Graph representation learning (GRL) methods, such as graph neural networks and graph transformer models, have been successfully used to analyze graph-structured data, mainly focusing on node classification and link prediction tasks. However, the existing studies mostly only consider local connectivity while ignoring long-range connectivity and the roles of nodes. In this paper, we propose Unified Graph Transformer Networks (UGT) that effectively integrate local and global structural information into fixed-length vector representations. First, UGT learns local structure by identifying the local substructures and aggregating features of the $k$-hop neighborhoods of each node. Second, we construct virtual edges, bridging distant nodes with structural similarity to capture the long-range dependencies. Third, UGT learns unified representations through self-attention, encoding structural distance and $p$-step transition probability between node pairs. Furthermore, we propose a self-supervised learning task that effectively learns transition probability to fuse local and global structural features, which could then be transferred to other downstream tasks. Experimental results on real-world benchmark datasets over various downstream tasks showed that UGT significantly outperformed baselines that consist of state-of-the-art models. In addition, UGT reaches the expressive power of the third-order Weisfeiler-Lehman isomorphism test (3d-WL) in distinguishing non-isomorphic graph pairs. The source code is available at https://github.com/NSLab-CUK/Unified-Graph-Transformer.
We propose an extension of the Contextual Graph Markov Model, a deep and probabilistic machine learning model for graphs, to model the distribution of edge features. Our approach is architectural, as we introduce an additional Bayesian network mapping edge features into discrete states to be used by the original model. In doing so, we are also able to build richer graph representations even in the absence of edge features, which is confirmed by the performance improvements on standard graph classification benchmarks. Moreover, we successfully test our proposal in a graph regression scenario where edge features are of fundamental importance, and we show that the learned edge representation provides substantial performance improvements against the original model on three link prediction tasks. By keeping the computational complexity linear in the number of edges, the proposed model is amenable to large-scale graph processing.
Existing studies for applying the mixup technique on graphs mainly focus on graph classification tasks, while the research in node classification is still under-explored. In this paper, we propose a novel mixup augmentation for node classification called Structural Mixup (S-Mixup). The core idea is to take into account the structural information while mixing nodes. Specifically, S-Mixup obtains pseudo-labels for unlabeled nodes in a graph along with their prediction confidence via a Graph Neural Network (GNN) classifier. These serve as the criteria for the composition of the mixup pool for both inter and intra-class mixups. Furthermore, we utilize the edge gradient obtained from the GNN training and propose a gradient-based edge selection strategy for selecting edges to be attached to the nodes generated by the mixup. Through extensive experiments on real-world benchmark datasets, we demonstrate the effectiveness of S-Mixup evaluated on the node classification task. We observe that S-Mixup enhances the robustness and generalization performance of GNNs, especially in heterophilous situations. The source code of S-Mixup can be found at \url{https://github.com/SukwonYun/S-Mixup}
Graph Neural Network (GNN) has demonstrated extraordinary performance in classifying graph properties. However, due to the selection bias of training and testing data (e.g., training on small graphs and testing on large graphs, or training on dense graphs and testing on sparse graphs), distribution deviation is widespread. More importantly, we often observe \emph{hybrid structure distribution shift} of both scale and density, despite of one-sided biased data partition. The spurious correlations over hybrid distribution deviation degrade the performance of previous GNN methods and show large instability among different datasets. To alleviate this problem, we propose \texttt{OOD-GMixup} to jointly manipulate the training distribution with \emph{controllable data augmentation} in metric space. Specifically, we first extract the graph rationales to eliminate the spurious correlations due to irrelevant information. Secondly, we generate virtual samples with perturbation on graph rationale representation domain to obtain potential OOD training samples. Finally, we propose OOD calibration to measure the distribution deviation of virtual samples by leveraging Extreme Value Theory, and further actively control the training distribution by emphasizing the impact of virtual OOD samples. Extensive studies on several real-world datasets on graph classification demonstrate the superiority of our proposed method over state-of-the-art baselines.
Graph Neural Networks (GNNs), despite achieving remarkable performance across different tasks, are theoretically bounded by the 1-Weisfeiler-Lehman test, resulting in limitations in terms of graph expressivity. Even though prior works on topological higher-order GNNs overcome that boundary, these models often depend on assumptions about sub-structures of graphs. Specifically, topological GNNs leverage the prevalence of cliques, cycles, and rings to enhance the message-passing procedure. Our study presents a novel perspective by focusing on simple paths within graphs during the topological message-passing process, thus liberating the model from restrictive inductive biases. We prove that by lifting graphs to path complexes, our model can generalize the existing works on topology while inheriting several theoretical results on simplicial complexes and regular cell complexes. Without making prior assumptions about graph sub-structures, our method outperforms earlier works in other topological domains and achieves state-of-the-art results on various benchmarks.
Implicit Graph Neural Networks (GNNs) have achieved significant success in addressing graph learning problems recently. However, poorly designed implicit GNN layers may have limited adaptability to learn graph metrics, experience over-smoothing issues, or exhibit suboptimal convergence and generalization properties, potentially hindering their practical performance. To tackle these issues, we introduce a geometric framework for designing implicit graph diffusion layers based on a parameterized graph Laplacian operator. Our framework allows learning the metrics of vertex and edge spaces, as well as the graph diffusion strength from data. We show how implicit GNN layers can be viewed as the fixed-point equation of a Dirichlet energy minimization problem and give conditions under which it may suffer from over-smoothing during training (OST) and inference (OSI). We further propose a new implicit GNN model to avoid OST and OSI. We establish that with an appropriately chosen hyperparameter greater than the largest eigenvalue of the parameterized graph Laplacian, DIGNN guarantees a unique equilibrium, quick convergence, and strong generalization bounds. Our models demonstrate better performance than most implicit and explicit GNN baselines on benchmark datasets for both node and graph classification tasks.
Graph classification is a crucial task in many real-world multimedia applications, where graphs can represent various multimedia data types such as images, videos, and social networks. Previous efforts have applied graph neural networks (GNNs) in balanced situations where the class distribution is balanced. However, real-world data typically exhibit long-tailed class distributions, resulting in a bias towards the head classes when using GNNs and limited generalization ability over the tail classes. Recent approaches mainly focus on re-balancing different classes during model training, which fails to explicitly introduce new knowledge and sacrifices the performance of the head classes. To address these drawbacks, we propose a novel framework called Retrieval Augmented Hybrid Network (RAHNet) to jointly learn a robust feature extractor and an unbiased classifier in a decoupled manner. In the feature extractor training stage, we develop a graph retrieval module to search for relevant graphs that directly enrich the intra-class diversity for the tail classes. Moreover, we innovatively optimize a category-centered supervised contrastive loss to obtain discriminative representations, which is more suitable for long-tailed scenarios. In the classifier fine-tuning stage, we balance the classifier weights with two weight regularization techniques, i.e., Max-norm and weight decay. Experiments on various popular benchmarks verify the superiority of the proposed method against state-of-the-art approaches.
Graph Machine Learning (GML) has numerous applications, such as node/graph classification and link prediction, in real-world domains. Providing human-understandable explanations for GML models is a challenging yet fundamental task to foster their adoption, but validating explanations for link prediction models has received little attention. In this paper, we provide quantitative metrics to assess the quality of link prediction explanations, with or without ground-truth. State-of-the-art explainability methods for Graph Neural Networks are evaluated using these metrics. We discuss how underlying assumptions and technical details specific to the link prediction task, such as the choice of distance between node embeddings, can influence the quality of the explanations.
Counterfactual examples have emerged as an effective approach to produce simple and understandable post-hoc explanations. In the context of graph classification, previous work has focused on generating counterfactual explanations by manipulating the most elementary units of a graph, i.e., removing an existing edge, or adding a non-existing one. In this paper, we claim that such language of explanation might be too fine-grained, and turn our attention to some of the main characterizing features of real-world complex networks, such as the tendency to close triangles, the existence of recurring motifs, and the organization into dense modules. We thus define a general density-based counterfactual search framework to generate instance-level counterfactual explanations for graph classifiers, which can be instantiated with different notions of dense substructures. In particular, we show two specific instantiations of this general framework: a method that searches for counterfactual graphs by opening or closing triangles, and a method driven by maximal cliques. We also discuss how the general method can be instantiated to exploit any other notion of dense substructures, including, for instance, a given taxonomy of nodes. We evaluate the effectiveness of our approaches in 7 brain network datasets and compare the counterfactual statements generated according to several widely-used metrics. Results confirm that adopting a semantic-relevant unit of change like density is essential to define versatile and interpretable counterfactual explanation methods.
The isomorphism problem is a fundamental problem in network analysis, which involves capturing both low-order and high-order structural information. In terms of extracting low-order structural information, graph isomorphism algorithms analyze the structural equivalence to reduce the solver space dimension, which demonstrates its power in many applications, such as protein design, chemical pathways, and community detection. For the more commonly occurring high-order relationships in real-life scenarios, the problem of hypergraph isomorphism, which effectively captures these high-order structural relationships, cannot be straightforwardly addressed using graph isomorphism methods. Besides, the existing hypergraph kernel methods may suffer from high memory consumption or inaccurate sub-structure identification, thus yielding sub-optimal performance. In this paper, to address the abovementioned problems, we first propose the hypergraph Weisfiler-Lehman test algorithm for the hypergraph isomorphism test problem by generalizing the Weisfiler-Lehman test algorithm from graphs to hypergraphs. Secondly, based on the presented algorithm, we propose a general hypergraph Weisfieler-Lehman kernel framework and implement two instances, which are Hypergraph Weisfeiler-Lehamn Subtree Kernel and Hypergraph Weisfeiler-Lehamn Hyperedge Kernel. In order to fulfill our research objectives, a comprehensive set of experiments was meticulously designed, including seven graph classification datasets and 12 hypergraph classification datasets. Results on hypergraph classification datasets show significant improvements compared to other typical kernel-based methods, which demonstrates the effectiveness of the proposed methods. In our evaluation, we found that our proposed methods outperform the second-best method in terms of runtime, running over 80 times faster when handling complex hypergraph structures.
Graph neural network (GNN) has been demonstrated powerful in modeling graph-structured data. However, despite many successful cases of applying GNNs to various graph classification and prediction tasks, whether the graph geometrical information has been fully exploited to enhance the learning performance of GNNs is not yet well understood. This paper introduces a new approach to enhance GNN by discrete graph Ricci curvature. Specifically, the graph Ricci curvature defined on the edges of a graph measures how difficult the information transits on one edge from one node to another based on their neighborhoods. Motivated by the geometric analogy of Ricci curvature in the graph setting, we prove that by inserting the curvature information with different carefully designed transformation function $\zeta$, several known computational issues in GNN such as over-smoothing can be alleviated in our proposed model. Furthermore, we verified that edges with very positive Ricci curvature (i.e., $\kappa_{i,j} \approx 1$) are preferred to be dropped to enhance model's adaption to heterophily graph and one curvature based graph edge drop algorithm is proposed. Comprehensive experiments show that our curvature-based GNN model outperforms the state-of-the-art baselines in both homophily and heterophily graph datasets, indicating the effectiveness of involving graph geometric information in GNNs.
The Dissemination Process Classification (DPC) is a popular application of temporal graph classification. The aim of DPC is to classify different spreading patterns of information or pestilence within a community represented by discrete-time temporal graphs. Recently, a reservoir computing-based model named Dynamical Graph Echo State Network (DynGESN) has been proposed for processing temporal graphs with relatively high effectiveness and low computational costs. In this study, we propose a novel model which combines a novel data augmentation strategy called snapshot merging with the DynGESN for dealing with DPC tasks. In our model, the snapshot merging strategy is designed for forming new snapshots by merging neighboring snapshots over time, and then multiple reservoir encoders are set for capturing spatiotemporal features from merged snapshots. After those, the logistic regression is adopted for decoding the sum-pooled embeddings into the classification results. Experimental results on six benchmark DPC datasets show that our proposed model has better classification performances than the DynGESN and several kernel-based models.
Traditional convolutional neural networks are limited to handling Euclidean space data, overlooking the vast realm of real-life scenarios represented as graph data, including transportation networks, social networks, and reference networks. The pivotal step in transferring convolutional neural networks to graph data analysis and processing lies in the construction of graph convolutional operators and graph pooling operators. This comprehensive review article delves into the world of graph convolutional neural networks. Firstly, it elaborates on the fundamentals of graph convolutional neural networks. Subsequently, it elucidates the graph neural network models based on attention mechanisms and autoencoders, summarizing their application in node classification, graph classification, and link prediction along with the associated datasets.
Brain graphs, which model the structural and functional relationships between brain regions, are crucial in neuroscientific and clinical applications involving graph classification. However, dense brain graphs pose computational challenges including high runtime and memory usage and limited interpretability. In this paper, we investigate effective designs in Graph Neural Networks (GNNs) to sparsify brain graphs by eliminating noisy edges. While prior works remove noisy edges based on explainability or task-irrelevant properties, their effectiveness in enhancing performance with sparsified graphs is not guaranteed. Moreover, existing approaches often overlook collective edge removal across multiple graphs. To address these issues, we introduce an iterative framework to analyze different sparsification models. Our findings are as follows: (i) methods prioritizing interpretability may not be suitable for graph sparsification as they can degrade GNNs' performance in graph classification tasks; (ii) simultaneously learning edge selection with GNN training is more beneficial than post-training; (iii) a shared edge selection across graphs outperforms separate selection for each graph; and (iv) task-relevant gradient information aids in edge selection. Based on these insights, we propose a new model, Interpretable Graph Sparsification (IGS), which enhances graph classification performance by up to 5.1% with 55.0% fewer edges. The retained edges identified by IGS provide neuroscientific interpretations and are supported by well-established literature.
Recent research has shown that alignment between the structure of graph data and the geometry of an embedding space is crucial for learning high-quality representations of the data. The uniform geometry of Euclidean and hyperbolic spaces allows for representing graphs with uniform geometric and topological features, such as grids and hierarchies, with minimal distortion. However, real-world graph data is characterized by multiple types of geometric and topological features, necessitating more sophisticated geometric embedding spaces. In this work, we utilize the Riemannian symmetric space of symmetric positive definite matrices (SPD) to construct graph neural networks that can robustly handle complex graphs. To do this, we develop an innovative library that leverages the SPD gyrocalculus tools \cite{lopez2021gyroSPD} to implement the building blocks of five popular graph neural networks in SPD. Experimental results demonstrate that our graph neural networks in SPD substantially outperform their counterparts in Euclidean and hyperbolic spaces, as well as the Cartesian product thereof, on complex graphs for node and graph classification tasks. We release the library and datasets at \url{https://github.com/andyweizhao/SPD4GNNs}.
Autonomous individuals establish a structural complex system through pairwise connections and interactions. Notably, the evolution reflects the dynamic nature of each complex system since it recodes a series of temporal changes from the past, the present into the future. Different systems follow distinct evolutionary trajectories, which can serve as distinguishing traits for system classification. However, modeling a complex system's evolution is challenging for the graph model because the graph is typically a snapshot of the static status of a system, and thereby hard to manifest the long-term evolutionary traits of a system entirely. To address this challenge, we suggest utilizing a heat-driven method to generate temporal graph augmentation. This approach incorporates the physics-based heat kernel and DropNode technique to transform each static graph into a sequence of temporal ones. This approach effectively describes the evolutional behaviours of the system, including the retention or disappearance of elements at each time point based on the distributed heat on each node. Additionally, we propose a dynamic time-wrapping distance GDTW to quantitatively measure the distance between pairwise evolutionary systems through optimal matching. The resulting approach, called the Evolution Kernel method, has been successfully applied to classification problems in real-world structural graph datasets. The results yield significant improvements in supervised classification accuracy over a series of baseline methods.
A pooling operation is essential for effective graph-level representation learning, where the node drop pooling has become one mainstream graph pooling technology. However, current node drop pooling methods usually keep the top-k nodes according to their significance scores, which ignore the graph diversity in terms of the node features and the graph structures, thus resulting in suboptimal graph-level representations. To address the aforementioned issue, we propose a novel plug-and-play score scheme and refer to it as MID, which consists of a \textbf{M}ultidimensional score space with two operations, \textit{i.e.}, fl\textbf{I}pscore and \textbf{D}ropscore. Specifically, the multidimensional score space depicts the significance of nodes through multiple criteria; the flipscore encourages the maintenance of dissimilar node features; and the dropscore forces the model to notice diverse graph structures instead of being stuck in significant local structures. To evaluate the effectiveness of our proposed MID, we perform extensive experiments by applying it to a wide variety of recent node drop pooling methods, including TopKPool, SAGPool, GSAPool, and ASAP. Specifically, the proposed MID can efficiently and consistently achieve about 2.8\% average improvements over the above four methods on seventeen real-world graph classification datasets, including four social datasets (IMDB-BINARY, IMDB-MULTI, REDDIT-BINARY, and COLLAB), and thirteen biochemical datasets (D\&D, PROTEINS, NCI1, MUTAG, PTC-MR, NCI109, ENZYMES, MUTAGENICITY, FRANKENSTEIN, HIV, BBBP, TOXCAST, and TOX21). Code is available at~\url{https://github.com/whuchuang/mid}.
Certifying the robustness of a graph-based machine learning model poses a critical challenge for safety. Current robustness certificates for graph classifiers guarantee output invariance with respect to the total number of node pair flips (edge addition or edge deletion), which amounts to an $l_{0}$ ball centred on the adjacency matrix. Although theoretically attractive, this type of isotropic structural noise can be too restrictive in practical scenarios where some node pairs are more critical than others in determining the classifier's output. The certificate, in this case, gives a pessimistic depiction of the robustness of the graph model. To tackle this issue, we develop a randomised smoothing method based on adding an anisotropic noise distribution to the input graph structure. We show that our process generates structural-aware certificates for our classifiers, whereby the magnitude of robustness certificates can vary across different pre-defined structures of the graph. We demonstrate the benefits of these certificates in both synthetic and real-world experiments.
Graph neural networks (GNNs) have emerged as a powerful model to capture critical graph patterns. Instead of treating them as black boxes in an end-to-end fashion, attempts are arising to explain the model behavior. Existing works mainly focus on local interpretation to reveal the discriminative pattern for each individual instance, which however cannot directly reflect the high-level model behavior across instances. To gain global insights, we aim to answer an important question that is not yet well studied: how to provide a global interpretation for the graph learning procedure? We formulate this problem as globally interpretable graph learning, which targets on distilling high-level and human-intelligible patterns that dominate the learning procedure, such that training on this pattern can recover a similar model. As a start, we propose a novel model fidelity metric, tailored for evaluating the fidelity of the resulting model trained on interpretations. Our preliminary analysis shows that interpretative patterns generated by existing global methods fail to recover the model training procedure. Thus, we further propose our solution, Graph Distribution Matching (GDM), which synthesizes interpretive graphs by matching the distribution of the original and interpretive graphs in the GNN's feature space as its training proceeds, thus capturing the most informative patterns the model learns during training. Extensive experiments on graph classification datasets demonstrate multiple advantages of the proposed method, including high model fidelity, predictive accuracy and time efficiency, as well as the ability to reveal class-relevant structure.
Graph structure expression plays a vital role in distinguishing various graphs. In this work, we propose a Structure-Sensitive Graph Dictionary Embedding (SS-GDE) framework to transform input graphs into the embedding space of a graph dictionary for the graph classification task. Instead of a plain use of a base graph dictionary, we propose the variational graph dictionary adaptation (VGDA) to generate a personalized dictionary (named adapted graph dictionary) for catering to each input graph. In particular, for the adaptation, the Bernoulli sampling is introduced to adjust substructures of base graph keys according to each input, which increases the expression capacity of the base dictionary tremendously. To make cross-graph measurement sensitive as well as stable, multi-sensitivity Wasserstein encoding is proposed to produce the embeddings by designing multi-scale attention on optimal transport. To optimize the framework, we introduce mutual information as the objective, which further deduces to variational inference of the adapted graph dictionary. We perform our SS-GDE on multiple datasets of graph classification, and the experimental results demonstrate the effectiveness and superiority over the state-of-the-art methods.