Cycle Analysis of Directed Acyclic Graphs

TL;DR

This paper analyzes the cyclic structure of directed networks by decomposing them into undirected graphs with metadata, identifying four classes of cycles, and introducing metrics to characterize DAGs and their minimal cycle bases.

Contribution

It introduces a novel approach to characterize directed acyclic graphs using minimal cycle bases and metrics, highlighting the stabilizing effect of transitive reduction.

Findings

Four classes of directed cycles identified.

Metrics effectively distinguish different DAG models.

Transitive reduction stabilizes minimal cycle basis properties.

Abstract

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.