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Structural Properties and Complexity of a New Network Class: Collatz Step Graphs

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  • Frank Emmert-Streib

Abstract

In this paper, we introduce a biologically inspired model to generate complex networks. In contrast to many other construction procedures for growing networks introduced so far, our method generates networks from one-dimensional symbol sequences that are related to the so called Collatz problem from number theory. The major purpose of the present paper is, first, to derive a symbol sequence from the Collatz problem, we call the step sequence, and investigate its structural properties. Second, we introduce a construction procedure for growing networks that is based on these step sequences. Third, we investigate the structural properties of this new network class including their finite scaling and asymptotic behavior of their complexity, average shortest path lengths and clustering coefficients. Interestingly, in contrast to many other network models including the small-world network from Watts & Strogatz, we find that CS graphs become ‘smaller’ with an increasing size.

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  • 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.
    • Handle: RePEc:plo:pone00:0056461
    DOI: 10.1371/journal.pone.0056461
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    References listed on IDEAS

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    1. Stanley, H.E. & Buldyrev, S.V. & Goldberger, A.L. & Goldberger, Z.D. & Havlin, S. & Mantegna, R.N. & Ossadnik, S.M. & Peng, C.-K. & Simons, M., 1994. "Statistical mechanics in biology: how ubiquitous are long-range correlations?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 205(1), pages 214-253.
    2. Mark D Humphries & Kevin Gurney, 2008. "Network ‘Small-World-Ness’: A Quantitative Method for Determining Canonical Network Equivalence," PLOS ONE, Public Library of Science, vol. 3(4), pages 1-10, April.
    3. Kim, Jongkwang & Wilhelm, Thomas, 2008. "What is a complex graph?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(11), pages 2637-2652.
    4. 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.
    5. Ebeling, Werner & Neiman, Alexander, 1995. "Long-range correlations between letters and sentences in texts," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 215(3), pages 233-241.
    6. Marcus Kaiser & Claus C Hilgetag, 2006. "Nonoptimal Component Placement, but Short Processing Paths, due to Long-Distance Projections in Neural Systems," PLOS Computational Biology, Public Library of Science, vol. 2(7), pages 1-11, July.
    7. Chandra, Anjan Kumar & Dasgupta, Subinay, 2005. "A small world network of prime numbers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 357(3), pages 436-446.
    8. Frank Emmert-Streib, 2011. "Parametric Construction of Episode Networks from Pseudoperiodic Time Series Based on Mutual Information," PLOS ONE, Public Library of Science, vol. 6(12), pages 1-12, December.
    9. 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.
    10. Frank Emmert-Streib & Matthias Dehmer, 2010. "Influence of the Time Scale on the Construction of Financial Networks," PLOS ONE, Public Library of Science, vol. 5(9), pages 1-9, September.
    11. Frank Emmert-Streib, 2010. "Statistic Complexity: Combining Kolmogorov Complexity with an Ensemble Approach," PLOS ONE, Public Library of Science, vol. 5(8), pages 1-6, August.
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    Cited by:

    1. José A. Tenreiro Machado & Alexandra Galhano & Daniel Cao Labora, 2021. "A Clustering Perspective of the Collatz Conjecture," Mathematics, MDPI, vol. 9(4), pages 1-14, February.

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