On anti-stochastic properties of unlabeled graphs
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We study vulnerability of a uniformly distributed random graph to an attack by an adversary who aims for a global change of the distribution while being able to make only a local change in the graph. We call a graph property A anti-stochastic if the probability that a random graph G satisfies A is small but, with high probability, there is a small perturbation transforming G into a graph satisfying A. While for labeled graphs such properties are easy to obtain from binary covering codes, the existence of anti-stochastic properties for unlabeled graphs is not so evident. If an admissible perturbation is either the addition or the deletion of one edge, we exhibit an anti-stochastic property that is satisfied by a random unlabeled graph of order n with probability (2+o(1))/n^2, which is as small as possible. We also express another anti-stochastic property in terms of the degree sequence of a graph. This property has probability (2+o(1))/(nln n), which is optimal up to factor of 2.
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