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Measuring the complexity of directed graphs: A polynomial-based approach.

Matthias Dehmer1,2,3, Zengqiang Chen2, Frank Emmert-Streib4,5

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This study introduces new graph measures for directed networks using graph polynomials and their positive zeros. These efficient measures are applicable to large-scale networks, complementing existing complexity measures.

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Area of Science:

  • Graph theory
  • Network analysis
  • Computational mathematics

Background:

  • Directed networks are prevalent in various scientific domains.
  • Existing complexity measures for directed graphs often lack scalability.
  • There is a need for efficient and scalable graph measures.

Purpose of the Study:

  • To define novel graph measures for directed networks.
  • To utilize graph polynomials based on node degrees.
  • To investigate the properties and efficiency of these new measures.

Main Methods:

  • Definition of graph polynomials using out- and in-degrees.
  • Derivation of graph measures from the positive zeros of these polynomials.
  • Analytical and numerical investigation of measure properties.

Main Results:

  • Novel graph measures based on graph polynomials were successfully defined.
  • The computational complexity of the proposed measures is polynomial, ensuring efficiency.
  • The measures demonstrate meaningful properties investigated through analysis and simulations.

Conclusions:

  • The proposed graph measures offer an efficient and scalable approach for analyzing directed networks.
  • This method complements existing literature by enabling large-scale application.
  • The novel measures provide valuable insights into network structure and complexity.