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1 results

  • 4 - Evolution

  • from Part II - Erdős–Rényi–Gilbert Model
  • Alan Frieze, Carnegie Mellon University, Pennsylvania, Michał Karoński, Adam Mickiewicz University, Poznań, Poland
  • Book:
    Random Graphs and Networks: A First Course
    Published online:
    02 March 2023
    Print publication:
    09 March 2023, pp 45-63

  • Summary

    In this chapter, we mainly explore how the typical component structure evolves as the number of edges m increases. The following statements should be qualified with the caveat, w.h.p. The evolution of Erdős–Rényi–Gilbert type random graphs has clearly distinguishable phases. The first phase, at the beginning of the evolution, can be described as a period when a random graph is a collection of small components which are mostly trees. Next, a random graph passes through a phase transition phase when a giant component, of order comparable with the order of random graph, starts to emerge.