# Unifying Invariant and Variant Features for Graph Out-of-Distribution via Probability of Necessity and Sufficiency

arXiv: https://arxiv.org/abs/2407.15273 · HTML: https://mlsanity.com/paper/2407.15273/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 75 references that are in the map.
- First posted on arXiv 21 Jul 2024.

## Builds on (most cited first, 30 of 51)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization](https://mlsanity.com/paper/1911.08731/) (2019)
- [Deep CORAL: Correlation Alignment for Deep Domain Adaptation](https://mlsanity.com/paper/1607.01719/) (2016)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [Out-of-Distribution Generalization via Risk Extrapolation (REx)](https://mlsanity.com/paper/2003.00688/) (2020)
- [GNNExplainer: Generating Explanations for Graph Neural Networks](https://mlsanity.com/paper/1903.03894/) (2019)
- [Towards Out-Of-Distribution Generalization: A Survey](https://mlsanity.com/paper/2108.13624/) (2021)
- [Self-Supervised Learning with Data Augmentations Provably Isolates Content from Style](https://mlsanity.com/paper/2106.04619/) (2021)
- [Discovering Invariant Rationales for Graph Neural Networks](https://mlsanity.com/paper/2201.12872/) (2022)
- [Interpretable and Generalizable Graph Learning via Stochastic Attention Mechanism](https://mlsanity.com/paper/2201.12987/) (2022)
- [Deep Stable Learning for Out-Of-Distribution Generalization](https://mlsanity.com/paper/2104.07876/) (2021)
- [Learning Causally Invariant Representations for Out-of-Distribution Generalization on Graphs](https://mlsanity.com/paper/2202.05441/) (2022)
- [GOOD: A Graph Out-of-Distribution Benchmark](https://mlsanity.com/paper/2206.08452/) (2022)
- [Causal Attention for Interpretable and Generalizable Graph Classification](https://mlsanity.com/paper/2112.15089/) (2021)
- [Uncertainty Aware Semi-Supervised Learning on Graph Data](https://mlsanity.com/paper/2010.12783/) (2020)
- [Probabilities of Causation: Bounds and Identification](https://mlsanity.com/paper/1301.3898/) (2013)
- [A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2002.03689/) (2020)
- [Learning and Evaluating Graph Neural Network Explanations based on Counterfactual and Factual Reasoning](https://mlsanity.com/paper/2202.08816/) (2022)
- [OOD-GNN: Out-of-Distribution Generalized Graph Neural Network](https://mlsanity.com/paper/2112.03806/) (2021)
- [Generalizing Graph Neural Networks on Out-Of-Distribution Graphs](https://mlsanity.com/paper/2111.10657/) (2021)

## Built on by (most cited first, 2 of 2)

- [AdvSynGNN: Structure-Adaptive Graph Neural Nets via Adversarial Synthesis and Self-Corrective Propagation](https://mlsanity.com/paper/2602.17071/) (2026)
- [Beyond Sufficiency: Time Series Explanation with Counterfactual Necessity](https://mlsanity.com/paper/2607.21573/) (2026)

## Abstract (arXiv)

Graph Out-of-Distribution (OOD), requiring that models trained on biased data generalize to the unseen test data, has considerable real-world applications. One of the most mainstream methods is to extract the invariant subgraph by aligning the original and augmented data with the help of environment augmentation. However, these solutions might lead to the loss or redundancy of semantic subgraphs and result in suboptimal generalization. To address this challenge, we propose exploiting Probability of Necessity and Sufficiency (PNS) to extract sufficient and necessary invariant substructures. Beyond that, we further leverage the domain variant subgraphs related to the labels to boost the generalization performance in an ensemble manner. Specifically, we first consider the data generation process for graph data. Under mild conditions, we show that the sufficient and necessary invariant subgraph can be extracted by minimizing an upper bound, built on the theoretical advance of the probability of necessity and sufficiency. To further bridge the theory and algorithm, we devise the model called Sufficiency and Necessity Inspired Graph Learning (SNIGL), which ensembles an invariant subgraph classifier on top of latent sufficient and necessary invariant subgraphs, and a domain variant subgraph classifier specific to the test domain for generalization enhancement. Experimental results demonstrate that our SNIGL model outperforms the state-of-the-art techniques on six public benchmarks, highlighting its effectiveness in real-world scenarios.