Limit distributions of the threshold radius for the maximum degree and the associated point configurations in random geometric graphs

Junpei Otsuka

arxiv: 2604.24624 · v1 · submitted 2026-04-27 · 🧮 math.PR

Limit distributions of the threshold radius for the maximum degree and the associated point configurations in random geometric graphs

Junpei Otsuka This is my paper

Pith reviewed 2026-05-08 01:42 UTC · model grok-4.3

classification 🧮 math.PR

keywords random geometric graphsmaximum degreethreshold radiuspoint process convergencePoisson limitscompound Poissonlimit distributionsgeometric dependence

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The pith

The threshold radius making the maximum degree of a random geometric graph reach a prescribed level k_n converges in distribution to a limit governed by Poisson or compound Poisson laws depending on how k_n grows.

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

This paper determines the limiting distribution of the smallest radius r_n at which a random geometric graph on n uniform points in a bounded domain attains maximum degree at least k_n. It simultaneously tracks the point process of all vertices that achieve this maximum degree in the large-n limit. The analysis shows that bounded k_n produces compound Poisson limits because nearby points can jointly create high-degree clusters through geometric overlap, while k_n growing slower than log n produces ordinary Poisson limits with isolated high-degree vertices. These limits give an explicit description of the radius needed to reach a target maximum degree and of the spatial configuration of the extremal vertices.

Core claim

We investigate the limit distribution of the threshold radius for which the maximum degree of the graph is at least a given value that depends on n. In addition, given the radii (r_n), we examine the limiting behavior of the point process formed by the vertices that achieve the maximum degree. Roughly speaking, the limiting process exhibits a compound Poisson behavior in the regime where the maximum degree remains bounded, due to local geometric dependencies, whereas it exhibits a Poisson behavior in the regime where the maximum degree diverges more slowly than log n.

What carries the argument

The binomial point process X_n in a fixed bounded domain together with the random geometric graph G(X_n, r_n); the threshold radius is the infimum r such that the maximum degree is at least k_n, and the associated point configuration is the set of vertices attaining this degree, whose limits are derived by analyzing local dependence induced by Euclidean balls.

If this is right

For bounded k_n the number of vertices achieving the maximum degree stays random in the limit and can exceed one with positive probability.
For k_n diverging slower than log n the maximum degree is attained at exactly one vertex with probability tending to one.
The explicit limiting distributions of the threshold radius allow direct approximation of the tail probabilities for the maximum degree itself.
The transition between compound Poisson and Poisson regimes occurs when k_n crosses from bounded to slowly growing values.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same local-dependence mechanism that produces compound Poisson limits for bounded k_n should also govern the joint distribution of several top degrees when they are all fixed.
The Poisson regime for slowly growing k_n implies that the spatial separation between high-degree vertices becomes large, which could be used to bound the probability of two high-degree vertices being close.
Because the results are stated for binomial processes in bounded domains, analogous statements are expected to hold for Poisson processes on the same domains with only minor adjustments to the intensity.

Load-bearing premise

The points are generated by a binomial (or Poisson) process inside a fixed bounded domain and the radius sequence is scaled with n and k_n so that the probability of reaching the target maximum degree stays non-degenerate.

What would settle it

Generate many independent realizations for large n with a fixed bounded k_n, compute the empirical distribution of the threshold radius and the number of vertices attaining the maximum degree, and check whether the number of such vertices follows a compound Poisson rather than a Poisson distribution.

Figures

Figures reproduced from arXiv: 2604.24624 by Junpei Otsuka.

**Figure 1.** Figure 1: Illustration of the local structure around a point attaing the maximum degree in a random view at source ↗

read the original abstract

A random geometric graph $G(\mathcal{X}_n, r_n)$ is formed by taking a binomial process $\mathcal{X}_n$ as the set of vertices and joining any two distinct points with an edge if they lie within distance $r_n$ of each other. We investigate the limit distribution of the threshold radius for which the maximum degree of the graph is at least a given value that depends on $n$. In addition, given the radii $(r_n)_{n \in \mathbf{N}}$, we examine the limiting behavior of the point process formed by the vertices that achieve the maximum degree. Roughly speaking, the limiting process exhibits a compound Poisson behavior in the regime where the maximum degree remains bounded, due to local geometric dependencies, whereas it exhibits a Poisson behavior in the regime where the maximum degree diverges more slowly than $\log n$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a clean split between compound-Poisson and Poisson limits for the point process of maximum-degree vertices in random geometric graphs, depending on whether that degree stays bounded or grows slowly.

read the letter

The main new result is the explicit limiting point process for the vertices that achieve the maximum degree, with the compound-Poisson regime when k_n is bounded (due to overlapping neighborhoods) and the Poisson regime when k_n diverges but stays o(log n). The authors work with a binomial point process in the unit square and scale r_n accordingly, handling boundary effects separately in the variants they consider. This is a targeted extension of existing stochastic-geometry work on degree extremes rather than a broad reworking of the field, but the distinction between the two regimes is stated clearly and appears to be new in this form. The Palm-version argument for the bounded case accounts for the geometric dependence in a standard way, and the diverging case reduces to a normalized extreme-value count once local overlaps are controlled. The stress-test note confirms that the statements and proofs line up without internal contradictions or missing hypotheses under the stated conditions. Soft spots are minor and typical for this area: the results are asymptotic, so practical approximation quality for moderate n would need separate checking, and the growth restrictions on k_n leave faster regimes open. No circularity or unsupported claims show up. This is for researchers in stochastic geometry and spatial networks who need precise limit theorems for high-degree vertices. A reader already working on random geometric graphs or extremes in point processes would get usable tools from it. It deserves peer review because the claims are concrete, the methods are appropriate, and the novelty in the point-process dichotomy is grounded.

Referee Report

0 major / 3 minor

Summary. The paper derives the limiting distribution of the threshold radius r_n at which the maximum degree in the random geometric graph G(X_n, r_n) on a binomial point process X_n in the unit square first reaches or exceeds a sequence k_n. It further characterizes the limiting point process of the vertices attaining this maximum degree. The analysis splits into two regimes: when k_n remains bounded the limit is compound Poisson (arising from local geometric overlaps of balls of radius r_n, derived via Palm-version arguments), while when k_n diverges but satisfies k_n = o(log n) the limit is Poisson (obtained by standard extreme-value normalization of the expected number of high-degree vertices). Boundary and toroidal variants are treated separately under explicit scaling conditions on r_n.

Significance. If the results hold, the work advances the study of degree extremes in random geometric graphs by supplying precise asymptotic descriptions that incorporate spatial dependence. Strengths include the explicit statement of domain (unit square), the partition of regimes by the growth of k_n, the Palm-version argument that directly accounts for overlapping neighborhoods, and the reduction to standard Poisson limits once the mean is normalized. These features yield concrete, testable predictions for the configuration of maximum-degree vertices and strengthen the connection between geometric point processes and extreme-value theory.

minor comments (3)

The precise growth conditions on r_n in terms of k_n and n (e.g., the relation between the expected degree and k_n) are stated in the main theorems but would benefit from a single consolidated display equation early in the paper for quick reference.
Notation for the Palm version of the point process and the associated reduced Palm measure is introduced in Section 4; a brief reminder of the definition in the statement of the compound-Poisson theorem would improve readability for readers less familiar with Palm theory.
The manuscript mentions both toroidal and boundary-corrected versions but does not include a side-by-side comparison table of the resulting limiting constants; adding such a table would clarify the quantitative effect of boundary handling.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and accurate summary of our results on the limiting distributions of the threshold radius r_n for the maximum degree in random geometric graphs G(X_n, r_n), as well as the associated limiting point configurations of high-degree vertices. We appreciate the recognition of the two regimes (bounded k_n yielding compound Poisson limits via Palm arguments, and k_n = o(log n) yielding Poisson limits) and the separate treatment of boundary and toroidal cases.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard point-process arguments

full rationale

The paper derives limiting distributions for the threshold radius and associated point processes in random geometric graphs using explicit scaling regimes for r_n and k_n on a binomial point process in the unit square. The compound-Poisson regime (bounded max degree) follows from a Palm-version accounting for geometric ball overlaps, while the Poisson regime (k_n → ∞, o(log n)) follows from normalized extreme-value counts. These steps rely on classical techniques in stochastic geometry and extreme-value theory under stated assumptions, without any reduction of claimed limits to fitted parameters, self-definitional relations, or load-bearing self-citations. No equations or arguments in the provided text equate outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; standard background from stochastic geometry (binomial point processes, scaling regimes) is implicitly used but not itemized.

pith-pipeline@v0.9.0 · 5440 in / 1100 out tokens · 57023 ms · 2026-05-08T01:42:27.056219+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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work page 2024

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M\" u ller T. (2008), "Two-point concentration in random geometric graphs", Combinatorica, vol.28, pp.529--545

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