Distribution of diameters for Erdős-Rényi random graphs

Physical Review. E
Alexander K Hartmann, M Mézard

Abstract

We study the distribution of diameters d of Erdős-Rényi random graphs with average connectivity c. The diameter d is the maximum among all the shortest distances between pairs of nodes in a graph and an important quantity for all dynamic processes taking place on graphs. Here we study the distribution P(d) numerically for various values of c, in the nonpercolating and percolating regimes. Using large-deviation techniques, we are able to reach small probabilities like 10^{-100} which allow us to obtain the distribution over basically the full range of the support, for graphs up to N=1000 nodes. For values c<1, our results are in good agreement with analytical results, proving the reliability of our numerical approach. For c>1 the distribution is more complex and no complete analytical results are available. For this parameter range, P(d) exhibits an inflection point, which we found to be related to a structural change of the graphs. For all values of c, we determined the finite-size rate function Φ(d/N) and were able to extrapolate numerically to N→∞, indicating that the large-deviation principle holds.

References

Sep 18, 1989·Physical Review Letters·A M Ferrenberg, R H Swendsen
Jan 6, 1992·Physical Review Letters·B A Berg, T Neuhaus
Apr 6, 2001·Physical Review Letters·F Wang, D P Landau
Jun 13, 2002·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Alexander K Hartmann
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Citations

Nov 30, 2019·Chaos·Yannick Feld, Alexander K Hartmann
Aug 17, 2018·Physical Review. E·Eytan KatzavAlexander K Hartmann
Jan 23, 2020·Physical Review. E·Wiebke Staffeldt, Alexander K Hartmann
Feb 20, 2020·Physical Review. E·Edgar Guzmán-GonzálezFernando L Metz
Sep 11, 2019·Physical Review. E·Bertrand Lacroix-A-Chez-ToineGrégory Schehr
Sep 11, 2019·Physical Review. E·Fernando L Metz, Isaac Pérez Castillo
Aug 17, 2020·Physical Review. E·Haggai BonneauEytan Katzav
Dec 18, 2020·Physical Review. E·Hendrik Schawe, Alexander K Hartmann
Feb 20, 2021·Physical Review. E·David S DeanHendrik Schawe

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