有向无圈图的圈分析
Cycle Analysis of Directed Acyclic Graphs
摘要：
在本文中，我们采用分解的一个有向网络作为一个无向图加上其相关的节点元数据，通过找到一个最小的无向图的循环基，并增加其组件的方向信息，以恢复在有向网络中发现的循环结构。我们发现，只有四类有向循环存在，它们可以完全区分的组织和数量的源-汇节点对和它们的反链结构。我们对有向无环图特别感兴趣，并介绍了一组使用有向无环图元数据信息来描述最小循环基的度量。特别是，我们用数值方法证明了传递约简稳定了我们引入的度量所测量的最小循环基的性质，同时保留了有向无环图的关键性质。这使得有向无环图和它们所代表的系统的度量一致。我们测量的四个模型的传递性减少有向无环图的最小循环基的特性，并表明，引入的度量是能够区分的模型，并敏感于其生成机制。
英文摘要：
In this paper, we employ the decomposition of a directed network as an undirected graph plus its associated node metadata to characterise the cyclic structure found in directed networks by finding a Minimal Cycle Basis of the undirected graph and augment its components with direction information. We show that only four classes of directed cycles exist, and that they can be fully distinguished by the organisation and number of source-sink node pairs and their antichain structure. We are particularly interested in Directed Acyclic Graphs and introduce a set of metrics that characterise the Minimal Cycle Basis using the Directed Acyclic Graphs metadata information. In particular, we numerically show that Transitive Reduction stabilises the properties of Minimal Cycle Bases measured by the metrics we introduced while retaining key properties of the Directed Acyclic Graph. This makes the metrics consistent characterisation of Directed Acyclic Graphs and the systems they represent. We measure the characteristics of the Minimal Cycle Bases of four models of Transitively Reduced Directed Acyclic Graphs and show that the metrics introduced are able to distinguish the models and are sensitive to their generating mechanisms.
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