In recent years, graph neural networks (GNNs) have emerged as a powerful neural architecture to learn vector representations of nodes and graphs in a supervised, end-to-end fashion. Up to now, GNNs have only been evaluated empirically-showing promising results. The following work investigates GNNs from a theoretical point of view and relates them to the 1-dimensional Weisfeiler-Leman graph isomorphism heuristic (1-WL). We show that GNNs have the same expressiveness as the 1-WL in terms of distinguishing non-isomorphic (sub-)graphs. Hence, both algorithms also have the same shortcomings. Based on this, we propose a generalization of GNNs, so-called k-dimensional GNNs (k-GNNs), which can take higher-order graph structures at multiple scales into account. These higher-order structures play an essential role in the characterization of social networks and molecule graphs. Our experimental evaluation confirms our theoretical findings as well as confirms that higher-order information is useful in the task of graph classification and regression.completing the equivalence. Since the power of the 1-WL has been completely characterized, see, e.g., (Arvind et al. 2015;Kiefer, Schweitzer, and Selman 2015), we can transfer these results to the case of GNNs, showing that both approaches have the same shortcomings.Going further, we leverage these theoretical relationships to propose a generalization of GNNs, called k-GNNs, which are neural architectures based on the k-dimensional WL algorithm (k-WL), which are strictly more powerful than GNNs. The key insight in these higher-dimensional variants is that they perform message passing directly between subgraph structures, rather than individual nodes. This higher-order form of message passing can capture structural information that is not visible at the node-level.Graph kernels based on the k-WL have been proposed in the past (Morris, Kersting, and Mutzel 2017). However, a key advantage of implementing higher-order message passing in GNNs-which we demonstrate here-is that we can design hierarchical variants of k-GNNs, which combine graph representations learned at different granularities in an end-to-end trainable framework. Concretely, in the presented hierarchical approach the initial messages in a k-GNN are based on the output of lower-dimensional k -GNN (with k < k), which allows the model to effectively capture graph structures of varying granularity. Many real-world graphs inherit a hierarchical structure-e.g., in a social network we must model both the ego-networks around individual nodes, as well as the coarse-grained relationships between entire communities, see, e.g., (Newman 2003)-and our experimental results demonstrate that these hierarchical k-GNNs are able to consistently outperform traditional GNNs on a variety of graph classification and regression tasks. Across twelve graph regression tasks from the QM9 benchmark, we find that our hierarchical model reduces the mean absolute error by 54.45% on average. For graph classification, we find that our hierarchical models...

We present Spline-based Convolutional Neural Networks (SplineCNNs), a variant of deep neural networks for irregular structured and geometric input, e.g., graphs or meshes. Our main contribution is a novel convolution operator based on B-splines, that makes the computation time independent from the kernel size due to the local support property of the B-spline basis functions. As a result, we obtain a generalization of the traditional CNN convolution operator by using continuous kernel functions parametrized by a fixed number of trainable weights. In contrast to related approaches that filter in the spectral domain, the proposed method aggregates features purely in the spatial domain. In addition, SplineCNN allows entire end-to-end training of deep architectures, using only the geometric structure as input, instead of handcrafted feature descriptors.For validation, we apply our method on tasks from the fields of image graph classification, shape correspondence and graph node classification, and show that it outperforms or pars state-of-the-art approaches while being significantly faster and having favorable properties like domainindependence. Our source code is available on GitHub 1 .Recently, a set of methods brought together under the term geometric deep learning [3] emerged, which aim to achieve this transfer by defining convolution operations for deep neural networks that can handle irregular input data. Existing work in this field can loosely be divided into two different subsets: the spectral and the spatial filtering approaches. The former is based on spectral graph theory [5], where eigenvalues of a graph's Laplacian matrix are interpreted as frequencies of node signals [22]. They are filtered in the spectral domain, analogously to Fourier domain filtering of traditional signals. The latter subset, the spatial approaches, perform convolution in local Euclidean neighborhoods w.r.t. local positional relations between points, represented for example as polar, spherical or Cartesian coordinates, as shown as examples in Figure 1.Contribution. We present Spline-based Convolutional Neural Networks (SplineCNNs), a variant of deep neural networks for irregular structured data. The main contribution is a trainable, spatial, continuous convolution kernel that leverages properties of B-spline bases to efficiently filter geometric input of arbitrary dimensionality. We show

We introduce PyTorch Geometric, a library for deep learning on irregularly structured input data such as graphs, point clouds and manifolds, built upon PyTorch. In addition to general graph data structures and processing methods, it contains a variety of recently published methods from the domains of relational learning and 3D data processing. PyTorch Geometric achieves high data throughput by leveraging sparse GPU acceleration, by providing dedicated CUDA kernels and by introducing efficient mini-batch handling for input examples of different size. In this work, we present the library in detail and perform a comprehensive comparative study of the implemented methods in homogeneous evaluation scenarios.

We present the OPEN GRAPH BENCHMARK (OGB), a diverse set of challenging and realistic benchmark datasets to facilitate scalable, robust, and reproducible graph machine learning (ML) research. OGB datasets are large-scale (up to 100+ million nodes and 1+ billion edges), encompass multiple important graph ML tasks, and cover a diverse range of domains, ranging from social and information networks to biological networks, molecular graphs, source code ASTs, and knowledge graphs. For each dataset, we provide a unified evaluation protocol using meaningful application-specific data splits and evaluation metrics. In addition to building the datasets, we also perform extensive benchmark experiments for each dataset. Our experiments suggest that OGB datasets present significant challenges of scalability to large-scale graphs and out-of-distribution generalization under realistic data splits, indicating fruitful opportunities for future research. Finally, OGB provides an automated end-to-end graph ML pipeline that simplifies and standardizes the process of graph data loading, experimental setup, and model evaluation. OGB will be regularly updated and welcomes inputs from the community. OGB datasets as well as data loaders, evaluation scripts, baseline code, and leaderboards are publicly available at https://ogb.stanford.edu.

Enabling effective and efficient machine learning (ML) over large-scale graph data (e.g., graphs with billions of edges) can have a huge impact on both industrial and scientific applications. However, community efforts to advance large-scale graph ML have been severely limited by the lack of a suitable public benchmark. For KDD Cup 2021, we present OGB Large-Scale Challenge (OGB-LSC), a collection of three realworld datasets for advancing the state-of-the-art in large-scale graph ML. OGB-LSC provides graph datasets that are orders of magnitude larger than existing ones and covers three core graph learning tasks-link prediction, graph regression, and node classification. Furthermore, OGB-LSC provides dedicated baseline experiments, scaling up expressive graph ML models to the massive datasets. We show that the expressive models significantly outperform simple scalable baselines, indicating an opportunity for dedicated efforts to further improve graph ML at scale. Our datasets and baseline code are released and maintained as part of our OGB initiative (Hu et al., 2020a). We hope OGB-LSC at KDD Cup 2021 can empower the community to discover innovative solutions for large-scale graph ML. 1