The Metric Dimension of Sparse Random Graphs

TL;DR

The paper extends the typical metric dimension results for $G(n,p)$ into the sparser regime $d \ge c\log n$ up to $\log^5 n$ by pairing an entropic lower bound with a Boll MitPra–style upper bound. It analyzes distance-shell growth via $V_t(w)$ and establishes concentration results for vertex degrees and shells, distinguishing two regimes via $\gamma = d^{t^*+1}/n$ that lead to different entropy and MD behaviors. When $\gamma=\Theta(1)$, MD scales as $\Theta(\log n)$ up to constants; when $\gamma=\omega(1)$, MD becomes super-logarithmic with bounds dependent on $\gamma$ and $d$, capturing a zig-zag behavior tied to the sparse connectivity. Overall, the work substantially narrows the typical MD in sparse random graphs and provides explicit bounds with entropic insights and expansion-based arguments.

Abstract

In 2013, Bollobás, Mitsche, and Pralat at 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 $d=pn$. 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 Bollobás, Mitsche, and Pralat, whereas our upper bound is similar to theirs.

Theorems & Definitions (34)

• Definition 2.1
• Theorem 2.2: BollMitPra
• Theorem 3.1
• Lemma 4.1
• Remark 4.2
• proof
• Definition 4.3
• Corollary 4.4
• Remark 4.5
• Lemma 4.6