Large-deviation properties of the largest 2-core component for random graphs
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Distributions of the size of the largest component of the 2-core for Erdos-Rényi (ER) random graphs with finite connectivity c and a finite number N of nodes are studied. The distributions are obtained basically over the full range of the support, with probabilities down to values as small as 10−320. This is achieved by using an artificial finite-temperature (Boltzmann) ensemble. The distributions for the 2-core resemble roughly the results obtained previously for the largest components of the full ER random graphs, but they are shifted to much smaller probabilities (c ≤ 1) or to smaller sizes (c > 1). The numerical data is compatible with a convergence of the rate function to a limiting shape, i.e., the large-deviations principle apparently holds.
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