# HOPE: Heterophily-Aware Open-Set Node Classification with Pseudo-Extrapolation

arXiv: https://arxiv.org/abs/2609.08685 · HTML: https://mlsanity.com/paper/2609.08685/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 36 references that are in the map.
- First posted on arXiv 8 Sep 2026.

## Builds on (most cited first, 26 of 26)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [A critical look at the evaluation of GNNs under heterophily: Are we really making progress?](https://mlsanity.com/paper/2302.11640/) (2023)
- [Label Propagation for Deep Semi-supervised Learning](https://mlsanity.com/paper/1904.04717/) (2019)
- [Graph Neural Networks for Graphs with Heterophily: A Survey](https://mlsanity.com/paper/2202.07082/) (2022)
- [CoMatch: Semi-supervised Learning with Contrastive Graph Regularization](https://mlsanity.com/paper/2011.11183/) (2020)
- [Unifying Graph Convolutional Neural Networks and Label Propagation](https://mlsanity.com/paper/2002.06755/) (2020)
- [New Benchmarks for Learning on Non-Homophilous Graphs](https://mlsanity.com/paper/2104.01404/) (2021)
- [LEGO-Learn: Label-Efficient Graph Open-Set Learning](https://mlsanity.com/paper/2410.16386/) (2024)
- [Graph Condensation for Open-World Graph Learning](https://mlsanity.com/paper/2405.17003/) (2024)
- [Raising the Bar in Graph OOD Generalization: Invariant Learning Beyond Explicit Environment Modeling](https://mlsanity.com/paper/2502.10706/) (2025)
- [From Few-Shot to Zero-Shot: Towards Generalist Graph Anomaly Detection](https://mlsanity.com/paper/2602.18793/) (2026)
- [Influence-Oriented Personalized Federated Learning](https://mlsanity.com/paper/2410.03315/) (2024)
- [When Heterophily Meets Heterogeneity: Challenges and a New Large-Scale Graph Benchmark](https://mlsanity.com/paper/2407.10916/) (2024)
- [FedCIGAR: A Personalized Reconstruction Approach for Federated Graph-level Anomaly Detection](https://mlsanity.com/paper/2605.09428/) (2026)
- [G2Pxy: Generative Open-Set Node Classification on Graphs with Proxy Unknowns](https://mlsanity.com/paper/2308.05463/) (2023)
- [Multi-Stage Verification-Centric Framework for Mitigating Hallucination in Multi-Modal RAG](https://mlsanity.com/paper/2507.20136/) (2025)
- [Towards Anomaly Detection on Relational Data](https://mlsanity.com/paper/2606.18621/) (2026)
- [DiRW: Path-Aware Digraph Learning for Heterophily](https://mlsanity.com/paper/2410.10320/) (2024)
- [ROGPL: Robust Open-Set Graph Learning via Region-Based Prototype Learning](https://mlsanity.com/paper/2402.18495/) (2024)

## Abstract (arXiv)

Standard open-set node classification methods rely on the homophily assumption, where connected nodes share labels. However, real-world graphs are often heterophilic, exposing the limitations of current methods and posing new challenges to open-set node classification. On the one hand, cross-class connectivity causes representations from different known or unknown classes to become intertwined after aggregation, undermining their discriminative capacity. On the other hand, structural mixture invalidates threshold-based open-set methods and cross-class feature interpolation, leading to unreliable unknown-class rejection. To address these challenges, we propose HOPE, a Heterophily-aware Open-set node classification method with Pseudo-Extrapolation. To adapt open-set graph neural networks (GNNs) to heterophilic scenarios, HOPE uses a structure-augmented feature initialization layer to capture multi-hop structural patterns. Meanwhile, we design a trustworthy neighborhood aggregation mechanism for standard GNNs to dynamically filter noisy cross-class neighbors. To enhance unknown-class rejection, we introduce a heterophily-guided pseudo-extrapolation strategy. It dynamically maintains known-class centers and extrapolates along cross-class neighborhood displacement directions, synthesizing pseudo-unknown proxies near structurally ambiguous regions. Finally, we optimize the network with joint classification and logit margin regularization, routing synthetic proxies into a dedicated rejection slot without imposing geometric margin constraints in the representation space. Extensive experiments on multiple datasets show that HOPE consistently outperforms state-of-the-art models, validating its effectiveness, robustness, and efficiency.