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k-Core organization of complex networks.

S N Dorogovtsev1, A V Goltsev, J F F Mendes

  • 1Departamento de Física da Universidade de Aveiro, 3810-193 Aveiro, Portugal and Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia.

Physical Review Letters
|February 21, 2006
PubMed
Summary

This study reveals how random network damage creates nested k-core structures. Networks with finite second-nearest neighbors exhibit hybrid transitions, while diverging neighbors lead to ultrarobust k-cores.

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Area of Science:

  • Network Science
  • Statistical Physics
  • Graph Theory

Background:

  • Understanding network resilience is crucial for various complex systems.
  • Random damage can significantly alter network architecture and function.
  • Characterizing substructures within networks helps predict their robustness.

Purpose of the Study:

  • To analytically describe the architecture of randomly damaged uncorrelated networks.
  • To identify and characterize k-cores and their emergence thresholds (bootstrap percolation).
  • To investigate the impact of second-nearest neighbors on k-core formation and network robustness.

Main Methods:

  • Analytical description of network architecture.
  • Identification of k-cores as successively enclosed substructures.

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  • Calculation of bootstrap percolation thresholds as birthpoints for k-cores.

    • Main Results:

    • Defined k-cores as the largest subgraphs with minimum degree k.
    • Determined the structure, sizes, and birthpoints of k-cores.
    • Showcased hybrid phase transitions for k-core emergence when the second-nearest neighbor number (zeta2) is finite.
    • Demonstrated the existence of infinite, ultrarobust k-core sequences when zeta2 diverges.

    Conclusions:

    • The k-core decomposition provides a framework for understanding damaged network architecture.
    • Network robustness against random damage depends critically on the behavior of second-nearest neighbors.
    • Diverging zeta2 leads to highly resilient network cores, suggesting potential for robust system design.