On the Rigidity of Sparse Random Graphs

TL;DR

This paper investigates the rigidity of sparse random graphs, showing that their 2-core is rigid in a broader range of edge probabilities and providing a polynomial-time canonical labeling algorithm with high probability.

Contribution

It extends the known range of edge probabilities for which the 2-core of sparse random graphs is rigid and introduces an efficient canonical labeling algorithm applicable in this range.

Findings

The 2-core of G(n,p) is rigid for rac{ ext{log} n}{n}+rac{ ext{log} n}{n} in the sparse regime.

A polynomial-time canonical labeling algorithm is proposed with an error rate approaching zero.

The results imply that G(n,p) graphs are reconstructible with high probability across all p.

Abstract

A graph with a trivial automorphism group is said to be rigid. Wright proved that for $\frac{\log n}{n}+\omega(\frac 1n)\leq p\leq \frac 12$ a random graph $G\in G(n,p)$ is rigid whp. It is not hard to see that this lower bound is sharp and for $p<\frac{(1-\epsilon)\log n}{n}$ with positive probability $\text{aut}(G)$ is nontrivial. We show that in the sparser case $\omega(\frac 1 n)\leq p\leq \frac{\log n}{n}+\omega(\frac 1n)$, it holds whp that $G$'s $2$-core is rigid. We conclude that for all $p$, a graph in $G(n,p)$ is reconstrutible whp. In addition this yields for $\omega(\frac 1n)\leq p\leq \frac 12$ a canonical labeling algorithm that almost surely runs in polynomial time with $o(1)$ error rate. This extends the range for which such an algorithm is currently known.