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Offdiagonal complexity: A computationally quick complexity measure for graphs and networks

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  • Claussen, Jens Christian

Abstract

A vast variety of biological, social, and economical networks shows topologies drastically differing from random graphs; yet the quantitative characterization remains unsatisfactory from a conceptual point of view. Motivated from the discussion of small scale-free networks, a biased link distribution entropy is defined, which takes an extremum for a power-law distribution. This approach is extended to the node–node link cross-distribution, whose nondiagonal elements characterize the graph structure beyond link distribution, cluster coefficient and average path length. From here a simple (and computationally cheap) complexity measure can be defined. This offdiagonal complexity (OdC) is proposed as a novel measure to characterize the complexity of an undirected graph, or network. While both for regular lattices and fully connected networks OdC is zero, it takes a moderately low value for a random graph and shows high values for apparently complex structures as scale-free networks and hierarchical trees. The OdC approach is applied to the Helicobacter pylori protein interaction network and randomly rewired surrogates.

Suggested Citation

  • Claussen, Jens Christian, 2007. "Offdiagonal complexity: A computationally quick complexity measure for graphs and networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 375(1), pages 365-373.
    • Handle: RePEc:eee:phsmap:v:375:y:2007:i:1:p:365-373
    DOI: 10.1016/j.physa.2006.08.067
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    References listed on IDEAS

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    1. Meyer-Ortmanns, Hildegard, 2004. "Functional complexity measure for networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 337(3), pages 679-690.
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    Cited by:

    1. Roanes-Lozano, Eugenio & Laita, Luis M. & Roanes-Macías, Eugenio & Wester, Michael J. & Ruiz-Lozano, José Luis & Roncero, Carlos, 2009. "Evolution of railway network flexibility: The Spanish broad gauge case," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(8), pages 2317-2332.
    2. Kim, Jongkwang & Wilhelm, Thomas, 2008. "What is a complex graph?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(11), pages 2637-2652.
    3. Tuğal, İhsan & Karcı, Ali, 2019. "Comparisons of Karcı and Shannon entropies and their effects on centrality of social networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 352-363.
    4. Frank Emmert-Streib, 2013. "Structural Properties and Complexity of a New Network Class: Collatz Step Graphs," PLOS ONE, Public Library of Science, vol. 8(2), pages 1-14, February.
    5. Lavanya Sivakumar & Matthias Dehmer, 2012. "Towards Information Inequalities for Generalized Graph Entropies," PLOS ONE, Public Library of Science, vol. 7(6), pages 1-14, June.
    6. Vahan Mkrtchyan & Hovhannes Sargsyan, 2018. "A tight lower bound for the hardness of clutters," Journal of Combinatorial Optimization, Springer, vol. 35(1), pages 21-25, January.

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