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arXiv · 1710.05680

Distribution of diameters for Erdös-Rényi random graphs

Alexander K. Hartmann·Marc Mézard
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Abstract

We study the distribution of diameters d of Erdös-Rényi random graphs with average connectivity c. The diameter d is the maximum among all 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 non-percolating and the percolating regime. Using large-deviations 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 Phi(d/N) and were able to extrapolate numerically to N->infinity, indicating that the large deviation principle holds.

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Alexander K. Hartmann, Marc Mézard. 2017-10-16. Distribution of diameters for Erdös-Rényi random graphs. https://doi.org/10.1103/physreve.97.032128

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