[Paper Review] On the edge metric dimension for the random graph

Nina Zubrilina|arXiv (Cornell University)|Dec 21, 2016

Graph Labeling and Dimension Problems7 references3 citations

TL;DR

This paper establishes the asymptotic edge metric dimension of the Erdős-Rényi random graph $G(n,p)$, showing that $\operatorname{edim}(G(n,p)) = (1+o(1))\frac{4\log n}{\log(1/q)}$, where $q = 1 - 2p(1-p)^2(2-p)$. Using the probabilistic method and edge-distinguishing properties of random vertex sets, the authors derive this result by analyzing the probability that a random vertex set fails to distinguish pairs of edges, extending prior work on metric dimension to the edge metric setting.

ABSTRACT

Let $G(V, E)$ be a connected simple undirected graph. In this paper we prove that the edge metric dimension (introduced by Kelenc, Tratnik and Yero) of the Erdős-Rényi random graph $G(n, p)$ is given by: $$ extrm{edim}(G(n, p)) = (1 + o(1))\frac{4\log(n)}{\log(1/q)},$$ where $q = 1 - 2p(1-p)^2(2-p)$.

Motivation & Objective

• To determine the asymptotic edge metric dimension of the Erdős-Rényi random graph $G(n,p)$ for constant $p \in (0,1)$.
• To extend the probabilistic method used in prior work on metric dimension to the edge metric dimension setting.
• To characterize the threshold at which a random vertex set of size $r$ asymptotically almost surely distinguishes all edge pairs in $G(n,p)$.
• To compare the edge metric dimension with the classical metric dimension, showing $\dim(G(n,p)) < \operatorname{edim}(G(n,p))$ a.a.s. for all constant $p$.

Proposed method

• Applying the probabilistic method to show that a random vertex set of size $r = (1+o(1))\frac{4\log n}{\log(1/q)}$ asymptotically almost surely forms an edge basis.
• Defining $q = 1 - 2p(1-p)^2(2-p)$ as the probability that a uniformly random vertex fails to distinguish a random pair of disjoint edges.
• Using edge pair classification into intersecting and disjoint types, and showing that disjoint pairs dominate the analysis asymptotically.
• Bounding the probability that a random vertex set fails to distinguish any edge pair using independence and exponential decay via $q^r$.
• Estimating the expected number of undistinguished edge pairs via the first and second moment methods, showing it tends to zero as $n \to \infty$.
• Applying the second moment method to show concentration of the number of edge-generating sets, proving existence with positive probability.

Experimental results

Research questions

• RQ1 What is the asymptotic edge metric dimension of the Erdős-Rényi random graph $G(n,p)$ for constant edge probability $p$?
• RQ2 How does the edge metric dimension of $G(n,p)$ compare to its classical metric dimension?
• RQ3 Can the probabilistic method be adapted to derive asymptotic bounds for edge metric dimension, similar to its use in classical metric dimension?
• RQ4 For which values of $p$ does $\operatorname{edim}(G(n,p)) > \dim(G(n,p))$ hold asymptotically almost surely?

Key findings

• The edge metric dimension of $G(n,p)$ is asymptotically $\operatorname{edim}(G(n,p)) = (1+o(1))\frac{4\log n}{\log(1/q)}$, where $q = 1 - 2p(1-p)^2(2-p)$.
• For constant $p \in (0,1)$, the edge metric dimension is asymptotically larger than the classical metric dimension, since $\frac{4}{\log(1/q)} > \frac{2}{\log(1/Q)}$ with $Q = p^2 + (1-p)^2$.
• The probability that a uniformly random vertex fails to distinguish a random pair of disjoint edges is asymptotically $q = 1 - 2p(1-p)^2(2-p)$.
• A random vertex set of size $r = (1+o(1))\frac{4\log n}{\log(1/q)}$ asymptotically almost surely distinguishes all edge pairs in $G(n,p)$.
• The expected number of edge-generating sets of size $r$ tends to zero as $n \to \infty$, confirming the tightness of the bound.
• The analysis confirms that $\operatorname{edim}(G(n,p))$ is strictly greater than $\dim(G(n,p))$ a.a.s. for all constant $p \in (0,1)$.