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Cycle analysis of Directed Acyclic Graphs

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  • Vasiliauskaite, Vaiva
  • Evans, Tim S.
  • Expert, Paul

Abstract

In this paper, we employ the decomposition of a directed network as an undirected graph plus its associated node meta-data to characterise the cyclic structure found in directed networks by finding a Minimal Cycle Basis of the undirected graph and augmenting 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 meta-data 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 a 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.

Suggested Citation

  • Vasiliauskaite, Vaiva & Evans, Tim S. & Expert, Paul, 2022. "Cycle analysis of Directed Acyclic Graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 596(C).
    • Handle: RePEc:eee:phsmap:v:596:y:2022:i:c:s0378437122001340
    DOI: 10.1016/j.physa.2022.127097
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    References listed on IDEAS

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    1. Gergely Palla & Imre Derényi & Illés Farkas & Tamás Vicsek, 2005. "Uncovering the overlapping community structure of complex networks in nature and society," Nature, Nature, vol. 435(7043), pages 814-818, June.
    2. Michele Coscia, 2018. "Using arborescences to estimate hierarchicalness in directed complex networks," PLOS ONE, Public Library of Science, vol. 13(1), pages 1-18, January.
    3. Clough, James R. & Evans, Tim S., 2016. "What is the dimension of citation space?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 448(C), pages 235-247.
    4. Coscia, Michele, 2018. "Using arborescences to estimate hierarchicalness in directed complex networks," Scholarly Articles 37140312, Harvard Kennedy School of Government.
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    Cited by:

    1. Király, Balázs & Borsos, István & Szabó, György, 2023. "Quantification and statistical analysis of topological features of recursive trees," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 617(C).

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