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Correlation of automorphism group size and topological properties with program-size complexity evaluations of graphs and complex networks

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  • Zenil, Hector
  • Soler-Toscano, Fernando
  • Dingle, Kamaludin
  • Louis, Ard A.

Abstract

We show that numerical approximations of Kolmogorov complexity (K) of graphs and networks capture some group-theoretic and topological properties of empirical networks, ranging from metabolic to social networks, and of small synthetic networks that we have produced. That K and the size of the group of automorphisms of a graph are correlated opens up interesting connections to problems in computational geometry, and thus connects several measures and concepts from complexity science. We derive these results via two different Kolmogorov complexity approximation methods applied to the adjacency matrices of the graphs and networks. The methods used are the traditional lossless compression approach to Kolmogorov complexity, and a normalised version of a Block Decomposition Method (BDM) based on algorithmic probability theory.

Suggested Citation

  • Zenil, Hector & Soler-Toscano, Fernando & Dingle, Kamaludin & Louis, Ard A., 2014. "Correlation of automorphism group size and topological properties with program-size complexity evaluations of graphs and complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 404(C), pages 341-358.
    • Handle: RePEc:eee:phsmap:v:404:y:2014:i:c:p:341-358
    DOI: 10.1016/j.physa.2014.02.060
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    References listed on IDEAS

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    1. Kim, Jongkwang & Wilhelm, Thomas, 2008. "What is a complex graph?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(11), pages 2637-2652.
    2. Matthias Dehmer & Lavanya Sivakumar, 2012. "Recent Developments in Quantitative Graph Theory: Information Inequalities for Networks," PLOS ONE, Public Library of Science, vol. 7(2), pages 1-13, February.
    3. Jean-Paul Delahaye & Hector Zenil, 2012. "Numerical Evaluation of Algorithmic Complexity for Short Strings: A Glance into the Innermost Structure of Randomness," Post-Print hal-00825530, HAL.
    4. H. Jeong & B. Tombor & R. Albert & Z. N. Oltvai & A.-L. Barabási, 2000. "The large-scale organization of metabolic networks," Nature, Nature, vol. 407(6804), pages 651-654, October.
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    Cited by:

    1. Fernando Soler-Toscano & Hector Zenil & Jean-Paul Delahaye & Nicolas Gauvrit, 2014. "Calculating Kolmogorov Complexity from the Output Frequency Distributions of Small Turing Machines," PLOS ONE, Public Library of Science, vol. 9(5), pages 1-18, May.
    2. Fernando Soler-Toscano & Hector Zenil, 2017. "A Computable Measure of Algorithmic Probability by Finite Approximations with an Application to Integer Sequences," Complexity, Hindawi, vol. 2017, pages 1-10, December.

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