The Metric Dimension of Sparse Random Graphs

2025-04-30

AI Summary

This study addresses the metric dimension of sparse random graphs $G(n,p)$ near the connectivity threshold and in the moderately sparse regime $clog n < pn le log^5 n$, closing a theoretical gap previously restricted to $pn ge log^5 n$. Methodologically, we develop a general information-theoretic lower bound framework based on entropy, overcoming the limitations of Suen’s inequality in extremely sparse regimes; further, we combine the first- and second-moment methods with refined probabilistic estimates to systematically characterize the asymptotic behavior of the metric dimension across this range. Our main contribution is establishing that the metric dimension grows as $Theta(log n)$, with tight upper and lower bounds. This work completes the spectral characterization of metric structure in Erdős–Rényi random graphs and provides new theoretical tools and benchmarks for analyzing geometric properties of sparse networks.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In 2013, Bollob'as, Mitsche, and Pralat at gave upper and lower bounds for the likely metric dimension of random ErdH{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.

Problem

• Extends metric dimension bounds to sparser random graphs
• Focuses on G(n,p) near connectivity transition (d=c log n)
• Improves lower bound technique using entropic argument

Innovation

• Extends metric dimension bounds to sparser graphs
• Uses entropic argument for lower bound
• Similar upper bound to prior work