Large-deviation properties of largest component for random graphs

TL;DR

This paper investigates the large-deviation properties of the largest component in random graphs and percolation, revealing phase transitions and distribution behaviors through numerical simulations.

Contribution

It introduces a numerical approach to study large deviations in graph ensembles, including cases without analytical solutions, and uncovers phase transitions in artificial ensembles.

Findings

Distribution shapes are consistent with analytical results for Erdős-Rényi graphs.

Percolation distributions are qualitatively similar but differ in shape and finite-size effects.

A first-order phase transition at low temperatures is observed in the artificial ensemble.

Abstract

Distributions of the size of the largest component, in particular the large-deviation tail, are studied numerically for two graph ensembles, for Erdoes-Renyi 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. The distributions for the Erdoes-Renyi 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.