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What is a complex graph?

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  • Kim, Jongkwang
  • Wilhelm, Thomas

Abstract

Many papers published in recent years show that real-world graphs G(n,m) (n nodes, m edges) are more or less “complex” in the sense that different topological features deviate from random graphs. Here we narrow the definition of graph complexity and argue that a complex graph contains many different subgraphs. We present different measures that quantify this complexity, for instance C1e, the relative number of non-isomorphic one-edge-deleted subgraphs (i.e. DECK size). However, because these different subgraph measures are computationally demanding, we also study simpler complexity measures focussing on slightly different aspects of graph complexity. We consider heuristically defined “product measures”, the products of two quantities which are zero in the extreme cases of a path and clique, and “entropy measures” quantifying the diversity of different topological features. The previously defined network/graph complexity measures Medium Articulation and Offdiagonal complexity (OdC) belong to these two classes. We study OdC measures in some detail and compare it with our new measures. For all measures, the most complex graph GCmax has a medium number of edges, between the edge numbers of the minimum and the maximum connected graph n−1

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  • Kim, Jongkwang & Wilhelm, Thomas, 2008. "What is a complex graph?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(11), pages 2637-2652.
    • Handle: RePEc:eee:phsmap:v:387:y:2008:i:11:p:2637-2652
    DOI: 10.1016/j.physa.2008.01.015
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    Cited by:

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    2. 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.
    3. Mehmet N. Aydin & N. Ziya Perdahci, 0. "Dynamic network analysis of online interactive platform," Information Systems Frontiers, Springer, vol. 0, pages 1-12.
    4. Mikołaj Morzy & Tomasz Kajdanowicz & Przemysław Kazienko, 2017. "On Measuring the Complexity of Networks: Kolmogorov Complexity versus Entropy," Complexity, Hindawi, vol. 2017, pages 1-12, November.
    5. da Cunha, Éverton Fernandes & da Fontoura Costa, Luciano, 2022. "On hypercomplex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 591(C).
    6. Mahmoud Saleh & Yusef Esa & Ahmed Mohamed, 2018. "Applications of Complex Network Analysis in Electric Power Systems," Energies, MDPI, vol. 11(6), pages 1-16, May.
    7. Glover, Fred & Lewis, Mark & Kochenberger, Gary, 2018. "Logical and inequality implications for reducing the size and difficulty of quadratic unconstrained binary optimization problems," European Journal of Operational Research, Elsevier, vol. 265(3), pages 829-842.
    8. Lin, Yun Hui & Wang, Yuan & Lee, Loo Hay & Chew, Ek Peng, 2021. "Consistency matters: Revisiting the structural complexity for supply chain networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 572(C).
    9. 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.
    10. 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.
    11. 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.
    12. Wang, Jiang & Yang, Chen & Wang, Ruofan & Yu, Haitao & Cao, Yibin & Liu, Jing, 2016. "Functional brain networks in Alzheimer’s disease: EEG analysis based on limited penetrable visibility graph and phase space method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 460(C), pages 174-187.
    13. Raducha, Tomasz & Gubiec, Tomasz, 2017. "Coevolving complex networks in the model of social interactions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 427-435.
    14. Mehmet N. Aydin & N. Ziya Perdahci, 2019. "Dynamic network analysis of online interactive platform," Information Systems Frontiers, Springer, vol. 21(2), pages 229-240, April.
    15. Nasrolahzadeh, Mahda & Mohammadpoory, Zeynab & Haddadnia, Javad, 2023. "Indices from visibility graph complexity of spontaneous speech signal: An efficient nonlinear tool for Alzheimer's disease diagnosis," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).

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    Keywords

    Network; Graph; Complexity;
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