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Distribution of diameters for Erdős-Rényi random graphs.

A K Hartmann1, M Mézard2

  • 1Institut für Physik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany.

Physical Review. E
|May 20, 2018
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Summary

We analyzed the diameter distribution in Erdős-Rényi random graphs, revealing key insights into graph structure and dynamics. Our findings provide a comprehensive understanding of graph diameters across different connectivity regimes.

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Area of Science:

  • Graph Theory
  • Statistical Physics
  • Network Science

Background:

  • The diameter of a graph is crucial for understanding dynamic processes.
  • Erdős-Rényi random graphs are fundamental models in network analysis.
  • Characterizing the distribution of graph diameters is an ongoing challenge.

Purpose of the Study:

  • To numerically investigate the distribution of diameters (P(d)) in Erdős-Rényi random graphs.
  • To explore this distribution across various average connectivity values (c).
  • To analyze behavior in both nonpercolating (c<1) and percolating (c>1) regimes.

Main Methods:

  • Numerical simulations of Erdős-Rényi random graphs up to N=1000 nodes.
  • Application of large-deviation techniques to probe probabilities as low as 10^-100.
  • Analysis of the finite-size rate function Φ(d/N) and extrapolation to infinite size (N→∞).

Main Results:

  • Numerical results for c<1 align well with existing analytical findings.
  • For c>1, a more complex diameter distribution P(d) was observed, featuring an inflection point.
  • The inflection point in P(d) for c>1 correlates with a structural change in the graphs.

Conclusions:

  • The study validates the numerical approach for analyzing graph diameters.
  • A significant structural change in Erdős-Rényi graphs for c>1 is identified through diameter distribution analysis.
  • The findings support the validity of the large-deviation principle for graph diameters in these models.