Graph autoencoders (GAEs) are self-supervised learning models that can learn meaningful representations of graph-structured data by reconstructing the input graph from a low-dimensional latent space. Over the past few years, GAEs have gained significant attention in academia and industry. In particular, the recent advent of GAEs with masked autoencoding schemes marks a significant advancement in graph self-supervised learning research. While numerous GAEs have been proposed, the underlying mechanisms of GAEs are not well understood, and a comprehensive benchmark for GAEs is still lacking. We revisit the GAEs studied in previous works and demonstrate how contrastive learning principles can be applied to GAEs. Motivated by these insights, we introduce lrGAE (left-right GAE), a general and powerful GAE framework that leverages contrastive learning principles to learn meaningful representations. Our proposed lrGAE not only facilitates a deeper understanding of GAEs but also sets a new benchmark for GAEs across diverse graph-based learning tasks. In the last years, self-supervised learning (SSL) has emerged as a powerful learning paradigm for learning graph representations, approaching, and sometimes even surpassing, the performance of supervised counterparts on many downstream tasks Hjelm et al. (2019); van den Oord et al. (2018). Compared with supervised learning, self-supervised learning gets equal or even better performance with limited or no-labeled data which saves much annotation time and plenty of resources. In a nutshell, SSL purely makes use of rich unlabeled data via well-designed pretext tasks that exploit the underlying structure and patterns in the data. Most recent approaches are shaped by the design of pretext tasks and architectural design, which has led to two lines of research: contrastive and non-contrastive learning Garrido et al. (2023); Balestriero & LeCun (2022). As one of the most successful and widespread SSL strategies, contrastive learning has first shown promising performance in vision representation learning Chen et al. (2020); Gao et al. (2021). It brings together embeddings of different views of the same image while pushing away the embeddings from different ones. Contrastive learning develops rapidly and has recently been applied to the graph learning domain because of the scarcity of graph datasets with labels.

We present a novel framework for learning on CW-complex structured data points. Recent advances have discussed CW-complexes as ideal learning representations for problems in cheminformatics. However, there is a lack of available machine learning methods suitable for learning on CW-complexes. In this paper we develop notions of convolution and attention that are well defined for CW-complexes. These notions enable us to create the first Hodge informed neural network that can receive a CW-complex as input. We illustrate and interpret this framework in the context of supervised prediction.

Graph neural networks (GNNs) have shown great success in learning from graph-based data. The key mechanism of current GNNs is message passing, where a node's feature is updated based on the information passing from its local neighbourhood. A limitation of this mechanism is that node features become increasingly dominated by the information aggregated from the neighbourhood as we use more rounds of message passing. Consequently, as the GNN layers become deeper, adjacent node features tends to be similar, making it more difficult for GNNs to distinguish adjacent nodes, thereby, limiting the performance of GNNs. In this paper, we propose conditional local feature encoding (CLFE) to help prevent the problem of node features being dominated by the information from local neighbourhood. The idea of our method is to extract the node hidden state embedding from message passing process and concatenate it with the nodes feature from previous stage, then we utilise linear transformation to form a CLFE based on the concatenated vector. The CLFE will form the layer output to better preserve node-specific information, thus help to improve the performance of GNN models. To verify the feasibility of our method, we conducted extensive experiments on seven benchmark datasets for four graph domain tasks: super-pixel graph classification, node classification, link prediction, and graph regression. The experimental results consistently demonstrate that our method improves model performance across a variety of baseline GNN models for all four tasks.

Graph Neural Networks (GNNs) are limited in their propagation operators. In many cases, these operators often contain non-negative elements only and are shared across channels, limiting the expressiveness of GNNs. Moreover, some GNNs suffer from over-smoothing, limiting their depth. On the other hand, Convolutional Neural Networks (CNNs) can learn diverse propagation filters, and phenomena like over-smoothing are typically not apparent in CNNs. In this paper, we bridge these gaps by incorporating trainable channel-wise weighting factors $\omega$ to learn and mix multiple smoothing and sharpening propagation operators at each layer. Our generic method is called $\omega$GNN, and is easy to implement. We study two variants: $\omega$GCN and $\omega$GAT. For $\omega$GCN, we theoretically analyse its behaviour and the impact of $\omega$ on the obtained node features. Our experiments confirm these findings, demonstrating and explaining how both variants do not over-smooth. Additionally, we experiment with 15 real-world datasets on node- and graph-classification tasks, where our $\omega$GCN and $\omega$GAT perform on par with state-of-the-art methods.

Graph convolutional networks (GCNs) and its variants are designed for unsigned graphs containing only positive links. Many existing GCNs have been derived from the spectral domain analysis of signals lying over (unsigned) graphs and in each convolution layer they perform low-pass filtering of the input features followed by a learnable linear transformation. Their extension to signed graphs with positive as well as negative links imposes multiple issues including computational irregularities and ambiguous frequency interpretation, making the design of computationally efficient low pass filters challenging. In this paper, we address these issues via spectral analysis of signed graphs and propose two different signed graph neural networks, one keeps only low-frequency information and one also retains high-frequency information. We further introduce magnetic signed Laplacian and use its eigendecomposition for spectral analysis of directed signed graphs. We test our methods for node classification and link sign prediction tasks on signed graphs and achieve state-of-the-art performances.

Graph neural networks (GNNs), which propagate the node features through the edges and learn how to transform the aggregated features under label supervision, have achieved great success in supervised feature extraction for both node-level and graph-level classification tasks. However, GNNs typically treat the graph structure as given and ignore how the edges are formed. This paper introduces a graph generative process to model how the observed edges are generated by aggregating the node interactions over a set of overlapping node communities, each of which contributes to the edges via a logical OR mechanism. Based on this generative model, we partition each edge into the summation of multiple community-specific weighted edges and use them to define community-specific GNNs. A variational inference framework is proposed to jointly learn a GNN based inference network that partitions the edges into different communities, these community-specific GNNs, and a GNN based predictor that combines community-specific GNNs for the end classification task. Extensive evaluations on real-world graph datasets have verified the effectiveness of the proposed method in learning discriminative representations for both node-level and graph-level classification tasks.

These models have recently been successfully applied in a variety of tasks such as computer vision and graphics Monti et al. (2017), recommender systems Ying et al. (2018), transportation Derrow-Pinion et al. (2021), computational chemistry (Gilmer et al., 2017), drug discovery Gaudelet et al. (2021), physics (Shlomi et al., 2020), and analysis of social networks (see Zhou et al. (2019); Bronstein et al. (2021) for additional applications). Several recent works proposed Graph ML models based on differential equations coming from physics Avelar et al. (2019); Poli et al. (2019b); Zhuang et al. (2020); Xhonneux et al. (2020b), including diffusion Chamberlain et al. (2021b) and wave Eliasof et al. (2021) equations and geometric equations such as Beltrami Chamberlain et al. (2021a) and Ricci Topping et al. (2021) flows. Such approaches allow not only to recover popular GNN models as discretization schemes for the underling differential equations, but also, in some cases, can address problems encountered in traditional GNNs such as oversmoothing Nt & Maehara (2019); Oono & Suzuki (2020) and bottlenecks Alon & Yahav (2021). In this paper, we propose a novel physically-inspired approach to learning on graphs. Our framework, termed GraphCON (Graph-Coupled Oscillator Network) builds upon suitable time-discretizations of a specific class of ordinary differential equations (ODEs) that model the dyanmics of a network of non-linear forced and damped oscillators, which are coupled via the adjacency structure of the underlying graph. Graph-coupled oscillators are often encountered in mechanical, electronic, and biological systems, and have been studied extensively Strogatz (2015), with a prominent example being functional circuits in the brain such as cortical columns Stiefel & Ermentrout (2016).

We propose a novel Graph Self-Attention module to enable Transformer models to learn graph representation. We aim to incorporate graph information, on the attention map and hidden representations of Transformer. To this end, we propose context-aware attention which considers the interactions between query, key and graph information. Moreover, we propose graph-embedded value to encode the graph information on the hidden representation. Our extensive experiments and ablation studies validate that our method successfully encodes graph representation on Transformer architecture. Finally, our method achieves state-of-the-art performance on multiple benchmarks of graph representation learning, such as graph classification on images and molecules to graph regression on quantum chemistry.

In many domains data is currently represented as graphs and therefore, the graph representation of this data becomes increasingly important in machine learning. Network data is, implicitly or explicitly, always represented using a graph shift operator (GSO) with the most common choices being the adjacency, Laplacian matrices and their normalisations. In this paper, a novel parametrised GSO (PGSO) is proposed, where specific parameter values result in the most commonly used GSOs and message-passing operators in graph neural network (GNN) frameworks. The PGSO is suggested as a replacement of the standard GSOs that are used in state-of-the-art GNN architectures and the optimisation of the PGSO parameters is seamlessly included in the model training. It is proved that the PGSO has real eigenvalues and a set of real eigenvectors independent of the parameter values and spectral bounds on the PGSO are derived. PGSO parameters are shown to adapt to the sparsity of the graph structure in a study on stochastic blockmodel networks, where they are found to automatically replicate the GSO regularisation found in the literature. On several real-world datasets the accuracy of state-of-theart GNN architectures is improved by the inclusion of the PGSO in both nodeand graph-classification tasks. Graph representation learning has attracted a significant research interest over the last years, mainly due to the structural complexity of real-world data and applications (Hamilton et al., 2017b; Wu et al., 2020). The topology of the observations plays a central role when performing machine learning tasks on graph structured data.

Recent studies have indicated that Graph Convolutional Networks (GCNs) act as a low pass filter in spectral domain and encode smoothed node representations. In this paper, we consider their opposite, namely Graph Deconvolutional Networks (GDNs) that reconstruct graph signals from smoothed node representations. We motivate the design of Graph Deconvolutional Networks via a combination of inverse filters in spectral domain and de-noising layers in wavelet domain, as the inverse operation results in a high pass filter and may amplify the noise. Based on the proposed GDN, we further propose a graph autoencoder framework that first encodes smoothed graph representations with GCN and then decodes accurate graph signals with GDN. We demonstrate the effectiveness of the proposed method on several tasks including unsupervised graph-level representation, social recommendation and graph generation. Autoencoders have demonstrated excellent performance on tasks such as unsupervised representation learning (Bengio, 2009) and de-noising (Vincent et al., 2010). Recently, several studies (Zeiler & Fergus, 2014; Long et al., 2015) have demonstrated that the performance of autoencoders can be further improved by encoding with Convolutional Networks and decoding with Deconvolutional Networks (Zeiler et al., 2010). Notably, Noh et al. (2015) present a novel symmetric architecture that provides a bottom-up mapping from input signals to latent hierarchical feature space with {convolution, pooling} operations and then maps the latent representation back to the input space with {deconvolution, unpooling} operations. While this architecture has been successful when processing features with structures existed in the Euclidean space (e.g., images), recently there has been a surging interest in applying such a framework on non-Euclidean data like graphs.