Invariant graph representation learning aims to learn the invariance among data from different environments for out-of-distribution generalization on graphs. As the graph environment partitions are usually expensive to obtain, augmenting the environment information has become the de facto approach. However, the usefulness of the augmented environment information has never been verified. In this work, we find that it is fundamentally impossible to learn invariant graph representations via environment augmentation without additional assumptions. Therefore, we develop a set of minimal assumptions, including variation sufficiency and variation consistency, for feasible invariant graph learning.

Invariant graph representation learning aims to learn the invariance among data from different environments for out-of-distribution generalization on graphs. As the graph environment partitions are usually expensive to obtain, augmenting the environment information has become the de facto approach. However, the usefulness of the augmented environment information has never been verified. In this work, we find that it is fundamentally impossible to learn invariant graph representations via environment augmentation without additional assumptions. Therefore, we develop a set of minimal assumptions, including variation sufficiency and variation consistency, for feasible invariant graph learning.

Recently, there has been great success in applying deep neural networks on graph structured data. Most work, however, focuses on either node-or graph-level supervised learning, such as node, link or graph classification or node-level unsupervised learning (e.g., node clustering). Despite its wide range of possible applications, graph-level unsupervised representation learning has not received much attention yet. This might be mainly attributed to the high representation complexity of graphs, which can be represented by n! equivalent adjacency matrices, where n is the number of nodes. In this work we address this issue by proposing a permutation-invariant variational autoencoder for graph structured data. Our proposed model indirectly learns to match the node order of input and output graph, without imposing a particular node order or performing expensive graph matching. We demonstrate the effectiveness of our proposed model for graph reconstruction, generation and interpolation and evaluate the expressive power of extracted representations for downstream graph-level classification and regression.

Machine learning provides a valuable tool for analyzing high-dimensional functional neuroimaging data, and is proving effective in predicting various neurological conditions, psychiatric disorders, and cognitive patterns. In functional magnetic resonance imaging (MRI) research, interactions between brain regions are commonly modeled using graph-based representations. The potency of graph machine learning methods has been established across myriad domains, marking a transformative step in data interpretation and predictive modeling. Yet, despite their promise, the transposition of these techniques to the neuroimaging domain has been challenging due to the expansive number of potential preprocessing pipelines and the large parameter search space for graph-based dataset construction. In this paper, we introduce NeuroGraph1, a collection of graph-based neuroimaging datasets, and demonstrated its utility for predicting multiple categories of behavioral and cognitive traits.

Recently, graph neural networks [42, 87] emerged as the most prominent (supervised) GRL architectures. Notable instances of this architecture include, e.g., [31, 47, 97], and the spectral approaches proposed in, e.g., [19, 30, 56, 72]--all of which descend from early work in [57, 69, 87, 90]. Recent extensions and improvements to the GNN framework include approaches to incorporate different local structures (around subgraphs), e.g., [2, 36, 50, 82, 106], novel techniques for pooling vertex representations in order perform graph classification, e.g., [20, 37, 108, 113], incorporating distance information [110], and non-euclidian geometry approaches [22]. Moreover, recently empirical studies on neighborhood aggregation functions for continuous vertex features [28], edge-based GNNs leveraging physical knowledge [4, 58], and sparsification methods [85] emerged. A survey of recent advancements in GNN techniques can be found, e.g., in [21, 103, 114]. Dasoulas et al. [29], Abboud et al. [1] investigate the connection between random coloring and universality. Recent works have extended GNNs' expressive power by encoding vertex identifiers [80, 98], adding random features [86], using higher-order topology as features [18], considering simplicial complexes [3, 14], encoding ego-networks [109], and encoding distance information [61]. Although these works increase the expressiveness of GNNs, their generalization abilities are understood to a lesser extent. Further, works such as Vignac et al. [98, Lemma 6] and the most recent Beaini et al. [11] and Bodnar et al. [14] prove the boost in expressiveness with a single pair of graphs, giving no insights into the extent of their expressive power or their generalization abilities. For clarity, throughout this work, we use the term GNNs to denote the class of message-passing architectures limited by the 1-WL algorithm, where the class of distinguishable graphs is well understood [5].

Graph Neural Networks (GNNs) have emerged as a dominant approach in graph representation learning, yet they often struggle to capture consistent similarity relationships among graphs. To capture similarity relationships, while graph kernel methods like the Weisfeiler-Lehman subtree (WL-subtree) and Weisfeiler-Lehman optimal assignment (WLOA) perform effectively, they are heavily reliant on predefined kernels and lack sufficient non-linearities. Our work aims to bridge the gap between neural network methods and kernel approaches by enabling GNNs to consistently capture relational structures in their learned representations. Given the analogy between the message-passing process of GNNs and WL algorithms, we thoroughly compare and analyze the properties of WL-subtree and WLOA kernels. We find that the similarities captured by WLOA at different iterations are asymptotically consistent, ensuring that similar graphs remain similar in subsequent iterations, thereby leading to superior performance over the WL-subtree kernel. Inspired by these findings, we conjecture that the consistency in the similarities of graph representations across GNN layers is crucial in capturing relational structures and enhancing graph classification performance. Thus, we propose a loss to enforce the similarity of graph representations to be consistent across different layers. Our empirical analysis verifies our conjecture and shows that our proposed consistency loss can significantly enhance graph classification performance across several GNN backbones on various datasets.

Most work, however, focuses on either node-or graph-level supervised learning, such as node, link or graph classification or node-level unsupervised learning (e.g., node clustering). Despite its wide range of possible applications, graph-level unsupervised representation learning has not received much attention yet. This might be mainly attributed to the high representation complexity ofgraphs, which can berepresented byn!equivalent adjacencymatrices, where n is the number of nodes. In this work we address this issue by proposing a permutation-invariant variational autoencoder for graph structured data.

Work done during an internship at Tencent AI Lab.