The Metric Dimension of Sparse Random Graphs

In 2013, Bollobás, Mitsche, and Prałat gave upper and lower bounds for the likely metric dimension of random Erdős-Rényi graphs $G(n,p)$ for a large range of expected degrees. However, their results only apply when $d=pn=\omega(\log^5 n)$, leaving open sparser random graphs with $d=O(\log^5 n)$ or $d=o(\log^5n)$. Here we provide upper and lower bounds on the likely metric dimension of $G(n,p)$ in a range of $d$ starting just above the connectivity transition, i.e., where $d=c \log n$ for some constant $c > 1$, up to $d=O(\log^5 n)$. Our lower bound technique is based on an entropic argument which is weaker but more general than the use of Suen's inequality by Bollobás, Mitsche, and Prałat, whereas our upper bound is similar to theirs.