近年来，基于Weisfeiler-Leman算法的算法和神经架构，是图形同构的着名启发式问题，它被成为具有图形和关系数据的（监督）机器学习的强大工具。在这里，我们全面概述了机器学习设置中的算法使用。我们讨论了理论背景，展示了如何将其用于监督的图形和节点分类，讨论最近的扩展，以及其与神经结构的连接。此外，我们概述了当前的应用和未来方向，以刺激研究。

近年来，基于Weisfeiler-Leman算法的算法和神经架构，是一个众所周知的Graph同构问题的启发式问题，它成为具有图形和关系数据的机器学习的强大工具。在这里，我们全面概述了机器学习设置中的算法的使用，专注于监督的制度。我们讨论了理论背景，展示了如何将其用于监督的图形和节点表示学习，讨论最近的扩展，并概述算法的连接（置换 - ）方面的神经结构。此外，我们概述了当前的应用和未来方向，以刺激进一步的研究。

近年来，图形神经网络（GNNS）被出现为一个强大的神经结构，以学习在监督的端到端时尚中的节点和图表的矢量表示。到目前为止，只有经验评估GNNS - 显示有希望的结果。以下工作从理论的角度调查了GNN，并将它们与1美元 - 二维韦斯美犬 - Leman Graph同构Heuristic（1美元-WL）相关联。我们表明GNNS在区分非同义（子）图表中，GNN具有与1美元-WL相同的表现力。因此，这两种算法也具有相同的缺点。基于此，我们提出了GNN的概括，所谓的$ k $ -dimensional gnns（$ k $ -gnns），这可以考虑多个尺度的高阶图结构。这些高阶结构在社交网络和分子图的表征中起重要作用。我们的实验评估证实了我们的理论调查结果，并确认了更高阶信息在图形分类和回归的任务中有用。

尽管（消息通话）图形神经网络在图形或一般关系数据上近似置换量等函数方面具有明显的局限性，但更具表现力的高阶图神经网络不会扩展到大图。他们要么在$ k $ - 订单张量子上操作，要么考虑所有$ k $ - 节点子图，这意味着在内存需求中对$ k $的指数依赖，并且不适合图形的稀疏性。通过为图同构问题引入新的启发式方法，我们设计了一类通用的，置换式的图形网络，与以前的体系结构不同，该网络在表达性和可伸缩性之间提供了细粒度的控制，并适应了图的稀疏性。这些体系结构与监督节点和图形级别的标准高阶网络以及回归体系中的标准高阶图网络相比大大减少了计算时间，同时在预测性能方面显着改善了标准图神经网络和图形内核体系结构。

在过去十年中，图形内核引起了很多关注，并在结构化数据上发展成为一种快速发展的学习分支。在过去的20年中，该领域发生的相当大的研究活动导致开发数十个图形内核，每个图形内核都对焦于图形的特定结构性质。图形内核已成功地成功地在广泛的域中，从社交网络到生物信息学。本调查的目标是提供图形内核的文献的统一视图。特别是，我们概述了各种图形内核。此外，我们对公共数据集的几个内核进行了实验评估，并提供了比较研究。最后，我们讨论图形内核的关键应用，并概述了一些仍有待解决的挑战。

最近出现了许多子图增强图神经网络（GNN），可证明增强了标准（消息通话）GNN的表达能力。但是，对这些方法之间的相互关系和weisfeiler层次结构的关系有限。此外，当前的方法要么使用给定尺寸的所有子图，要随机均匀地对其进行采样，或者使用手工制作的启发式方法，而不是学习以数据驱动的方式选择子图。在这里，我们提供了一种统一的方法来研究此类体系结构，通过引入理论框架并扩展了亚图增强GNN的已知表达结果。具体而言，我们表明，增加子图的大小总是会增加表达能力，并通过将它们与已建立的$ k \ text { - } \ Mathsf {Wl} $ hierArchy联系起来，从而更好地理解其局限性。此外，我们还使用最近通过复杂的离散概率分布进行反向传播的方法探索了学习对子图进行采样的不同方法。从经验上讲，我们研究了不同子图增强的GNN的预测性能，表明我们的数据驱动体系结构与非DATA驱动的亚图增强图形神经网络相比，在标准基准数据集上提高了对标准基准数据集的预测准确性，同时减少了计算时间。

Knowledge graphs, modeling multi-relational data, improve numerous applications such as question answering or graph logical reasoning. Many graph neural networks for such data emerged recently, often outperforming shallow architectures. However, the design of such multi-relational graph neural networks is ad-hoc, driven mainly by intuition and empirical insights. Up to now, their expressivity, their relation to each other, and their (practical) learning performance is poorly understood. Here, we initiate the study of deriving a more principled understanding of multi-relational graph neural networks. Namely, we investigate the limitations in the expressive power of the well-known Relational GCN and Compositional GCN architectures and shed some light on their practical learning performance. By aligning both architectures with a suitable version of the Weisfeiler-Leman test, we establish under which conditions both models have the same expressive power in distinguishing non-isomorphic (multi-relational) graphs or vertices with different structural roles. Further, by leveraging recent progress in designing expressive graph neural networks, we introduce the $k$-RN architecture that provably overcomes the expressiveness limitations of the above two architectures. Empirically, we confirm our theoretical findings in a vertex classification setting over small and large multi-relational graphs.

消息传递神经网络（MPNNS）是由于其简单性和可扩展性而大部分地进行图形结构数据的深度学习的领先架构。不幸的是，有人认为这些架构的表现力有限。本文提出了一种名为Comifariant Subgraph聚合网络（ESAN）的新颖框架来解决这个问题。我们的主要观察是，虽然两个图可能无法通过MPNN可区分，但它们通常包含可区分的子图。因此，我们建议将每个图形作为由某些预定义策略导出的一组子图，并使用合适的等分性架构来处理它。我们为图同构同构同构造的1立维Weisfeiler-Leman（1-WL）测试的新型变体，并在这些新的WL变体方面证明了ESAN的表达性下限。我们进一步证明，我们的方法增加了MPNNS和更具表现力的架构的表现力。此外，我们提供了理论结果，描述了设计选择诸如子图选择政策和等效性神经结构的设计方式如何影响我们的架构的表现力。要处理增加的计算成本，我们提出了一种子图采样方案，可以将其视为我们框架的随机版本。关于真实和合成数据集的一套全面的实验表明，我们的框架提高了流行的GNN架构的表现力和整体性能。

In recent years, graph neural networks (GNNs) have emerged as a promising tool for solving machine learning problems on graphs. Most GNNs are members of the family of message passing neural networks (MPNNs). There is a close connection between these models and the Weisfeiler-Leman (WL) test of isomorphism, an algorithm that can successfully test isomorphism for a broad class of graphs. Recently, much research has focused on measuring the expressive power of GNNs. For instance, it has been shown that standard MPNNs are at most as powerful as WL in terms of distinguishing non-isomorphic graphs. However, these studies have largely ignored the distances between the representations of nodes/graphs which are of paramount importance for learning tasks. In this paper, we define a distance function between nodes which is based on the hierarchy produced by the WL algorithm, and propose a model that learns representations which preserve those distances between nodes. Since the emerging hierarchy corresponds to a tree, to learn these representations, we capitalize on recent advances in the field of hyperbolic neural networks. We empirically evaluate the proposed model on standard node and graph classification datasets where it achieves competitive performance with state-of-the-art models.

Pre-publication draft of a book to be published byMorgan & Claypool publishers. Unedited version released with permission. All relevant copyrights held by the author and publisher extend to this pre-publication draft.

Machine learning on graphs is an important and ubiquitous task with applications ranging from drug design to friendship recommendation in social networks. The primary challenge in this domain is finding a way to represent, or encode, graph structure so that it can be easily exploited by machine learning models. Traditionally, machine learning approaches relied on user-defined heuristics to extract features encoding structural information about a graph (e.g., degree statistics or kernel functions). However, recent years have seen a surge in approaches that automatically learn to encode graph structure into low-dimensional embeddings, using techniques based on deep learning and nonlinear dimensionality reduction. Here we provide a conceptual review of key advancements in this area of representation learning on graphs, including matrix factorization-based methods, random-walk based algorithms, and graph neural networks. We review methods to embed individual nodes as well as approaches to embed entire (sub)graphs. In doing so, we develop a unified framework to describe these recent approaches, and we highlight a number of important applications and directions for future work.

Graph Neural Networks (GNNs) are an effective framework for representation learning of graphs. GNNs follow a neighborhood aggregation scheme, where the representation vector of a node is computed by recursively aggregating and transforming representation vectors of its neighboring nodes. Many GNN variants have been proposed and have achieved state-of-the-art results on both node and graph classification tasks. However, despite GNNs revolutionizing graph representation learning, there is limited understanding of their representational properties and limitations. Here, we present a theoretical framework for analyzing the expressive power of GNNs to capture different graph structures. Our results characterize the discriminative power of popular GNN variants, such as Graph Convolutional Networks and GraphSAGE, and show that they cannot learn to distinguish certain simple graph structures. We then develop a simple architecture that is provably the most expressive among the class of GNNs and is as powerful as the Weisfeiler-Lehman graph isomorphism test. We empirically validate our theoretical findings on a number of graph classification benchmarks, and demonstrate that our model achieves state-of-the-art performance. * Equal contribution. † Work partially performed while in Tokyo, visiting Prof. Ken-ichi Kawarabayashi.

Deep learning has revolutionized many machine learning tasks in recent years, ranging from image classification and video processing to speech recognition and natural language understanding. The data in these tasks are typically represented in the Euclidean space. However, there is an increasing number of applications where data are generated from non-Euclidean domains and are represented as graphs with complex relationships and interdependency between objects. The complexity of graph data has imposed significant challenges on existing machine learning algorithms. Recently, many studies on extending deep learning approaches for graph data have emerged. In this survey, we provide a comprehensive overview of graph neural networks (GNNs) in data mining and machine learning fields. We propose a new taxonomy to divide the state-of-the-art graph neural networks into four categories, namely recurrent graph neural networks, convolutional graph neural networks, graph autoencoders, and spatial-temporal graph neural networks. We further discuss the applications of graph neural networks across various domains and summarize the open source codes, benchmark data sets, and model evaluation of graph neural networks. Finally, we propose potential research directions in this rapidly growing field.

Recently, there has been an increasing interest in (supervised) learning with graph data, especially using graph neural networks. However, the development of meaningful benchmark datasets and standardized evaluation procedures is lagging, consequently hindering advancements in this area. To address this, we introduce the TUDATASET for graph classification and regression. The collection consists of over 120 datasets of varying sizes from a wide range of applications. We provide Python-based data loaders, kernel and graph neural network baseline implementations, and evaluation tools. Here, we give an overview of the datasets, standardized evaluation procedures, and provide baseline experiments. All datasets are available at www.graphlearning.io. The experiments are fully reproducible from the code available at www.github.com/chrsmrrs/tudataset.

消息传递神经网络（MPNNs）是格拉夫神经网络（GNN）的一个常见的类型，其中，每个节点的表示是通过聚集从表示其直接邻居（消息）类似于一个星形图案递归计算。 MPNNs的呼吁是有效的，可扩展的，怎么样，曾经它们的表现是由一阶Weisfeiler雷曼同构测试（1-WL）的上界。对此，之前的作品提出在可扩展性的成本极富表现力的模型，有时泛化性能。我们的工作表示这两个政权：我们介绍抬升任何MPNN更加传神，具有可扩展性有限的开销，大大提高了实用性能的总体框架。我们从星星图案一般的子模式（例如，K-egonets）在MPNNs扩展本地聚合实现这一点：在我们的框架中，每个节点表示被计算为周边诱发子的编码，而不是唯一的近邻编码（即一个明星）。我们选择子编码器是一个GNN（主要是MPNNs，考虑到可扩展性）来设计用作一个包装掀任何GNN的总体框架。我们把我们提出的方法GNN-AK（GNN为核心），作为框架用GNNS更换内核类似于卷积神经网络。从理论上讲，我们表明，我们的框架比1和2-WL确实更强大，并且不超过3-WL那么强大。我们还设计子取样策略，可大大降低内存占用和提高速度的同时保持性能。我们的方法将大利润率多家知名图形ML任务新的国家的最先进的性能;具体地，0.08 MAE锌，74.79％和86.887％的准确度上CIFAR10和分别PATTERN。

许多现代神经架构的核心的卷积运算符可以有效地被视为在输入矩阵和滤波器之间执行点产品。虽然这很容易适用于诸如图像的数据，其可以在欧几里德空间中表示为常规网格，延伸卷积操作者以在图形上工作，而是由于它们的不规则结构而被证明更具有挑战性。在本文中，我们建议使用图形内部产品的图形内核，即在图形上计算内部产品，以将标准卷积运算符扩展到图形域。这使我们能够定义不需要计算输入图的嵌入的完全结构模型。我们的架构允许插入任何类型和数量的图形内核，并具有在培训过程中学到的结构面具方面提供一些可解释性的额外益处，类似于传统卷积神经网络中的卷积掩模发生的事情。我们执行广泛的消融研究，调查模型超参数的影响，我们表明我们的模型在标准图形分类数据集中实现了竞争性能。

组合优化是运营研究和计算机科学领域的一个公认领域。直到最近，它的方法一直集中在孤立地解决问题实例，而忽略了它们通常源于实践中的相关数据分布。但是，近年来，人们对使用机器学习，尤其是图形神经网络（GNN）的兴趣激增，作为组合任务的关键构件，直接作为求解器或通过增强确切的求解器。GNN的电感偏差有效地编码了组合和关系输入，因为它们对排列和对输入稀疏性的意识的不变性。本文介绍了对这个新兴领域的最新主要进步的概念回顾，旨在优化和机器学习研究人员。

随着图表和图表学习的开发，已经提出了许多优越的方法来处理图形结构学习的可扩展性和过度厚度问题。但是，大多数策略都是基于实践经验而不是理论分析而设计的。在本文中，我们使用连接到所有现有顶点的特定虚拟节点，而不会影响原始顶点和边缘属性。我们进一步证明，这种虚拟节点可以帮助构建有效的单态边缘到vertex变换，并呈现呈呈倒数，以恢复原始图。这也表明，添加虚拟节点可以保留本地和全局结构，以更好地图表表示。我们扩展了具有虚拟节点的图形内核和图形神经网络，并在图形分类和子图同构匹配任务上进行实验。经验结果表明，以虚拟节点为输入的图表显着增强了图形结构学习，并且使用其边缘到vertex图也可以实现相似的结果。我们还讨论了神经网络中假人的表达能力的增长。

图形内核是历史上最广泛使用的图形分类任务的技术。然而，由于图的手工制作的组合特征，这些方法具有有限的性能。近年来，由于其性能卓越，图形神经网络（GNNS）已成为与下游图形相关任务的最先进的方法。大多数GNN基于消息传递神经网络（MPNN）框架。然而，最近的研究表明，MPNN不能超过Weisfeiler-Lehman（WL）算法在图形同构术中的力量。为了解决现有图形内核和GNN方法的限制，在本文中，我们提出了一种新的GNN框架，称为\ Texit {内核图形神经网络}（Kernnns），该框架将图形内核集成到GNN的消息传递过程中。通过卷积神经网络（CNNS）中的卷积滤波器的启发，KERGNNS采用可训练的隐藏图作为绘图过滤器，该绘图过滤器与子图组合以使用图形内核更新节点嵌入式。此外，我们表明MPNN可以被视为Kergnns的特殊情况。我们将Kergnns应用于多个与图形相关的任务，并使用交叉验证来与基准进行公平比较。我们表明，与现有的现有方法相比，我们的方法达到了竞争性能，证明了增加GNN的表现能力的可能性。我们还表明，KERGNNS中的训练有素的图形过滤器可以揭示数据集的本地图形结构，与传统GNN模型相比，显着提高了模型解释性。

Graphs are ubiquitous in nature and can therefore serve as models for many practical but also theoretical problems. For this purpose, they can be defined as many different types which suitably reflect the individual contexts of the represented problem. To address cutting-edge problems based on graph data, the research field of Graph Neural Networks (GNNs) has emerged. Despite the field's youth and the speed at which new models are developed, many recent surveys have been published to keep track of them. Nevertheless, it has not yet been gathered which GNN can process what kind of graph types. In this survey, we give a detailed overview of already existing GNNs and, unlike previous surveys, categorize them according to their ability to handle different graph types and properties. We consider GNNs operating on static and dynamic graphs of different structural constitutions, with or without node or edge attributes. Moreover, we distinguish between GNN models for discrete-time or continuous-time dynamic graphs and group the models according to their architecture. We find that there are still graph types that are not or only rarely covered by existing GNN models. We point out where models are missing and give potential reasons for their absence.