Entropy and Distributed Source Coding of Connected Soft Random Geometric Graphs

Carl P. Dettmann; Oliver Baker

arxiv: 2605.04703 · v1 · submitted 2026-05-06 · 💻 cs.IT · math.IT· math.PR

Entropy and Distributed Source Coding of Connected Soft Random Geometric Graphs

Oliver Baker , Carl P. Dettmann This is my paper

Pith reviewed 2026-05-08 16:40 UTC · model grok-4.3

classification 💻 cs.IT math.ITmath.PR

keywords distributed source codingSlepian-Wolfsoft random geometric graphsentropyasymptotic equipartition propertyrandom binningconnectivity thresholdinformation theory

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The pith

Connected soft random geometric graphs admit the classical Slepian-Wolf rate region for distributed compression.

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

The paper proves that the Slepian-Wolf rate region from information theory applies directly to soft random geometric graphs once they lie above the connectivity threshold. It does so by first establishing new limit theorems and asymptotic equipartition properties for the entropy of these graphs, then using those properties to justify random binning by a finite number of independent encoders that each compress separate sections. A sympathetic reader cares because this shows that geometric random graphs modeling wireless or sensor networks can be compressed in a fully distributed way without losing the optimal joint-entropy rates that hold for arbitrary sources.

Core claim

We consider the distributed compression of Soft Random Geometric Graphs (SRGGs) above the connectivity threshold. We establish the Slepian-Wolf rate region for the SRGG in the setting where there are a finite number of encoders compressing sections of the graph independently. To do so, we prove novel limit theorems and asymptotic equipartition properties for the SRGG and its entropy, which allow us to use random binning techniques for distributed compression.

What carries the argument

The asymptotic equipartition property and accompanying limit theorems for the entropy of connected SRGGs, which justify applying random binning to achieve the Slepian-Wolf rates.

Load-bearing premise

The soft random geometric graph must sit above the connectivity threshold for the new entropy limit theorems and asymptotic equipartition property to hold.

What would settle it

A direct computation showing that the entropy of finite SRGG samples fails to concentrate or satisfy the asymptotic equipartition property even when the graph is connected would prevent random binning from attaining the claimed rate region.

read the original abstract

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

The paper derives the Slepian-Wolf rate region for finite encoders on connected soft random geometric graphs by proving new limit theorems and an AEP for their entropy.

read the letter

The main point is that this work gives the Slepian-Wolf rate region for a finite number of encoders that each compress independent sections of a soft random geometric graph, as long as the graph sits above the connectivity threshold. They reach this by first establishing new limit theorems and an asymptotic equipartition property for the SRGG entropy, then feeding those into standard random binning arguments. The result is a clean characterization of the achievable rate region under that spatial model. What the paper does well is take a structured source that arises in geometric networks and show how the usual distributed compression machinery applies once the right ergodic properties are in place. The connectivity condition is used to guarantee the mixing needed for the AEP, which lets the classical Slepian-Wolf proof go through without extra fitting or self-reference. If the limit theorems are solid, this is a useful specialization that could inform coding schemes for sensor or ad-hoc networks where node locations induce the correlations. The soft spot is the reliance on those novel theorems. The abstract sketches the strategy but does not display the actual derivations, convergence rates, or explicit error bounds, so the soundness rests on whether the proofs handle the soft connection probabilities and finite-n effects correctly. The connectivity-threshold assumption is plausible and not circular, but it does narrow the scope to the connected regime. No load-bearing gaps are visible from the description, and the citation pattern appears to build on standard Slepian-Wolf and random geometric graph results without inventing entities. This is for readers working in network information theory or geometric probability who want a concrete rate region for this source class. A specialist looking for extensions of distributed coding to spatially correlated graphs would get direct value from the derivation. I would send it to peer review so the limit theorems can be checked in detail.

Referee Report

0 major / 2 minor

Summary. The manuscript considers distributed source coding of Soft Random Geometric Graphs (SRGGs) above the connectivity threshold. It claims to establish the Slepian-Wolf rate region for the case of a finite number of encoders that independently compress sections of the graph. The approach proceeds by first proving novel limit theorems and asymptotic equipartition properties (AEP) for the SRGG and its entropy, which then justify the application of standard random binning arguments to obtain the rate region.

Significance. If the claimed limit theorems and AEP hold, the result would provide a concrete extension of the classical Slepian-Wolf theorem to a geometrically constrained random-graph source model. This could be relevant for distributed compression in sensor networks or geometric data settings. The strategy is the standard one once the AEP is granted, and the connectivity-threshold hypothesis is a plausible prerequisite for the required ergodicity; no internal inconsistency is apparent from the abstract and high-level argument.

minor comments (2)

The abstract and introduction would benefit from a brief explicit statement of the main theorem (including the precise form of the rate region) rather than only describing the strategy.
Notation for the SRGG entropy and the sections compressed by each encoder should be introduced with a dedicated preliminary subsection to improve readability for readers unfamiliar with the model.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and the recommendation to accept. We appreciate the recognition that the limit theorems and AEP, if established, would provide a meaningful extension of the Slepian-Wolf theorem to this geometrically constrained source model.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds by first establishing novel limit theorems and an asymptotic equipartition property (AEP) for the entropy of connected SRGGs above the connectivity threshold, then invoking the classical random-binning argument to obtain the Slepian-Wolf rate region for a finite number of independent encoders. This is the standard information-theoretic route once an AEP is granted; the connectivity-threshold hypothesis is an external prerequisite for ergodicity rather than a self-referential definition or fitted input. No equation reduces a claimed prediction to a parameter fit by construction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The central claim therefore retains independent content beyond its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the SRGG is above the connectivity threshold and on standard information-theoretic tools; no free parameters or invented entities are indicated in the abstract.

axioms (1)

domain assumption SRGG lies above the connectivity threshold
Invoked to guarantee the novel limit theorems and AEP hold for the entropy.

pith-pipeline@v0.9.0 · 5359 in / 1176 out tokens · 54000 ms · 2026-05-08T16:40:53.129070+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

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Gray, R. , journal=. A new class of lower bounds to information rates of stationary sources via conditional rate-distortion functions , year=

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Entropy of spatial network ensembles , author =. Phys.\ Rev.\ E , volume =. 2018 , month =

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Information-spectrum methods in information theory , author=. 2003 , address=

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Entropy of labeled versus unlabeled networks , author =. Phys.\ Rev.\ E , volume =. 2022 , month =

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Penrose, Mathew , title =. 2003 , address=

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Abbe, Emmanuel , month =. 2016 , booktitle =

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work page 2006

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Compression of Graphical Structures: Fundamental Limits, Algorithms, and Experiments , year=

Choi, Yongwook and Szpankowski, Wojciech , journal=. Compression of Graphical Structures: Fundamental Limits, Algorithms, and Experiments , year=

work page

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Compression with Unlabeled Graph Side Information , year=

Nikpey, Hesam and Sarkar, Saswati and Bidokhti, Shirin Saeedi , booktitle=. Compression with Unlabeled Graph Side Information , year=

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arXiv preprint arXiv:2309.14464 , year=

Rate-Distortion Function of the Stochastic Block Model , author=. arXiv preprint arXiv:2309.14464 , year=

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Wafula, Martin Wachiye and Vippathalla, Praneeth Kumar and Coon, Justin and Badiu, Mihai-Alin , booktitle=. Rate-Distortion Function of the Stochastic Block Model , year=

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Badiu, Mihai-Alin and Coon, Justin P. , booktitle=. On the Distribution of Random Geometric Graphs , year=

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Compression of Preferential Attachment Graphs , year=

Łuczak, Tomasz and Magner, Abram and Szpankowski, Wojciech , booktitle=. Compression of Preferential Attachment Graphs , year=

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Compression of dynamic graphs generated by a duplication model , author=. Algorithmica , volume=. 2020 , month=

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Łuczak, Tomasz and Magner, Abram and Szpankowski, Wojciech , title =. Random Struct.\ Alg. , volume =. doi:https://doi.org/10.1002/rsa.20842 , year =

work page doi:10.1002/rsa.20842

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2012 , publisher=

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work page 2012

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Can smooth graphons in several dimensions be represented by smooth graphons on [0, 1]? , author=. Examples and Counterexamples , volume=. 2021 , publisher=

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Journal of Combinatorial Theory, Series B , volume=

Limits of dense graph sequences , author=. Journal of Combinatorial Theory, Series B , volume=. 2006 , publisher=

work page 2006

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Graphons, cut norm and distance, couplings and rearrangements , author=. New York Journal of Mathematics Monographs , year=

work page

[24] [24]

arXiv preprint arXiv:2601.15194v1 , year=

Entropy of Soft Random Geometric Graphs in General Geometries , author=. arXiv preprint arXiv:2601.15194v1 , year=

work page arXiv

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The Annals of Applied Probability , volume=

Connectivity of soft random geometric graphs , author=. The Annals of Applied Probability , volume=

work page

[26] [26]

IEEE Transactions on Information Theory , volume=

A Universal Lossless Compression Method Applicable to Sparse Graphs and Heavy-Tailed Sparse Graphs , author=. IEEE Transactions on Information Theory , volume=. 2022 , publisher=

work page 2022

[27] [27]

and Orlitsky, A

Alon, N. and Orlitsky, A. , journal=. A lower bound on the expected length of one-to-one codes , year=

work page

[28] [28]

2018 16th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt) , pages=

On the conditional entropy of wireless networks , author=. 2018 16th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt) , pages=. 2018 , organization=

work page 2018

[29] [29]

2024 IEEE International Symposium on Information Theory (ISIT) , pages=

On the lossy compression of spatial networks , author=. 2024 IEEE International Symposium on Information Theory (ISIT) , pages=. 2024 , organization=

work page 2024

[30] [30]

IEEE Transactions on Information Theory , volume=

Distributed compression of graphical data , author=. IEEE Transactions on Information Theory , volume=. 2021 , publisher=

work page 2021

[31] [31]

Physical Review E , volume=

Entropy of labeled versus unlabeled networks , author=. Physical Review E , volume=. 2022 , publisher=

work page 2022