SIAM Review, Volume 63, Issue 4, Page 823-823, January 2021.
The Education section in this issue presents two papers. The first paper, “Understanding Graph Embedding Methods and Their Applications,” is written by Mengjia Xu. Graph theory is part of discrete mathematics, whose origin is attributed to the paper of Leonhard Euler, “Seven Bridges of Königsberg,” published in 1736. With the rise of data science and machine learning, graphs have found applications in very many diverse fields. Next to applications in knots and group theory, graphs provide a popular modeling framework in reliability theory, in analysis of molecular networks, images, brain networks, protein--protein interaction networks, social networks, banking-institution networks, transportation, computer and wireless networks, and many others. Modern applications of graph theory to large-scale complex networks require efficient methods to store and analyze large graphs. This paper surveys techniques for creating low-dimensional, dense, and possibly continuous models of high-dimensional sparse graphs, so that the structural properties of the graphs are maximally preserved. Three major approaches to graph embedding methods exist in the literature: methods based on matrix factorization, on random walks, or on neural networks. The latter two frameworks are the main focus of this paper. The author outlines the key ideas and steps in the graph embedding techniques, pointing to the articles where the specifics are discussed. Structural preservation is assessed via quantifying proximity preservation, neighbor-based node similarity, similarity based on the adjacency matrix, and other criteria. In most applications, the systems represented by graphs evolve in time. Therefore, the development of dynamic models, which would properly reflect networks changing in time, is of great interest. Some ideas for dynamic embeddings are surveyed in this paper as well. The last part of the paper visits several popular applications: analysis of social networks, evaluation of the impact of scientific papers based on their citations, models of the brain operations, and genomic networks. The paper includes a substantial literature review and would be useful to anyone who looks for an introduction to the numerical techniques for compact representations of large graphs. The second paper, “A Precise and Reliable Multivariable Chain Rule,” by Raymond Boute, is directed to instructors of undergraduate students. It aims to clarify imprecise language and presentations pertaining to the notion of derivatives, to the process of calculating them, as well as to the notation used in this context. The paper pays special attention to derivatives of a composition of functions. A reliable formulation of the multivariable chain rule is presented and illustrated in the context of two examples. The discussion starts with clarifications of the notions of a function, its restriction to a certain domain, and its co-domain. The author analyzes various expressions occurring in the context of differentiation and comments on proper notation and careful delineation between numbers and functional expressions. Further comments pertain to the use of dimensional analysis in verifying the calculations. The examples demonstrate how the chain rule facilitates changing variables without errors and how it conveys structure to the calculations. The author concludes that “adhering to better practices than in the textbook can be problematic for instructors. However, even when some flawed notation is considered `common,' it is in the best interest of the students to provide a better alternative.”

中文翻译：

---

教育

SIAM 评论，第 63 卷，第 4 期，第 823-823 页，2021 年 1 月。
本期的教育部分介绍了两篇论文。第一篇论文“理解图嵌入方法及其应用”由 Mengjia Xu 撰写。图论是离散数学的一部分，其起源归因于 Leonhard Euler 于 1736 年发表的论文“柯尼斯堡的七座桥”。随着数据科学和机器学习的兴起，图已在许多不同的领域中得到应用。除了在结和群论中的应用，图还提供了可靠性理论中流行的建模框架，用于分析分子网络、图像、大脑网络、蛋白质-蛋白质相互作用网络、社交网络、银行-机构网络、交通、计算机和无线网络，以及许多其他。图论在大规模复杂网络中的现代应用需要有效的方法来存储和分析大型图。本文调查了创建高维稀疏图的低维、密集和可能连续模型的技术，以便最大限度地保留图的结构特性。文献中存在三种主要的图嵌入方法：基于矩阵分解、随机游走或神经网络的方法。后两个框架是本文的主要关注点。作者概述了图嵌入技术的关键思想和步骤，并指出了讨论细节的文章。结构保留是通过量化邻近保留、基于邻居的节点相似性、基于邻接矩阵的相似性和其他标准来评估的。在大多数应用程序中，由图表示的系统随时间演变。因此，开发能够正确反映网络随时间变化的动态模型具有重要意义。本文还对动态嵌入的一些想法进行了调查。论文的最后一部分访问了几个流行的应用程序：社交网络分析、基于引用的科学论文影响评估、大脑操作模型和基因组网络。这篇论文包含了大量的文献综述，对于任何想要介绍大图的紧凑表示的数值技术的人来说都是有用的。第二篇论文“精确可靠的多变量链规则”由雷蒙德·鲍特 (Raymond Boute) 撰写，面向本科生的教师。它旨在澄清与导数概念、计算它们的过程以及在此上下文中使用的符号有关的不精确语言和表述。本文特别关注函数组合的导数。在两个例子的上下文中提出并说明了多变量链式法则的可靠公式。讨论首先澄清函数的概念，它对某个域的限制，以及它的共同域。作者分析了微分上下文中出现的各种表达式，并评论了数字和函数表达式之间的正确符号和仔细划分。进一步的评论涉及在验证计算中使用尺寸分析。这些示例演示了链式法则如何促进无错误地更改变量以及它如何将结构传达给计算。作者总结道：“坚持比教科书更好的做法对教师来说可能是个问题。然而，即使某些有缺陷的符号被认为是‘常见的’，提供更好的替代方案也符合学生的最佳利益。”