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While tokenized graph Transformers have demonstrated strong performance in node classification tasks, their reliance on a limited subset of nodes with high similarity scores for constructing token sequences overlooks valuable information from other nodes, hindering their ability to fully harness graph information for learning optimal node representations. To address this limitation, we propose a novel graph Transformer called GCFormer. Unlike previous approaches, GCFormer develops a hybrid token generator to create two types of token sequences, positive and negative, to capture diverse graph information. And a tailored Transformer-based backbone is adopted to learn meaningful node representations from these generated token sequences. Additionally, GCFormer introduces contrastive learning to extract valuable information from both positive and negative token sequences, enhancing the quality of learned node representations. Extensive experimental results across various datasets, including homophily and heterophily graphs, demonstrate the superiority of GCFormer in node classification, when compared to representative graph neural networks (GNNs) and graph Transformers.

Recently, the emerging graph Transformers have made significant advancements for node classification on graphs. In most graph Transformers, a crucial step involves transforming the input graph into token sequences as the model input, enabling Transformer to effectively learn the node representations. However, we observe that existing methods only express partial graph information of nodes through single-type token generation. Consequently, they require tailored strategies to encode additional graph-specific features into the Transformer to ensure the quality of node representation learning, limiting the model flexibility to handle diverse graphs. To this end, we propose a new graph Transformer called NTFormer to address this issue. NTFormer introduces a novel token generator called Node2Par, which constructs various token sequences using different token elements for each node. This flexibility allows Node2Par to generate valuable token sequences from different perspectives, ensuring comprehensive expression of rich graph features. Benefiting from the merits of Node2Par, NTFormer only leverages a Transformer-based backbone without graph-specific modifications to learn node representations, eliminating the need for graph-specific modifications. Extensive experiments conducted on various benchmark datasets containing homophily and heterophily graphs with different scales demonstrate the superiority of NTFormer over representative graph Transformers and graph neural networks for node classification.

Graph pooling methods have been widely used on downsampling graphs, achieving impressive results on multiple graph-level tasks like graph classification and graph generation. An important line called node dropping pooling aims at exploiting learnable scoring functions to drop nodes with comparatively lower significance scores. However, existing node dropping methods suffer from two limitations: (1) for each pooled node, these models struggle to capture long-range dependencies since they mainly take GNNs as the backbones; (2) pooling only the highest-scoring nodes tends to preserve similar nodes, thus discarding the affluent information of low-scoring nodes. To address these issues, we propose a Graph Transformer Pooling method termed GTPool, which introduces Transformer to node dropping pooling to efficiently capture long-range pairwise interactions and meanwhile sample nodes diversely. Specifically, we design a scoring module based on the self-attention mechanism that takes both global context and local context into consideration, measuring the importance of nodes more comprehensively. GTPool further utilizes a diversified sampling method named Roulette Wheel Sampling (RWS) that is able to flexibly preserve nodes across different scoring intervals instead of only higher scoring nodes. In this way, GTPool could effectively obtain long-range information and select more representative nodes. Extensive experiments on 11 benchmark datasets demonstrate the superiority of GTPool over existing popular graph pooling methods.

The emerging graph Transformers have achieved impressive performance for graph representation learning over graph neural networks (GNNs). In this work, we regard the self-attention mechanism, the core module of graph Transformers, as a two-step aggregation operation on a fully connected graph. Due to the property of generating positive attention values, the self-attention mechanism is equal to conducting a smooth operation on all nodes, preserving the low-frequency information. However, only capturing the low-frequency information is inefficient in learning complex relations of nodes on diverse graphs, such as heterophily graphs where the high-frequency information is crucial. To this end, we propose a Signed Attention-based Graph Transformer (SignGT) to adaptively capture various frequency information from the graphs. Specifically, SignGT develops a new signed self-attention mechanism (SignSA) that produces signed attention values according to the semantic relevance of node pairs. Hence, the diverse frequency information between different node pairs could be carefully preserved. Besides, SignGT proposes a structure-aware feed-forward network (SFFN) that introduces the neighborhood bias to preserve the local topology information. In this way, SignGT could learn informative node representations from both long-range dependencies and local topology information. Extensive empirical results on both node-level and graph-level tasks indicate the superiority of SignGT against state-of-the-art graph Transformers as well as advanced GNNs., Comment: Submitted to a conference

Graph Transformers, emerging as a new architecture for graph representation learning, suffer from the quadratic complexity on the number of nodes when handling large graphs. To this end, we propose a Neighborhood Aggregation Graph Transformer (NAGphormer) that treats each node as a sequence containing a series of tokens constructed by our proposed Hop2Token module. For each node, Hop2Token aggregates the neighborhood features from different hops into different representations, producing a sequence of token vectors as one input. In this way, NAGphormer could be trained in a mini-batch manner and thus could scale to large graphs. Moreover, we mathematically show that compared to a category of advanced Graph Neural Networks (GNNs), called decoupled Graph Convolutional Networks, NAGphormer could learn more informative node representations from multi-hop neighborhoods. In addition, we propose a new data augmentation method called Neighborhood Augmentation (NrAug) based on the output of Hop2Token that augments simultaneously the features of neighborhoods from global as well as local views to strengthen the training effect of NAGphormer. Extensive experiments on benchmark datasets from small to large demonstrate the superiority of NAGphormer against existing graph Transformers and mainstream GNNs, and the effectiveness of NrAug for further boosting NAGphormer., Comment: 14pages, 5 figures. arXiv admin note: text overlap with arXiv:2206.04910

By incorporating the graph structural information into Transformers, graph Transformers have exhibited promising performance for graph representation learning in recent years. Existing graph Transformers leverage specific strategies, such as Laplacian eigenvectors and shortest paths of the node pairs, to preserve the structural features of nodes and feed them into the vanilla Transformer to learn the representations of nodes. It is hard for such predefined rules to extract informative graph structural features for arbitrary graphs whose topology structure varies greatly, limiting the learning capacity of the models. To this end, we propose an adaptive graph Transformer, termed Multi-Neighborhood Attention based Graph Transformer (MNA-GT), which captures the graph structural information for each node from the multi-neighborhood attention mechanism adaptively. By defining the input to perform scaled-dot product as an attention kernel, MNA-GT constructs multiple attention kernels based on different hops of neighborhoods such that each attention kernel can capture specific graph structural information of the corresponding neighborhood for each node pair. In this way, MNA-GT can preserve the graph structural information efficiently by incorporating node representations learned by different attention kernels. MNA-GT further employs an attention layer to learn the importance of different attention kernels to enable the model to adaptively capture the graph structural information for different nodes. Extensive experiments are conducted on a variety of graph benchmarks, and the empirical results show that MNA-GT outperforms many strong baselines., Comment: 8 pages, 4 figures, 5 tables, submitted to a conference of 2023

The decoupled Graph Convolutional Network (GCN), a recent development of GCN that decouples the neighborhood aggregation and feature transformation in each convolutional layer, has shown promising performance for graph representation learning. Existing decoupled GCNs first utilize a simple neural network (e.g., MLP) to learn the hidden features of the nodes, then propagate the learned features on the graph with fixed steps to aggregate the information of multi-hop neighborhoods. Despite effectiveness, the aggregation operation, which requires the whole adjacency matrix as the input, is involved in the model training, causing high training cost that hinders its potential on larger graphs. On the other hand, due to the independence of node attributes as the input, the neural networks used in decoupled GCNs are very simple, and advanced techniques cannot be applied to the modeling. To this end, we further liberate the aggregation operation from the decoupled GCN and propose a new paradigm of GCN, termed Neighborhood Convolutional Network (NCN), that utilizes the neighborhood aggregation result as the input, followed by a special convolutional neural network tailored for extracting expressive node representations from the aggregation input. In this way, the model could inherit the merit of decoupled GCN for aggregating neighborhood information, at the same time, develop much more powerful feature learning modules. A training strategy called mask training is incorporated to further boost the model performance. Extensive results demonstrate the effectiveness of our model for the node classification task on diverse homophilic graphs and heterophilic graphs., Comment: 8 pages, 4 figures, 2 tables, submitted to a conference of 2023

Node classification on attributed networks is a semi-supervised task that is crucial for network analysis. By decoupling two critical operations in Graph Convolutional Networks (GCNs), namely feature transformation and neighborhood aggregation, some recent works of decoupled GCNs could support the information to propagate deeper and achieve advanced performance. However, they follow the traditional structure-aware propagation strategy of GCNs, making it hard to capture the attribute correlation of nodes and sensitive to the structure noise described by edges whose two endpoints belong to different categories. To address these issues, we propose a new method called the itshape Propagation with Adaptive Mask then Training (PAMT). The key idea is to integrate the attribute similarity mask into the structure-aware propagation process. In this way, PAMT could preserve the attribute correlation of adjacent nodes during the propagation and effectively reduce the influence of structure noise. Moreover, we develop an iterative refinement mechanism to update the similarity mask during the training process for improving the training performance. Extensive experiments on four real-world datasets demonstrate the superior performance and robustness of PAMT.

The graph Transformer emerges as a new architecture and has shown superior performance on various graph mining tasks. In this work, we observe that existing graph Transformers treat nodes as independent tokens and construct a single long sequence composed of all node tokens so as to train the Transformer model, causing it hard to scale to large graphs due to the quadratic complexity on the number of nodes for the self-attention computation. To this end, we propose a Neighborhood Aggregation Graph Transformer (NAGphormer) that treats each node as a sequence containing a series of tokens constructed by our proposed Hop2Token module. For each node, Hop2Token aggregates the neighborhood features from different hops into different representations and thereby produces a sequence of token vectors as one input. In this way, NAGphormer could be trained in a mini-batch manner and thus could scale to large graphs. Moreover, we mathematically show that as compared to a category of advanced Graph Neural Networks (GNNs), the decoupled Graph Convolutional Network, NAGphormer could learn more informative node representations from the multi-hop neighborhoods. Extensive experiments on benchmark datasets from small to large are conducted to demonstrate that NAGphormer consistently outperforms existing graph Transformers and mainstream GNNs. Code is available at https://github.com/JHL-HUST/NAGphormer., Comment: Accepted by ICLR 2023

Multi-output Gaussian processes (MOGPs) are an extension of Gaussian Processes (GPs) for predicting multiple output variables (also called channels, tasks) simultaneously. In this paper we use the convolution theorem to design a new kernel for MOGPs, by modeling cross channel dependencies through cross convolution of time and phase delayed components in the spectral domain. The resulting kernel is called Multi-Output Convolution Spectral Mixture (MOCSM) kernel. Results of extensive experiments on synthetic and real-life datasets demonstrate the advantages of the proposed kernel and its state of the art performance. MOCSM enjoys the desirable property to reduce to the well known Spectral Mixture (SM) kernel when a single-channel is considered. A comparison with the recently introduced Multi-Output Spectral Mixture kernel reveals that this is not the case for the latter kernel, which contains quadratic terms that generate undesirable scale effects when the spectral densities of different channels are either very close or very far from each other in the frequency domain., Comment: 12 pages, 48 figures. arXiv admin note: text overlap with arXiv:1808.01132