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Degree distribution and assortativity in line graphs of complex networks

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  • Wang, Xiangrong
  • Trajanovski, Stojan
  • Kooij, Robert E.
  • Van Mieghem, Piet

Abstract

Topological characteristics of links of complex networks influence the dynamical processes executed on networks triggered by links, such as cascading failures triggered by links in power grids and epidemic spread due to link infection. The line graph transforms links in the original graph into nodes. In this paper, we investigate how graph metrics in the original graph are mapped into those for its line graph. In particular, we study the degree distribution and the assortativity of a graph and its line graph. Specifically, we show, both analytically and numerically, the degree distribution of the line graph of an Erdős–Rényi graph follows the same distribution as its original graph. We derive a formula for the assortativity of line graphs and indicate that the assortativity of a line graph is not linearly related to its original graph. Additionally, line graphs of various graphs, e.g. Erdős–Rényi graphs, scale-free graphs, show positive assortativity. In contrast, we find certain types of trees and non-trees whose line graphs have negative assortativity.

Suggested Citation

  • Wang, Xiangrong & Trajanovski, Stojan & Kooij, Robert E. & Van Mieghem, Piet, 2016. "Degree distribution and assortativity in line graphs of complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 445(C), pages 343-356.
    • Handle: RePEc:eee:phsmap:v:445:y:2016:i:c:p:343-356
    DOI: 10.1016/j.physa.2015.10.109
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    References listed on IDEAS

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    1. van Dam, E.R. & Haemers, W.H., 2002. "Which Graphs are Determined by their Spectrum?," Discussion Paper 2002-66, Tilburg University, Center for Economic Research.
    2. van Dam, E.R. & Kooij, R.E., 2006. "The Minimal Spectral Radius of Graphs with a Given Diameter," Discussion Paper 2006-102, Tilburg University, Center for Economic Research.
    3. H. Wang & W. Winterbach & P. Mieghem, 2011. "Assortativity of complementary graphs," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 83(2), pages 203-214, September.
    4. Krawczyk, M.J. & Muchnik, L. & Mańka-Krasoń, A. & Kułakowski, K., 2011. "Line graphs as social networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(13), pages 2611-2618.
    5. Sergey V. Buldyrev & Roni Parshani & Gerald Paul & H. Eugene Stanley & Shlomo Havlin, 2010. "Catastrophic cascade of failures in interdependent networks," Nature, Nature, vol. 464(7291), pages 1025-1028, April.
    6. P. Van Mieghem & H. Wang & X. Ge & S. Tang & F. A. Kuipers, 2010. "Influence of assortativity and degree-preserving rewiring on the spectra of networks," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 76(4), pages 643-652, August.
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    Cited by:

    1. Mohd-Zaid, Fairul & Kabban, Christine M. Schubert & Deckro, Richard F. & White, Edward D., 2017. "Parameter specification for the degree distribution of simulated Barabási–Albert graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 465(C), pages 141-152.
    2. Chu, J. & Nadarajah, S., 2018. "Estimating order statistics of network degrees," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 869-885.

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