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Cluster Tails for Critical Power-Law Inhomogeneous Random Graphs.

Remco van der Hofstad1, Sandra Kliem2, Johan S H van Leeuwaarden1

  • 11Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.

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|July 2, 2019
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Summary
This summary is machine-generated.

This study analyzes the largest cluster size in random graphs with infinite third moment degrees. We determine the probability of large cluster sizes, extending previous findings for Erdős-Rényi graphs.

Keywords:
Critical random graphsExponential tiltingInhomogeneous networksLarge deviationsPower-law degreesThinned Lévy processes

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Area of Science:

  • Graph Theory
  • Probability Theory
  • Statistical Physics

Background:

  • Previous work established the scaling limit of cluster sizes for rank-1 inhomogeneous random graphs with finite variance and infinite third moment degrees.
  • The established limit shows that rescaled cluster sizes converge to a sequence of decreasing random variables when degrees follow a power law.

Purpose of the Study:

  • To investigate the tail probabilities of the rescaled largest cluster in rank-1 inhomogeneous random graphs.
  • To extend existing results on largest cluster size distributions to graphs with infinite third moment degrees.

Main Methods:

  • Utilizing large deviation theory to analyze the tail behavior of the largest cluster.
  • Employing weak convergence arguments to establish the probabilistic limits.
  • Extending techniques from Erdős-Rényi random graph analysis to inhomogeneous models.

Main Results:

  • The study provides explicit characterization of the probability that the largest cluster size exceeds a given threshold.
  • Results are derived for rank-1 inhomogeneous random graphs where node degrees exhibit infinite third moments.

Conclusions:

  • This research offers a deeper understanding of the extreme cluster sizes in complex random graph structures.
  • The findings contribute to the theory of random graphs with heavy-tailed degree distributions.