The Metric Dimension of Sparse Random Graphs

In 2013, Bollob\'as, Mitsche, and Pralat at gave upper and lower bounds for the likely metric dimension of random Erd\H{o}s-R\'enyi graphs $G(n,p)$ for a large range of expected degrees $d=pn$. However, their results only apply when $d \ge \log^5 n$, leaving open sparser random graphs with $d < \log^5 n$. Here we provide upper and lower bounds on the likely metric dimension of $G(n,p)$ from just above the connectivity transition, i.e., where $d=pn=c \log n$ for some $c > 1$, up to $d=\log^5 n$. Our lower bound technique is based on an entropic argument which is more general than the use of Suen's inequality by Bollob\'as, Mitsche, and Pralat, whereas our upper bound is similar to theirs.