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Asymptotic Degree Distributions in Random Threshold Graphs.

Armand M Makowski1, Siddharth Pal2

  • 1Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA.

Entropy (Basel, Switzerland)
|June 26, 2026
PubMed
Summary
This summary is machine-generated.

In random threshold graphs, the fraction of nodes with a specific degree converges to a random variable, not a fixed value. This finding impacts how network degree distributions are analyzed and compared to other models.

Keywords:
asymptotic analysisdegree distributionrandom graphsrandom threshold graphs

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Area of Science:

  • Network Science
  • Statistical Physics
  • Graph Theory

Background:

  • Homogeneous random graphs, specifically random threshold graphs, are analyzed in the many-node regime.
  • The study assumes a weak condition on the fitness variable distribution, satisfied by the exponential distribution.

Purpose of the Study:

  • To investigate the limiting degree distributions in random threshold graphs under a specific scaling.
  • To clarify the behavior of nodal versus network-wide degree distributions in these graphs.

Main Methods:

  • Analysis of limiting degree distributions for homogeneous random graphs.
  • Mathematical derivation under a weak assumption on the fitness variable distribution.

Main Results:

  • The nodal degree distribution converges to a limiting probability mass function (pmf).
  • However, the fraction of nodes with degree 'd' converges in distribution to a random variable Π(d), not deterministically.
  • The distribution of Π(d) is characterized by its characteristic function.

Conclusions:

  • Empirical node distributions cannot proxy the limiting nodal pmf in these graphs.
  • Network-wide and nodal degree distributions can differ significantly even in homogeneous graphs.
  • Random threshold graphs with exponential fitness are not directly comparable to Barabási-Albert models as scale-free alternatives.