Multiset Metric Dimension of Binomial Random Graphs

Austin Eide; Pawel Pralat

REVIEW 1 major objections 5 minor 1 cited by

Multiset Metric Dimension of Binomial Random Graphs

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Pith's one-line read This paper proves explicit power-law bounds for the multiset metric dimension of $G(n,p)$: at most $n^{y_4+o(1)}$ for $x\le 1/8$, at least $n^{y_1+o(1)}$ for $x\le 1/2$, and infinite for $x>1/2$.

desk verdict First bounds for the multiset metric dimension of G(n,p), but the upper bound at reciprocal points relies on an unproved extension of a cited expansion lemma. read the letter →

arxiv 2507.11686 v1 pith:6ZI2FSDH submitted 2025-07-15 math.CO cs.DM

Austin Eide , Pawel Pralat This is my paper · ORCID

classification math.COcs.DM MSC 05C8005C12

keywords multisetmetricdimensionbinomialrandomgraphdistancesresolvingsetexpansionprofilediameter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper determines, up to polynomial factors, how large a multiset resolving set must be in the binomial random graph $G(n,p)$ when the average degree is $d=(n-1)p=\Theta(n^x)$. For $x\in(0,1/8]$ a resolving set of size $n^{y_4+o(1)}$ exists with high probability, where $y_4$ solves $f_x(y)=4$; for $x\in(0,1/2]$ every resolving set has size at least $n^{y_1+o(1)}$, where $y_1$ solves $f_x(y)=1$; and for $x>1/2$ no finite resolving set exists. The exponent function is $f_x(y)=\sum_{i=0}^{\lfloor 1/x\rfloor}\max\{ix+y-1,0\}$, so both thresholds are explicitly computable from $x$. The payoff is a concrete answer to when random graphs can be identified by unordered distance lists, with an application to low-dimensional graph embeddings and source localization.

What carries the argument

The argument is carried by a typical expansion profile for spheres in $G(n,p)$ (Lemma 2.1, taken from the literature): with high probability $|S_i(V')|=(1+o(1))|V'|d^i$ for $i\le i^*$, and the top shell has size roughly $(1-e^{-|V'|c})n$. The upper bound combines this profile with a random subset $R$ chosen with probability $r/n$: for a fixed pair $v,w$, the shell-intersection counts $|S_i^R(v)\setminus S_i(w)|$ and $|S_i^R(w)\setminus S_i(v)|$ are conditionally binomial with mean $\Theta(d^i r/n)$, and the maximum point mass of a binomial variable bounds the chance that the two coincide. Multiplying these failure probabilities over levels produces exactly $n^{-f_x(y)/2+o(1)}$. The lower bound counts signatures: for any $R$ of size $n^y$, a typical vertex has at most $n^{f_x(y)+o(1)}$ possible multiset signatures, so if $f_x(y)<1$ two vertices must collide.

What would settle it

Simulate $G(n,p)$ at, say, $n=10^6$ and $d=n^{1/8}$; for every pair $(v,w)$ compute $|S_{i^*+1}(w)\setminus N_{i^*+1}(v)|$ and compare it with $(1+o(1))e^{-c}(1-e^{-c})n$. Finding one pair whose shell size deviates by more than a small multiple of the predicted value would violate the expansion lemma and remove the basis for the upper bound. Alternatively, compute $\beta_{\mathrm{ms}}$ exactly for small $n$ and check whether it ever exceeds the $n^{y_4}$ bound in the stated regime.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem 1.4, is that with high probability the multiset metric dimension of $G(n,p)$ obeys $\beta_{\mathrm{ms}}(G(n,p))\le n^{y_4+O(\log^{-1}n)}$ for $x\le 1/8$, $\beta_{\mathrm{ms}}(G(n,p))\ge n^{y_1+O(\log^{-1}n)}$ for $x\le 1/2$, and $\beta_{\mathrm{ms}}(G(n,p))=\infty$ for $x>1/2$, where $d=(n-1)p=n^{x+O(\log^{-1}n)}$ and $y_4,y_1$ are the unique solutions in $(0,1)$ of $f_x(y)=4$ and $f_x(y)=1$. The upper bound is proved by sampling a random set $R$ of size roughly $n^y$ and showing that with probability $1-o(n^{-2})$ it distinguishes any fixed pair of vertices; the lower bound counts the number of distinct multiset signatures available to a set of size $n^y$ and shows it is too small once $f_x(y)<1$.

Load-bearing premise

The load-bearing premise is the quoted expansion lemma asserting that every shell of $G(n,p)$ has its predicted size for all pairs of vertices at every level up to the top; the paper notes that only a 'mild extension' is supplied for the top level, and if that extension fails the upper-bound construction cannot control the final coordinate of the multiset signature.

Editorial extensions

If this is right

For any fixed $x\le 1/8$, random graphs with average degree $\Theta(n^x)$ have a multiset resolving set of size $O(n^{y_4(x)})$ with high probability; since the diameter is $O(1/x)$, this gives an embedding of the graph into $\mathbb{R}^{O(1/x)}$ using multiset distances.
For any fixed $x\le 1/2$, every multiset resolving set of such a graph has size $\Omega(n^{y_1(x)})$ with high probability.
When $x>1/2$, the graph has diameter 2 with high probability, and the paper invokes the known fact that non-path diameter-2 graphs have no finite multiset resolving set, so $\beta_{\mathrm{ms}}(G(n,p))=\infty$.
The upper-bound proof depends only on the expansion profile, so the authors observe that it extends from $d=n^{x+o(1)}$ to the full regime $d=\omega(\log n)$.
The two bounds leave a polynomial gap and leave open whether $\beta_{\mathrm{ms}}(G(n,p))$ is finite for $1/8<x\le 1/2$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the quoted expansion lemma holds as stated, the upper-bound construction should transfer to every average degree $d=\omega(\log n)$, not just $d=n^{x+o(1)}$, yielding finite resolving sets in a much wider sparse regime than Theorem 1.4 states.
The zig-zag behaviour of $y_1$ and $y_4$ at $x=1/k$ suggests that any eventual sharp exponent for $\beta_{\mathrm{ms}}(G(n,p))$ will also jump at reciprocal integers, tracking the drop in diameter from $k+1$ to $k$.
The same signature-counting method could be adapted to other random graph models with known shell profiles, such as random geometric graphs, to produce analogous multiset-dimension exponents.
Comparing $\beta_{\mathrm{ms}}$ with the already studied metric dimension of random graphs would quantify how much locating power is lost when ordered distance vectors are collapsed to multisets.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: worth refereeing, but the paper as written is conditional on an external fact. It's the first study of the multiset metric dimension on binomial random graphs, and the main results are plausible and cleanly stated. The catch is that the upper bound at exactly the reciprocal points x = 1/k depends on an unproved 'mild extension' of a lemma from [18]; without that extension, the union bound in Theorem 3.1 falls apart.

What's new: the piecewise-linear function f_x(y) is a nice device, encoding the shell structure and explaining the zig-zag behavior at reciprocal integers. The lower bound via counting over typical vertices is solid, and the upper bound via a random set and Chernoff estimates is standard but carefully executed. The authors connect well to earlier work on metric dimension and the diameter of G(n,p). The exposition is clear.

The soft spot is exactly what the stress-test note identifies. Lemma 2.1 is quoted from [18], and for level i*+1 they say 'a mild extension is required' without stating or proving it. That extension is load-bearing at x = 1/k for k ≥ 8. Without a Θ(n) private shell at that level, the worst-case failure probability becomes n^{-(f_x(y)-y)/2} = n^{-(4-y4)/2}, which is far larger than n^{-2} because y4 < 1. So the final coordinate estimate fails and the union bound no longer holds. This is not a fatal flaw by itself; the extension is likely true. But it needs a precise statement and either a proof or a citation. The omitted proofs of Lemmas 1.1 and 3.2 are minor, as they are elementary.

If I were the editor, I would send this to a referee with the explicit task of verifying the extension of [18, Lemma 5.3] to the i*+1 case. Assuming that goes through, the paper is a solid contribution. The audience is researchers in random graphs and metric dimension. I'd maybe bring it to the reading group, mainly to discuss the f_x threshold behavior, and I would cite it if I were working in the area. It deserves a serious referee.

Referee Report

1 major / 5 minor

Summary. The paper studies the multiset metric dimension β_ms(G(n,p)) of the binomial random graph in the regime d=(n-1)p=n^{x+O(log^{-1} n)} for fixed x∈(0,1). The main result, Theorem 1.4, states that w.h.p. β_ms(G(n,p)) ≤ n^{y4+O(log^{-1} n)} for x∈(0,1/8], β_ms(G(n,p)) ≥ n^{y1+O(log^{-1} n)} for x∈(0,1/2], and β_ms(G(n,p))=∞ for x>1/2, where y4 and y1 are the unique solutions to f_x(y)=4 and f_x(y)=1, respectively, with f_x(y)=∑_{i=0}^{⌊1/x⌋} max{ix+y-1,0}. The upper bound is proved by a probabilistic construction of a resolving set using a random subset R of size about n^y, combined with a union bound over vertex pairs; the lower bound uses a counting argument over possible multiset signatures of 'typical' vertices, under the assumption that the diameter is known and that the graph satisfies a typical expansion profile V.

Significance. If the result is correct, it provides the first bounds for the multiset metric dimension of binomial random graphs, with explicit, computable exponents that are derived from the analysis rather than fitted to simulations. The paper also clearly identifies the gap between the lower and upper bounds, states the open problem of finiteness for 1/8<x≤1/2, and gives a concrete motivation via low-dimensional embeddings. The authors are transparent about the reliance on an extension of a known expansion lemma, which is a significant weakness. The central proof strategy is standard and largely coherent, and the numerical values for y1 at small reciprocal points are a useful concrete illustration.

major comments (1)

[Section 2, Corollary 2.2 and proof of Theorem 3.1 (x=1/k case)] The upper bound at reciprocal points x=1/k with k≥8 rests on the estimate |S_{i*+1}(w)\setminus N_{i*+1}(v)| = Θ(n) for every pair v,w, which is stated as Corollary 2.2(iv). This is precisely the 'mild extension' of [18, Lemma 5.3] that the paper flags but neither states nor proves. The estimate is load-bearing: in stage i*+1 of the proof, the failure probability is bounded by O(n^{-y/2}) only because the binomial has s_{i*+1}=Θ(n) trials; if only the trivial bound s_{i*+1}≤n is available, the product of the coordinate-wise failure probabilities at y=y4 becomes n^{-(4-y4)/2}, which is far larger than n^{-2}, so the union bound over all pairs fails. The same unproved extension underlies Corollary 2.2(iii), which is used in the lower bound for x=1/k to control i-atypical vertices at i=k. The authors should supply a precise statement and a proof of the needed extension (or a complete reference that covers it) before the stated upper and lower bounds can be considered established for x=1/k.

minor comments (5)

[Section 1.4, Lemma 1.3] The proof that f_x(1)>4 for x≤1/8 uses the inequality (⌊1/x⌋+1)⌊1/x⌋·x/2 > (1/x)^2/2, which is false in general; for example, at x=0.112 the left side is about 4.03 and the right side about 39.9. The lemma is nevertheless true, since for m=⌊1/x⌋≥8 we have f_x(1)=m(m+1)x/2 > m/2 ≥ 4, but the written derivation should be corrected.
[Section 1.4 and Section 3] The proofs of Lemma 1.1 and Lemma 3.2 are omitted. Both are elementary, but the paper should either include the short derivations or provide a precise reference; omitting them is unnecessary in a formal paper.
[Proof of Theorem 3.1, x=1/k case] The expression Pr(m_R(v)=m_R(w) | V') uses an undefined event V'; this should presumably be V, the expansion event defined in Section 2.
[Proof of Theorem 3.1, stage 0] The sentence 'We condition on neither vertex being in r' should read 'in R'; the symbol r is used for the parameter n^y, not for the random set.
[Section 5] There is a typo: 'muiltiset' should be 'multiset' in the concluding paragraph.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the y1/y4 exponents are derived from the proof's own union-bound and signature-counting estimates, and the flagged 'mild extension' of the expansion lemma is a rigor gap, not a circular step.

full rationale

The exponents y4 and y1 are not fitted inputs; they are solved from conditions that the analysis itself generates. For the upper bound, the authors bound, stage by stage, the probability that a random set R fails to distinguish a fixed pair v,w by products of binomial probability maxima of order n^{-(1/2)max{ix+y-1,0}}. Summing these exponents gives the function f_x(y), and y4 is defined by the equation f_x(y)=4 so that the union bound over all pairs closes. For the lower bound, the counting argument shows that typical vertices have at most n^{f_x(y)+o(1)} possible multiset signatures with respect to a candidate resolving set of size n^y; y1 is defined by f_x(y)=1 so that this count is less than 0.49n, forcing a collision by pigeonhole. Thus both threshold exponents are derived from the proof's own failure-probability and counting estimates, not assumed from data or relabeled as predictions. The central external input is Lemma 2.1, the typical expansion profile of G(n,p), quoted from [3] and [18]; although [3] includes a present co-author, the lemma is a published, independently stated result about shell sizes and is not equivalent to the multiset-metric-dimension conclusion. The paper itself flags a limitation in Section 2: for the level i*+1 case, it says 'a mild extension is required' of [18, Lemma 5.3]. This unproved extension is used in Corollary 2.2(iv) and in the reciprocal-x case of Theorem 3.1, making the upper bound at x=1/k conditional on that external fact. This is a correctness or completeness risk, not circularity, because the required shell-size estimate is an input condition rather than a restatement of the target theorem. The omitted proofs of Lemmas 1.1 and 3.2 are elementary and not load-bearing. No step in the paper reduces, by construction or by self-citation, to its own inputs; therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim depends on two external probabilistic facts: the shell-expansion estimates (Lemma 2.1) and the diameter estimates (Lemma 4.1). Neither is proved in the text; the first requires an unstated 'mild extension' of a cited lemma. No free parameters are fitted to data; the exponents y1 and y4 are determined by solving f_x(y)=1 or 4 and are not ad hoc. No new entities are introduced.

assumptions (3)

domain assumption Typical expansion property V holds w.h.p. for d=ω(log n) (Lemma 2.1)
Quoted from [3] and [18]; not proved in the paper. The paper states a mild extension is required for the level i*+1 case, which is not supplied.
standard math Diameter of G(n,p) satisfies the conditions of Lemma 4.1 (from [2, Corollary 10.12])
Used in the lower bound to fix the number of coordinates in the signature, and used for the x>1/2 case.
standard math Chernoff bounds and Stirling's approximation
Used throughout for concentration and for Lemma 3.2; the paper gives the Chernoff inequalities explicitly and cites Stirling.

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Pith. "Pith review of Multiset Metric Dimension of Binomial Random Graphs." pith.science (2026). https://pith.science/paper/6ZI2FSDH

@misc{pith2026250711686,
  author       = {Pith},
  title        = {Pith review of: Multiset Metric Dimension of Binomial Random Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6ZI2FSDH}},
  note         = {Machine review of arXiv:2507.11686}
}

abstract

For a graph $G = (V,E)$ and a subset $R \subseteq V$, we say that $R$ is \textit{multiset resolving} for $G$ if for every pair of vertices $v,w$, the \textit{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 \textit{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 Vitr\'{i}k 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 $d = (n-1)p = \Theta(n^{x})$ for fixed $x \in (0,1)$.

Figures

Figures reproduced from arXiv: 2507.11686 by the authors.

**Figure 1.** Plots of the points (x, y4) satisfying fx(y4) = 4 (blue) for 0 < x ≤ 1 8 and (x, y1) satisfying fx(y1) = 1 (red) for 0 < x ≤ 1 2 . and Vi ∗+1(V ′ ) = |Si(V ′ )| = 1 − e −|V ′ |c − |V ′ |d i ∗ n + O γ + log n √ n n . Let V := \ V ′ :|V | ′∈{1,2} i \∗+1 i=0 Vi(V ′ ). All of our results rely on the following lemma. We note that while our main results on the multiset metric dimension only apply to random grap… view at source ↗

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Multiset resolvability parameters in graphs: A survey with new results and open problems
math.CO 2026-07 accept novelty 5.0 of 10

Multiset resolvability parameters are surveyed; sharp outer-multiset lower bounds for diameter-two and join graphs are proved, and block graphs with local multiset dimension two are characterized.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [18]

Sequential metric dimension for random graphs

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work page 2021

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work page 2020

[3] [2]

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work page 1998

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B´ ela Bollob´ as, Dieter Mitsche, and Pawe l Pra lat. Metric dimension for random graphs. The Electronic Journal of Combinatorics , 20(4):P52, 2013

work page 2013

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Novi H Bong and Yuqing Lin. Some properties of the multiset dimension of graphs. Electron. J. Graph Theory Appl. , 9(1):215–221, 2021

work page 2021

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Josep D ´ ıaz, Harrison Hartle, and Cristopher Moore. The metric dimension of sparse random graphs. arXiv preprint arXiv:2504.21244 , 2025

work page Pith review arXiv 2025

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Local- ization game for random graphs

Andrzej Dudek, Sean English, Alan Frieze, Calum MacRury, and Pawe l Pra lat. Local- ization game for random graphs. Discrete Applied Mathematics , 309:202–214, 2022

work page 2022

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A note on the localization number of random graphs: diameter two case

Andrzej Dudek, Alan Frieze, and Wesley Pegden. A note on the localization number of random graphs: diameter two case. Discrete Applied Mathematics , 254:107–112, 2019

work page 2019