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2 results

  • 7 - Small-World Phenomena in Configuration Models

  • from Part III - Small-World Properties of Random Graphs
  • Remco van der Hofstad, Technische Universiteit Eindhoven, The Netherlands
  • Book:
    Random Graphs and Complex Networks
    Published online:
    08 February 2024
    Print publication:
    08 February 2024, pp 289-325

  • Summary

    In this chapter we investigate the distance structure of the configuration model by investigating its typical distances and its diameter. We adapt the path-counting techniques in Chapter 6 to the configuration model, and obtain typical distances from the “giant is almost local” proof. To understand the ultra-small distances for infinite-variance degree configuration models, we investigate the generation growth of infinite-mean branching processes. The relation to branching processes informally leads to the power-iteration technique that allows one to deduce typical distance results in random graphs in a relatively straightforward way.

  • 4 - Connected Components in Configuration Models

  • from Part II - Connected Components in Random Graphs
  • Remco van der Hofstad, Technische Universiteit Eindhoven, The Netherlands
  • Book:
    Random Graphs and Complex Networks
    Published online:
    08 February 2024
    Print publication:
    08 February 2024, pp 138-188

  • Summary

    In this chapter we investigate the local limit of the configuration model, we identify when it has a giant component and find its size and degree structure. We give two proofs, one based on a “the giant is almost local” argument, and another based on a continuous-time exploration of the connected components in the configuration model. Further results include its connectivity transition.