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 $(2+o(1))/(n\ln n)$, which is optimal up to factor of 2.

翻译：我们研究均匀分布的随机图在面对一个对手攻击时的脆弱性，该对手旨在通过仅对图进行局部改变来实现分布的全局变化。我们称一个图性质 $A$ 为反随机的，如果随机图 $G$ 满足 $A$ 的概率很小，但以高概率而言，存在一个小的扰动将 $G$ 转变为满足 $A$ 的图。虽然对于标号图，此类性质容易通过二元覆盖编码获得，但无标号图的反随机性质的存在性并不那么明显。如果允许的扰动是添加或删除一条边，我们展示了一个反随机性质，该性质由阶为 $n$ 的随机无标号图以概率 $(2+o(1))/n^2$ 满足，这是尽可能小的概率。我们还根据图的度序列表达了另一个反随机性质。该性质的概率为 $(2+o(1))/(n\ln n)$，在因子2范围内是最优的。