Title: Breadth, Depth, and Flux of Course-Prerequisite Networks Show Chinese title

Title: 课程先修条件网络的广度、深度和通量

Abstract: Course-prerequisite networks (CPNs) are directed acyclic graphs that model complex academic curricula by representing courses as nodes and dependencies between them as directed links. These networks are indispensable tools for visualizing, studying, and understanding curricula. For example, CPNs can be used to detect important courses, improve advising, guide curriculum design, analyze graduation time distributions, and quantify the strength of knowledge flow between different university departments. However, most CPN analyses to date have focused only on micro- and meso-scale properties. To fill this gap, we define and study three new global CPN measures: breadth, depth, and flux. All three measures are invariant under transitive reduction and are based on the concept of topological stratification, which generalizes topological ordering in directed acyclic graphs. These measures can be used for macro-scale comparison of different CPNs. We illustrate the new measures numerically by applying them to three real and synthetic CPNs from three universities: the Cyprus University of Technology, the California Institute of Technology, and Johns Hopkins University. The CPN data analyzed in this paper are publicly available in a GitHub repository. Show Chinese abstract

Abstract: 课程先修网络（CPNs）是有向无环图，通过将课程表示为节点，它们之间的依赖关系表示为有向链接来建模复杂的学术课程。 这些网络是可视化、研究和理解课程不可或缺的工具。 例如，CPNs可用于检测重要的课程，改进指导，指导课程设计，分析毕业时间分布，并量化不同大学院系之间知识流的强度。 然而，迄今为止大多数CPN分析仅集中在微观和介观尺度的特性上。 为了弥补这一差距，我们定义并研究了三种新的全局CPN度量：广度、深度和通量。 所有三个度量在传递约简下都是不变的，并且基于拓扑分层的概念，该概念推广了有向无环图中的拓扑排序。 这些度量可用于不同CPN的宏观尺度比较。 我们通过将它们应用于来自三所大学的三个真实和合成的CPN来数值地说明这些新度量： 塞浦路斯技术大学，加州理工学院， 约翰霍普金斯大学。 本文中分析的CPN数据可在GitHub仓库中公开获得。