On anti-stochastic properties of unlabeled graphs

TL;DR

This paper investigates anti-stochastic properties of unlabeled graphs, showing how small local changes can significantly alter the probability of certain global properties, with implications for graph vulnerability.

Contribution

It demonstrates the existence of optimal anti-stochastic properties for unlabeled graphs and characterizes their probabilities, extending prior concepts from labeled graphs.

Findings

Existence of anti-stochastic property with probability (2+o(1))/n^2

Another anti-stochastic property related to degree sequence with probability (2+o(1))/(n ln n)

These properties are optimal or near-optimal in probability bounds

Abstract

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.