The Relay Random Tree: A Stochastic Geometry Approach of Multihop Relay in an Urban Visibility Setting

Fran\c{c}ois Baccelli; Paul Rax

arxiv: 2605.03851 · v1 · submitted 2026-05-05 · 🧮 math.PR

The Relay Random Tree: A Stochastic Geometry Approach of Multihop Relay in an Urban Visibility Setting

Paul Rax , Fran\c{c}ois Baccelli This is my paper

Pith reviewed 2026-05-07 04:00 UTC · model grok-4.3

classification 🧮 math.PR

keywords relay random treestochastic geometrymultihop relayline of sight connectionseternal family treesgeometric random graphsurban environmentrandom trees

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The pith

A one-dimensional urban model of multihop line-of-sight relays forms a relay random tree that exemplifies eternal family tree classes.

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

The paper generalizes a stochastic geometry model of line-of-sight connections in cities by permitting arbitrary building height distributions and multiple relay hops. Buildings are represented as points along a line with independent random heights, and relays on rooftops extend connections beyond direct visibility. Analysis via geometric random graphs reveals that the resulting multihop structure is a relay random tree, providing a computable example that distinguishes the classes in the eternal family trees classification. This matters because it connects practical wireless relay problems to abstract random tree theory, allowing better understanding of connection loads and network structure.

Core claim

The LoS multihop relay geometry in the one-dimensional urban setting with arbitrary building height distributions constitutes a relay random tree. This random tree serves as a concrete, computationally tractable instance that highlights the different classes of the Eternal Family Trees classification in geometric random graph theory, while also addressing structural issues such as the total load on individual relays.

What carries the argument

The Relay Random Tree, constructed from a Poisson point process of buildings on the line with random heights, where edges represent direct or relayed LoS connections, organizing the propagation paths into a tree structure that maps to EFT classes.

If this is right

Relay loads can be analyzed using the properties of the specific EFT class realized by the height distribution.
Multi-hop connectivity probabilities and distances become derivable from the tree structure for any height distribution.
The model provides explicit examples of different EFT classes through variation of the building height distribution.
Total load on a relay is determined by the number of descendants in the random tree.

Where Pith is reading between the lines

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

This approach suggests that similar classifications might apply in two-dimensional city models if the visibility graph can be shown to be tree-like.
Network designers could use the EFT class to predict average relay loads without simulating each city layout.
Extending to correlated building heights might reveal new tree classes or break the existing classification.
The computational tractability could enable real-time optimization of relay placements in urban planning.

Load-bearing premise

The urban environment reduces to a one-dimensional line of independent buildings with heights drawn from a fixed but arbitrary distribution.

What would settle it

A simulation or measurement showing that the multihop LoS connection graph in the 1D model with random heights fails to produce tree structures classifiable under the EFT framework, or that the load distributions deviate from those predicted by the tree classes.

Figures

Figures reproduced from arXiv: 2605.03851 by Fran\c{c}ois Baccelli, Paul Rax.

**Figure 1.** Figure 1: The situation considered in [LB25]. From this situation, two angles are then defined : – The transmissive blockage angle ΘT , which is the right blockage angle seen from (X+, H+), or in other words the enhanced blockage angle obtained through the use of transmissive RIS on the right blocking building. – The reflexive blockage angle ΘR, which is right blockage angle seen from (X−, H−), that is the enhanced … view at source ↗

**Figure 2.** Figure 2: The shade can be seen as the part of the half plane that is in the shadow of (a, b) from a light source situated in (x1, h1). This notion is the natural one for the model described here. Indeed, the shade of a building from a point corresponds to the locations of R × R+ of which this building blocks direct transmission from the said point. It also allows us to reformulate the definition of blocking buildin… view at source ↗

**Figure 3.** Figure 3: The convexity of the shades implies this non-increasing character of the shades: Indeed, since the blocking building τ (x, y) as well as all buildings to its right are in the shade S (x,y) τ(x,y) , the shade S τ(x,y) τ(τ(x,y)) is included in S (x,y) τ(x,y) . Having defined the quantities of interest for a fixed landscape, let us now look at the types of landscapes we consider: 2.2. Random setting. — In the… view at source ↗

**Figure 4.** Figure 4: The blockage sequence seen from the origin. The main result of this section is the following: Theorem 3.1. — Evolution of blockage. Let N be a fixed integer. The sequence (Xn, Hn)0≤n≤N has density gN with respect to the product measure µ O N i=1 (Leb ⊗ L) (we consider here the Lebesgue measure on R+), which is given by gN : (xn, hn)0≤n≤N 7→ λ N exp −λ view at source ↗

**Figure 5.** Figure 5: Asking for having two points in the dashed areas such that every building between xi−1 + εi−1 and xi is below the line between the two is weaker than asking that all buildings between the two wanted x−coordinates are under the line of slope t min i and starting from (xi−1 + εi−1, hi−1). Now, thanks to these notations and observations, we can estimate the probability of Ξ: Lower bound on P(Ξ). We have the f… view at source ↗

**Figure 6.** Figure 6: The shadow-cutting building can be seen as the one on which the shadow of b stops. Remark 10. — If one is to consider scheme τ1, then the definitions slightly change: as we see, the main characteristic of the shadow-cutting-building is that it is ”the first to not be relayed by (α, β). Hence, the proper definition for τ1 are found by interchanging the large and strict inequalities: RS1(α, β) := x, y) ∈ R… view at source ↗

**Figure 7.** Figure 7: The buildings that are eventually relayed by b are those located between b and its shadow-cutting building. From this characterisation, we can deduce the following for schemes τ and τ2: Let b1 = (x1, h1) and b2 = (x2, h2) be two buildings, with b1 to the left of b2 (x1 ≤ x2). Then, since the sequence of the bases of the buildings τ (n) (b1) (let us denote this sequence (x n 1 ) goes to infinity, this means… view at source ↗

**Figure 8.** Figure 8: Representation of the three points above. Now, let us show that if a point is in the reverse shade of (α, β), but left of the shadow-cutting building, then its blocking building is either on the left of the blocking building (αsc, βsc) or the said shadow-cutting building itself. Let (x, y) ∈ RS(α, β) such that x < αsc be given. From Proposition 2.1, we deduce the following: – The angle ˜θ between (x, y) an… view at source ↗

**Figure 9.** Figure 9: One of the main differences with the infinite range setting, that will be the source of the more complex results, can be seen as the fact that the shades in this context are non convex Remark 14. — One can notice that for this specific definition of range, we can retrieve Corollary 2.1: the shades of the blocking buildings are still decreasing for the inclusion. However, this time they are not only charact… view at source ↗

**Figure 10.** Figure 10: The reverse shade is characterized by a succession of upwards jumps and then increasing slopes. Remark 15. — This difference of the reverse shades also changes the characterisation of eventual relays: our result mainly hinged on the fact any building on the left of the blocking building is eventually relayed by a building outside of the reverse shade (or in other words on a kind of inclusion of reverse s… view at source ↗

**Figure 11.** Figure 11: In the previous figure, considering only the present buildings, the relaying zone would look as above. Proposition 5.1. — Evolution of blockage: horizontal finite range. Recall the notations of Theorem 3.1. Let N be a fixed integer. The restriction of the law of (Xn, Hn)0≤n≤N to R×R n+1 + , or in other words, the restriction of the law of the evolution of blockage to the event of always finding a blocking… view at source ↗

**Figure 12.** Figure 12: In the case of Euclidian balls, the presence of tall buildings outside of the ball can prevent relaying further right. Here, the blockage angle is lower than the angle between the origin and a building in range due to this phenomenon. Definition 5.5. — Visibility, relaying scheme. Let L be a landscape. Let (a1, b1) and (a2, b2) be two buildings of L. We say that (a2, b2) is visible from (a1, b1) if it is … view at source ↗

read the original abstract

In a recent work (Lee, Baccelli $'25$), a one dimensional stochastic geometry model was introduced to study Line of Sight (LoS) connections using Reconfigurable Intelligent Surfaces (RIS), in the context of non terrestrial networks. In this model, signal can be propagated in a urban environment, with buildings acting as obstacles with RIS (which, for the scope of this present article can essentially be thought of as relays) on their rooftops, relaying the connection. The present paper extends this model by both allowing arbitrary distributions for the buildings heights, and considering multi-hop connections. Those generalities also lead to considering structural problems linked to the total load of a relay. Furthermore, studying this Line of Sight connection geometry at the light of geometric random graph theory, we show that it constitutes a computationally well understood example that highlights the different classes of the Eternal Family Trees (EFTs) classification.

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 extends the authors' 1D LoS relay model to arbitrary heights and multi-hop paths, then maps the resulting tree onto the EFT taxonomy as a concrete geometric example.

read the letter

The main point is that this paper takes the 2025 Lee-Baccelli one-dimensional model for line-of-sight connections through rooftop relays and adds two things: arbitrary distributions for building heights and multi-hop paths. It then frames the whole setup as a geometric random graph whose relay tree realizes different classes in the Eternal Family Trees classification, with some analysis of the load each relay carries from the paths that use it. The arbitrary heights keep the visibility rules probabilistic but still tractable, and the multi-hop version requires tracking how connections chain along the line without creating cycles or infinite branches. If the derivations are clean, this supplies a specific, simulatable instance that makes the EFT taxonomy less abstract. That is the useful part for someone already following stochastic geometry in wireless settings or the random-graph side of EFTs. The model stays internal to the 1D line with independent point obstacles, which is stated up front rather than hidden, so the claims about EFT classes are about properties inside this geometry, not a claim that real cities behave the same way. The load calculations appear to come from counting paths through each point, which is a natural extension but needs checking for how overlapping multi-hop routes are handled without overcounting. The EFT link is presented as an observation that the structure falls into known classes in a computationally friendly way; it builds directly on the authors' earlier definitions, so the step is incremental rather than a reorganization of the taxonomy itself. This is the sort of paper that belongs in a specialized stochastic-geometry or wireless-networks venue. A reader who wants an explicit, analyzable example of how visibility rules produce particular tree classes would get something concrete from it. The base model is already published, the extensions look consistent on the surface, and the geometric arguments are the kind that can be verified by a referee. I would send it to peer review rather than desk-reject.

Referee Report

2 major / 3 minor

Summary. The manuscript extends the one-dimensional stochastic geometry model from Lee and Baccelli (2025) for Line-of-Sight (LoS) connections in urban settings using relays on building rooftops. It generalizes the building height distribution to arbitrary distributions and incorporates multi-hop relaying, leading to an analysis of the relay tree's structural properties, including the total load on individual relays. By interpreting the resulting connection geometry through geometric random graph theory, the paper positions this model as a computationally tractable example that illustrates various classes within the Eternal Family Trees (EFT) classification.

Significance. If the technical claims hold, this work contributes a specific, analyzable instance of a geometric random graph arising from a visibility-constrained relay model. It links practical considerations in multihop wireless networks (such as relay load) to the abstract EFT taxonomy, potentially offering insights into connectivity structures in stochastic geometry. The allowance for arbitrary height distributions enhances the model's flexibility within the 1D framework, and the multi-hop extension addresses a natural generalization. Strengths include the focus on a well-defined model that may admit closed-form or simulatable properties.

major comments (2)

[§3] §3 (Multi-hop load analysis): The total load on a relay is central to the structural claims, yet its precise definition (e.g., as the number of descendant connections or the measure of traffic routed through a given relay) is not stated before the analysis begins. This definition is load-bearing for any subsequent results on finiteness or moments of the load, and must be given explicitly, perhaps via a recursive construction of the tree.
[§5] §5 (EFT classification): The assertion that the relay random tree 'highlights the different classes' of EFTs requires a concrete mapping. Which specific EFT class (or classes) does the model realize for a given height distribution, and which geometric random graph property (e.g., the visibility kernel or offspring distribution) establishes the identification? A theorem or proposition with this mapping is needed to substantiate the central link to the EFT taxonomy.

minor comments (3)

[Abstract] Abstract: The citation format 'Lee, Baccelli $'25$' is inconsistent with standard mathematical style; use 'Lee and Baccelli (2025)' and ensure the full reference appears in the bibliography.
[Introduction] Introduction: Explicitly delineate the incremental contributions relative to the 2025 base model, including which results on single-hop LoS are carried over unchanged and which are new under arbitrary heights and multi-hop.
[Model] Notation: Define the building height distribution (e.g., its cdf or pdf) at the first appearance in the model section rather than assuming it is understood from the prior work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments, which help improve the clarity of the presentation. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§3] §3 (Multi-hop load analysis): The total load on a relay is central to the structural claims, yet its precise definition (e.g., as the number of descendant connections or the measure of traffic routed through a given relay) is not stated before the analysis begins. This definition is load-bearing for any subsequent results on finiteness or moments of the load, and must be given explicitly, perhaps via a recursive construction of the tree.

Authors: We agree that an explicit definition of the total load is necessary before the analysis in Section 3. While the manuscript describes the load in the context of the multi-hop relay tree and its structural properties, we acknowledge that a formal definition is not provided at the outset. We will revise the beginning of Section 3 to include a precise definition: the total load on a relay is the number of its descendant connections in the relay random tree. This will be introduced via an explicit recursive construction of the tree, starting from the root and proceeding according to the LoS visibility rules determined by the building height distribution. We will also clarify the interpretation in terms of routed traffic. This revision will be incorporated in the next version of the manuscript. revision: yes
Referee: [§5] §5 (EFT classification): The assertion that the relay random tree 'highlights the different classes' of EFTs requires a concrete mapping. Which specific EFT class (or classes) does the model realize for a given height distribution, and which geometric random graph property (e.g., the visibility kernel or offspring distribution) establishes the identification? A theorem or proposition with this mapping is needed to substantiate the central link to the EFT taxonomy.

Authors: We thank the referee for this observation. The manuscript positions the relay random tree as a computationally tractable geometric random graph that illustrates different classes in the EFT taxonomy, with the connection arising from the visibility kernel induced by the arbitrary building height distribution and the resulting offspring distribution in the tree. However, we agree that a more explicit mapping would strengthen the central claim. We will add a new proposition in Section 5 that provides this concrete mapping: for a given height distribution, the model realizes specific EFT classes determined by the geometric random graph property consisting of the visibility kernel (which governs the LoS connection probabilities) and the induced offspring distribution. The proposition will detail how variations in the height distribution lead to different classes within the EFT classification. This addition will be made in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper explicitly defines its object as a 1D stochastic geometry model extending the Lee-Baccelli 2025 construction to arbitrary building-height distributions and multi-hop relays, then analyzes the resulting relay tree as a geometric random graph that realizes distinct EFT classes. This mapping is presented as an independent observation obtained after the model is specified, not as a redefinition or tautology. The 1D line with independent point obstacles is the stated modeling choice rather than a fitted or derived quantity, and the EFT link is asserted via geometric random graph theory applied to the constructed process. No equation reduces a claimed prediction to a fitted input by construction, no uniqueness theorem is imported solely from overlapping-author prior work to forbid alternatives, and no ansatz is smuggled via self-citation. The central claim concerns internal properties of the defined random tree and is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model inherits standard Poisson point process assumptions for building locations and an arbitrary but independent height distribution. No new particles or forces are postulated. The EFT classification link relies on existing random-graph theory rather than new axioms.

free parameters (1)

building height distribution
Arbitrary distribution chosen by the modeler; the paper claims results hold for any such distribution.

axioms (2)

domain assumption Buildings are modeled as points on a line with independent heights drawn from a fixed distribution.
Stated in the extension of the 2025 model; enables the 1D visibility process.
domain assumption Relays are located on rooftops and can forward signals only when line-of-sight exists.
Core modeling choice carried over from prior work.

pith-pipeline@v0.9.0 · 5460 in / 1452 out tokens · 48480 ms · 2026-05-07T04:00:11.835050+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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