Propagate, Then Sharpen: Post-Hoc Refinement of Frozen Node Classifiers
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.
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BibTeX
@misc{bentdal2026propagate,
title = {{Propagate, Then Sharpen: Post-Hoc Refinement of Frozen Node Classifiers}},
author = {Preben Johnsen Bentdal and Nello Blaser and Xue-Cheng Tai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35080},
eprint = {2609.35080},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35080},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Propagate, Then Sharpen: Post-Hoc Refinement of Frozen Node Classifiers
AU - Preben Johnsen Bentdal
AU - Nello Blaser
AU - Xue-Cheng Tai
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35080
UR - https://arxiv.org/abs/2609.35080
AB - 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.
ER -