84	0.983	0.907	0.129	0.037
	PtS	2.11	1.591	0.932	0.158	0.050
2	Frozen $Q$	3.55	3.514	2.112	0.302	0.047
	APPNP	1.47	2.108	1.752	0.178	0.075
	PtS	2.92	3.088	1.692	0.254	0.093

Appendix

Table 20: Calibration of the frozen prediction, APPNP and PtS with an MLP backbone, before and after temperature scaling fitted on validation nodes.

T

is the fitted temperature and accuracy drift is the change in accuracy caused by scaling. Values are means over WikiCS, Cora-TAPE, PubMed-TAPE, TAPE-arxiv23, ogbn-arxiv and ogbn-products

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Sep 3, 2026, cs.LGJ/K move · Enter open · S save

Selective Hypergraph Refinement for Frozen Graph Clustering

Zimo Si

Department of Mathematics, Faculty of Science, University of Macau · Macao SAR, China

Hypergraphs Refinement

Open paper AskPDF arXiv

Existing graph-clustering methods typically improve clustering performance by optimizing model parameters and node representations. Effective means of further improving the clustering results of an already trained and frozen model, however, remain limited. We study post-processing for frozen graph clustering. After checkpoint fixation, the procedure uses no labels and updates neither model parameters, node representations, nor the original graph structure. Instead, it exploits an attribute hypergraph to supplement higher-order relations that ordinary graphs cannot readily express, thereby refining existing cluster assignments. Because global hypergraph refinement can yield both performance gains and erroneous updates, we propose Selective Hypergraph Refinement (SHR). The method generates candidate residual directions from the hypergraph and evaluates their reliability using graph structure, node attributes, and matched-null evidence. It updates only nodes with sufficient support and otherwise retains their original assignments. Further analysis shows that whether a node changes cluster is jointly governed by its native assignment gap and the directional strength of the refinement. In a controlled common-suite evaluation, 13 of 15 backbone-dataset cells had a positive mean macro gain, one produced exact no-action, and one was negative. The cell-equal macro gain was 0.066 pp (95% bootstrap CI, [0.030, 0.107] pp), while only 0.209% of hard assignments changed on average. A broader 15-combination native-interface evaluation yielded a macro gain of 0.137 pp at a mean change ratio of 0.375%. These results indicate that frozen clustering outputs retain a limited but measurable refinement space after training. The effect is heterogeneous across backbone-dataset pairs, and broader coverage also increases exposure to negative transfer.

Abstract

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Selective Hypergraph Refinement for Frozen Graph Clustering

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