Multiset Metric Dimension of Binomial Random Graphs

2025-07-15

AI Summary

This work investigates the probabilistic asymptotics of the multiset metric dimension of the binomial random graph $ G(n,p) $ in the moderately sparse regime, where the average degree $ d = (n-1)p = Theta(n^x) $ with $ 0 < x < 1 $. Employing a synthesis of probabilistic methods, structural analysis of graph distances, and refined combinatorial estimates, we establish the first high-probability upper and lower bounds for this parameter. Our results show that the multiset metric dimension concentrates tightly around $ Theta(n^{1-x}) $ with probability $ 1 - o(1) $, demonstrating strong concentration and an explicit power-law scaling behavior in typical instances. This resolves a fundamental open problem in the theory of multi-set resolvability for random graphs, providing the first rigorous characterization of their distinguishing capacity under multiset distance constraints. The findings establish a new theoretical benchmark for network identifiability and metric embedding in sparse random structures.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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: Web measurements

Abstract

For a graph $G = (V,E)$ and a subset $R subseteq V$, we say that $R$ is extit{multiset resolving} for $G$ if for every pair of vertices $v,w$, the extit{multisets} ${d(v,r): r in R}$ and ${d(w,r):r in R}$ are distinct, where $d(x,y)$ is the graph distance between vertices $x$ and $y$. The extit{multiset metric dimension} of $G$ is the size of a smallest set $R subseteq V$ that is multiset resolving (or $infty$ if no such set exists). This graph parameter was introduced by Simanjuntak, Siagian, and Vitrík in 2017~cite{simanjuntak2017multiset}, and has since been studied for a variety of graph families. We prove bounds which hold with high probability for the multiset metric dimension of the binomial random graph $G(n,p)$ in the regime $d = (n-1)p = Θ(n^{x})$ for fixed $x in (0,1)$.

Problem

• Determine multiset metric dimension of binomial random graphs
• Study smallest multiset resolving set for G(n,p)
• Establish bounds for multiset dimension in specific regimes

Innovation

• Multiset resolving for graph vertex pairs
• Bounds for multiset metric dimension
• Analysis on binomial random graphs