# MoN10: Tenth Mathematics of Networks meeting, 16th September, Loughborough University

## Alexander Hartmann (Oldenburg) - Large deviation properties of random graphs

The large-deviation properties of different types of random graphs are studied
using numerical simulations.

First, distributions of the size of the largest
component, in particular the large-deviation tail,
are studied numerically for two graph ensembles,
for Erdős-Rényi random graphs
with finite connectivity and for two-dimensional bond percolation.
Probabilities as small as $10^{-180}$ are accessed
using an artificial finite-temperature (Boltzmann)
ensemble and applications of the Wang-Landau algorithm.
The distributions for the Erdős-Rényi ensemble
agree well with previously obtained analytical
results. The results for the percolation problem, where no analytical results
are available, are qualitatively similar, but the shapes of the distributions
are somehow different and the finite-size
corrections are sometimes much larger. Furthermore, for both problems,
a first-order phase transition
at low temperatures $T$ within the artificial ensemble is found in the
percolating regime, respectively.

Second, the distributions of the diameter are presented. Here,
partial analytic results are available
from previous studies for Erdős-Rényi random graphs in the small
connectivity region. The numerical results follow a Gumbel distribution
and agree well with the analytics. For higher
connectivities, where no analytic results are available,
the simulation results show that the distributions
are qualitatively different from the low connectivity region. This is
also connected to a first-order phase transition within the associated
finite-temperature ensemble.

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Contact:
Keith Briggs
()
or
Richard G. Clegg (richard@richardclegg.org)