Conditional molecular generation, aiming to generate 2D and 3D molecules that satisfy given properties, has achieved remarkable progress, thanks to the advances in deep generative models such as graph diffusion. However, existing methods generally assume that the given conditions for training and testing are consistent, failing to handle the realistic challenge when there exist distribution shifts between training and testing conditions. Invariant learning is a mainstream paradigm for addressing distribution shifts, but fusing invariant learning principles with conditional molecular generation faces three core challenges: (1) existing invariant learning methods focus on discriminative tasks and cannot be directly adapted to molecule generative tasks; (2) how to distinguish between invariant subgraph and variant subgraph of a molecule graph, which is treated as an integrated input; (3) how to fuse invariant subgraphs, variant subgraphs, and property conditions for effective generation. To tackle these challenges, we propose Invariant Conditional MOLecular generation (IC-MOL), a framework that combines invariant learning with graph diffusion to improve the generalization ability of conditional molecular generation under distribution shifts. Specifically, we first disentangle molecular graphs into invariant and variant subgraphs while maintaining SE(3) equivariance, an important inductive bias for molecular generation. On this basis, we further design a two-phase graph diffusion generation model. In the first phase, we generate an invariant molecular consistent with the target property. In the second phase, we propose a cross-attention mechanism to fuse variant subgraph representations and property conditions to guide the generation of complete molecules while maintaining property alignment. Extensive experiments on the benchmark dataset show that IC-MOL consistently outperforms state-of-the-art baselines across six property conditions under distribution shifts.

Graph Transformers (GTs) have demonstrated great effectiveness across various graph analytical tasks. However, the existing GTs focus on training and testing graph data originated from the same distribution, but fail to generalize under distribution shifts. Graph invariant learning, aiming to capture generalizable graph structural patterns with labels under distribution shifts, is potentially a promising solution, but how to design attention mechanisms and positional and structural encodings (PSEs) based on graph invariant learning principles remains challenging. To solve these challenges, we introduce graph out-of-distribution generalized Transformer (GOODFormer), aiming to learn generalized graph representations by capturing invariant relationships between predictive graph structures and labels through jointly optimizing three modules. Specifically, we first develop a GT-based entropy-guided invariant subgraph disentangler to separate invariant and variant subgraphs while preserving the sharpness of the attention function. Next, we design an evolving subgraph positional and structural encoder to effectively and efficiently capture the encoding information of dynamically changing subgraphs during training. Finally, we propose an invariant learning module utilizing subgraph node representations and encodings to derive graph representations that can generalize to unseen test graphs. We also provide theoretical justifications for our method. Extensive experiments on benchmark datasets demonstrate the superiority of our method over state-of-the-art baselines under distribution shifts.

Graph representation learning encounters great difficulties under distribution shifts. This challenge has aroused considerable interest in graph out-of-distribution (OOD) generalization and detection, which can effectively handle covariate and semantic shifts, respectively. However, real-world graph tasks often involve complex unlabeled wild data with both covariate and semantic shifts, motivating a critical question: can we design a unified framework for joint graph OOD generalization and detection? Invariant graph learning, which extracts stable relationships between features and labels, offers a promising candidate for joint OOD generalization and detection, but faces three critical challenges (1) how to model invariant subgraphs with unlabeled data, (2) how to ensure graph representations benefit both tasks, and (3) how to integrate labeled and unlabeled data under proper invariance principles. To solve these challenges, we introduce Unified Graph Out-Of-Distribution generalization and detection framework (UniGOOD) with three tailored components. Specifically, to capture subgraphs without relying on labels, we first propose the distributional invariant subgraph generator to model subgraph conditional distributions. Next, to enable generalization and reliable detection, we propose the cross-invariant-subgraph spectral contrastive learning module to learn invariant representations from subgraph distributions. Finally, for accurate subgraph discovery across labeled and unlabeled graphs, we design the triple-population invariance regularizer to enforce the invariance principle through spectral graph theory. We prove that our method theoretically ensures accurate invariant subgraphs, enabling effective OOD generalization and detection. Experiments show that UniGOOD outperforms state-of-the-art baselines for both graph OOD generalization and detection tasks.