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arxiv: 2605.04794 · v1 · submitted 2026-05-06 · 💻 cs.IT · eess.SP· math.IT

Distance Distributions Between Nodes in Concentric Disk-Annulus or Sphere-Shell Regions

Nicholas Vaiopoulos , Alexander Vavoulas , Harilaos G. Sandalidis , Konstantinos K. Delibasis This is my paper

Pith reviewed 2026-05-08 16:27 UTC · model grok-4.3

classification 💻 cs.IT eess.SPmath.IT
keywords distance distributionconcentric geometriesdisk-annulussphere-shellrandom waypoint modelprobability density functionwireless networksgeometric probability
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The pith

Closed-form expressions exist for the distance PDF between nodes in a disk-annulus or sphere-shell geometry

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

The paper derives closed-form expressions for the probability density function of the distance between two nodes located in an inner disk or sphere and an outer annulus or spherical shell. It handles two cases: independent uniform random placements in each region and a static outer node paired with an inner node following the stationary random waypoint distribution. These formulas matter for wireless network analysis because they turn geometric probability questions into direct calculations rather than repeated simulations. A reader would care if they need to evaluate coverage, interference, or connectivity in layered annular or shell-shaped deployment areas without numerical methods.

Core claim

The authors derive closed-form PDFs for the Euclidean distance between a node uniformly distributed in a disk (or sphere) and another in the surrounding annulus (or shell), as well as for the case where the inner node follows the stationary random waypoint distribution while the outer node remains static. These expressions are obtained through integration over the possible positions in polar or spherical coordinates, providing exact analytical results rather than approximations or numerical methods.

What carries the argument

Integration of geometric probability over concentric annular regions in 2D polar or 3D spherical coordinates to obtain the distance PDF from the area or volume elements

If this is right

  • Performance metrics such as outage probability or average path loss in concentric wireless regions can be computed directly from the distance PDF.
  • The formulas distinguish the effect of random waypoint mobility from uniform placement on inter-node distance statistics.
  • The 3D sphere-shell case follows from the same integration approach used for the 2D disk-annulus case.
  • System-level evaluations of heterogeneous networks with annular coverage zones no longer require Monte Carlo sampling of distances.

Where Pith is reading between the lines

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

  • The same integration technique could be tested on non-circular boundaries or multiple concentric layers if the resulting integrals remain evaluable in closed form.
  • Special cases such as vanishing annulus width should recover known distance distributions inside a single disk or sphere.
  • The expressions may support connectivity analysis in ring-shaped environments like circular highways or stadium perimeters.

Load-bearing premise

Node positions are statistically independent and follow either uniform distribution across the inner and outer regions or the stationary random waypoint model in the inner region.

What would settle it

Generate thousands of node position pairs using the uniform or random waypoint model in a disk of radius R inside an annulus from R to R+W, compute the empirical distance distribution, and check whether it matches the paper's closed-form PDF expression within sampling error.

Figures

Figures reproduced from arXiv: 2605.04794 by Alexander Vavoulas, Harilaos G. Sandalidis, Konstantinos K. Delibasis, Nicholas Vaiopoulos.

Figure 1
Figure 1. Figure 1: PDFs for a 2-D network: {R1, R2} = {1, 2}m in (a), (c) and {1, 3.5}m in (b), (d); analytical (red), beta (blue), and Monte Carlo (histograms). V = (4/3)π(R3 2 − R3 1 ). Here, θ1(ρ, r) and θ2(ρ, r) denote the angular limits determined by the intersections of the sphere of radius r with the inner and outer boundaries of the spherical shell, respectively. The corresponding conditional PDFs for the annulus or … view at source ↗
read the original abstract

This letter derives closed-form expressions for the probability density function of the distance between two nodes located in heterogeneous concentric geometries, namely a disk or sphere and a surrounding annulus or spherical shell. Two scenarios are considered: (i) both nodes are independently distributed in different regions, disk or sphere and annulus or shell, and (ii) one node is static in the outer region while the other follows the stationary distribution of the random waypoint model in the inner region. The resulting expressions provide a tractable analytical tool for performance evaluation in concentric wireless regions.

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.

Referee Report

0 major / 2 minor

Summary. The paper derives closed-form expressions for the PDF of the Euclidean distance between two nodes in concentric disk-annulus (2D) and sphere-shell (3D) regions. It treats two cases: independent uniform distributions across the inner and outer regions, and one fixed outer node with the inner node drawn from the stationary distribution of the random waypoint model.

Significance. If the derivations hold, the closed-form PDFs (involving elementary functions such as arccos, logs, and polynomials obtained via area/volume integrals) provide a tractable analytical tool for performance evaluation in wireless networks with heterogeneous concentric node placements. This is a strength, as it enables exact analysis of metrics like coverage or interference without simulation or numerical integration.

minor comments (2)
  1. [Abstract] The abstract and introduction could more explicitly contrast the two scenarios (uniform vs. RWP) with a short table of the resulting PDF forms to aid quick reference.
  2. Verify that the 3D sphere-shell integrals reduce correctly to the 2D disk-annulus case when the height dimension is collapsed; this would strengthen the unified presentation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our derivations for closed-form distance PDFs in concentric geometries. We appreciate the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives closed-form PDFs for Euclidean distances via direct integration over the intersection areas/volumes of annuli or spherical shells defined by the distance constraint r, using standard uniform or RWP stationary distributions. These steps rely on elementary geometric probability (arc-cos, logs, polynomials) without any fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations that reduce the central result to its own inputs. The RWP distribution is invoked as a known external result, and all expressions are obtained from first-principles area/volume integrals that remain independent of the target PDF. No step equates the claimed output to the input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions for node placement and mobility in wireless modeling, with no free parameters fitted to data, no invented entities, and no ad-hoc axioms beyond common geometric probability.

axioms (1)
  • domain assumption Nodes are independently and uniformly distributed in their respective regions or follow the stationary random waypoint model in the inner region.
    Standard modeling choice in wireless network analysis for random node locations.

pith-pipeline@v0.9.0 · 5401 in / 1121 out tokens · 39512 ms · 2026-05-08T16:27:29.681874+00:00 · methodology

discussion (0)

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Reference graph

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