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Master equation for the degree distribution of a Duplication and Divergence network

Author

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  • Sudbrack, Vítor
  • Brunnet, Leonardo G.
  • de Almeida, Rita M.C.
  • Ferreira, Ricardo M.
  • Gamermann, Daniel

Abstract

Network growth as described by the Duplication–Divergence model proposes a simple general idea for the evolution dynamics of natural networks. In particular it is an alternative to the well known Barabási–Albert model when applied to protein–protein interaction networks. In this work we derive a master equation for the node degree distribution of networks growing via Duplication and Divergence and we obtain an expression for the total number of links and for the degree distribution as a function of the number of nodes. Using algebra tools we investigate the degree distribution asymptotic behavior. Analytic results show that the network nodes average degree converges if the total mutation rate is greater than 0.5 and diverges otherwise. Treating original and duplicated node mutation rates as independent parameters has no effect on this result. However, difference in these parameters results in a slower rate of convergence and in different degree distributions. The more different these parameters are, the denser the tail of the distribution. We compare the solutions obtained with simulated networks. These results are in good agreement with the expected values from the derived expressions. The method developed is a robust tool to investigate other models for network growing dynamics.

Suggested Citation

  • Sudbrack, Vítor & Brunnet, Leonardo G. & de Almeida, Rita M.C. & Ferreira, Ricardo M. & Gamermann, Daniel, 2018. "Master equation for the degree distribution of a Duplication and Divergence network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 588-598.
    • Handle: RePEc:eee:phsmap:v:509:y:2018:i:c:p:588-598
    DOI: 10.1016/j.physa.2018.06.066
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    References listed on IDEAS

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    1. Pagani, Giuliano Andrea & Aiello, Marco, 2013. "The Power Grid as a complex network: A survey," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(11), pages 2688-2700.
    2. Ferreira, Ricardo M. & de Almeida, Rita M.C. & Brunnet, Leonardo G., 2017. "Analytic solutions for links and triangles distributions in finite Barabási–Albert networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 466(C), pages 105-110.
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    Cited by:

    1. Gamermann, D. & Triana-Dopico, J. & Jaime, R., 2019. "A comprehensive statistical study of metabolic and protein–protein interaction network properties," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 534(C).

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