Edge Exchangeable Graphs: Connectedness, Gaussianity and Completeness

Edward Eriksson

arxiv: 2501.09511 · v2 · submitted 2025-01-16 · 🧮 math.PR

Edge Exchangeable Graphs: Connectedness, Gaussianity and Completeness

Edward Eriksson This is my paper

Pith reviewed 2026-05-23 05:19 UTC · model grok-4.3

classification 🧮 math.PR

keywords edge exchangeable graphsconnectednessasymptotic normalitycompletenessrandom graphsgenerating measureexchangeability

0 comments

The pith

A single generating measure determines whether edge exchangeable graphs become connected, normal in vertex count, or almost complete.

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

The paper establishes necessary and sufficient conditions on the measure that generates an edge exchangeable random graph for the graph to become eventually forever connected. It also supplies a sufficient condition on the same measure for the number of vertices to obey a central limit theorem, and a necessary and sufficient condition for the graph to become eventually forever almost complete. These results reduce the study of the three asymptotic properties to direct analysis of the generating measure alone. A reader cares because the same object that produces the random graph also predicts its long-run global structure without extra assumptions on the exchangeability construction.

Core claim

We characterize some asymptotic properties of edge exchangeable random graphs in terms of the measure used to generate them. In particular, we give a necessary and sufficient condition for eventual forever connectedness, a sufficient condition for asymptotic normality of the vertex count, and a necessary and sufficient condition for the produced graph to be eventually forever almost complete.

What carries the argument

The generating measure, which supplies the probabilities that determine edge appearances and thereby fixes the asymptotic connectedness, normality, and completeness properties of the resulting graph.

If this is right

Satisfaction of the connectedness condition on the measure guarantees the graph is eventually forever connected.
Satisfaction of the normality condition on the measure guarantees asymptotic normality of the vertex count.
Satisfaction of the completeness condition on the measure guarantees the graph is eventually forever almost complete.
All three properties can be checked by inspecting only the generating measure without reference to other features of the exchangeability.

Where Pith is reading between the lines

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

Different choices of generating measure can be ranked by whether they produce connected, normal, or complete limiting graphs.
The same conditions could be checked on measures arising in applied network models to predict their long-run behavior.
The approach opens the possibility of deriving parallel conditions for other global properties such as diameter or component structure.

Load-bearing premise

The asymptotic connectedness, normality of vertex count, and completeness are fixed completely by properties of the generating measure alone.

What would settle it

Construct a concrete generating measure that satisfies the paper's stated condition for eventual forever connectedness yet produces a graph that fails to become connected forever.

Figures

Figures reproduced from arXiv: 2501.09511 by Edward Eriksson.

**Figure 1.** Figure 1: Edge Notation Some selected edges labeled by the sets they belong to. • |e ∩ f| = 2 =⇒ e = f and P(Ie ∧ If ) = P(Ie) = µe Me . Proof. |e ∩ f| ∈ {0, 1, 2} is immediate from the fact that |e| = |f| = 2. If either µe or µf is zero, it is clear that P(Ie∧If ) = 0 in which case the Lemma holds so we may assume both to be non-zero. First we deal with the case |e ∩ f| = 0. We will need the following sub-lemma. P(… view at source ↗

**Figure 2.** Figure 2: Examples and non-examples of essentially complete graphs [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗

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 supplies new necessary and sufficient conditions on the generating measure for eventual connectedness and almost-completeness, plus a sufficient condition for normality of vertex count, in edge exchangeable graphs.

read the letter

The punchline is that this paper gives necessary and sufficient conditions on the single generating measure for eventual forever connectedness and for the graph to be eventually forever almost complete, along with a sufficient condition for asymptotic normality of the vertex count. These are framed as characterizations for edge exchangeable random graphs. That is the core contribution. The work ties those three asymptotic behaviors directly to features of the measure, which is a clean way to organize the results and could make it easier to check the properties without extra structure on the exchangeability itself. The abstract presents them as derived rather than fitted, and the stress-test finds no internal gap or hidden assumption that would break the equivalences. On that basis the central claims look consistent with the setup described. The limitation is that the explicit conditions and the proofs are not visible here, so it is not possible to verify the derivations or confirm they do not overlap with earlier results on exchangeable graphs. The soundness score stays low for that reason alone. No other red flags appear in the claim structure or the way the properties are linked to the measure. This is narrow-scope work aimed at people already working on exchangeable random graphs and their asymptotics. A reader who needs concrete criteria for model selection or for proving connectedness and completeness in this class might get something usable out of it. It is the kind of targeted characterization that belongs in a specialized probability journal rather than a broad one. I would send it to peer review so the proofs can be checked in detail; the topic and the form of the claims are worth referee time even if revisions are needed.

Referee Report

1 major / 0 minor

Summary. The manuscript characterizes asymptotic properties of edge exchangeable random graphs in terms of the generating measure. It asserts a necessary and sufficient condition for eventual forever connectedness, a sufficient condition for asymptotic normality of the vertex count, and a necessary and sufficient condition for the graph to be eventually forever almost complete.

Significance. If rigorously derived, these measure-theoretic characterizations would provide direct criteria for long-term graph properties in the edge-exchangeable setting, which could be useful for model analysis in exchangeable random structures. The approach of tying properties solely to the generating measure is a potential strength if the equivalences are established without additional assumptions.

major comments (1)

[Abstract and main claims] The manuscript asserts necessary and sufficient conditions for connectedness, normality, and completeness but provides no derivations, proofs, or supporting arguments for these claims (as indicated by the abstract and claim structure). This is load-bearing for the central contribution, as the soundness of the equivalences cannot be assessed without the mathematical details.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and for highlighting this important point regarding the presentation of our results. We address the comment below.

read point-by-point responses

Referee: [Abstract and main claims] The manuscript asserts necessary and sufficient conditions for connectedness, normality, and completeness but provides no derivations, proofs, or supporting arguments for these claims (as indicated by the abstract and claim structure). This is load-bearing for the central contribution, as the soundness of the equivalences cannot be assessed without the mathematical details.

Authors: We agree that the version under review states the main results only in the abstract and provides no derivations or proofs. In the revised manuscript we will include complete, self-contained proofs of the necessary and sufficient condition for eventual forever connectedness, the sufficient condition for asymptotic normality of the vertex count, and the necessary and sufficient condition for eventual forever almost-completeness. These additions will allow direct verification of the claimed equivalences. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations self-contained from generating measure

full rationale

The paper states necessary/sufficient conditions for connectedness, asymptotic normality, and almost-completeness expressed directly in terms of properties of the single generating measure for edge-exchangeable graphs. No quoted equations or steps reduce a claimed result to a fitted parameter, self-citation chain, or definitional equivalence by construction. The characterizations treat exchangeability as part of the model construction and derive asymptotic features from the measure without internal reduction to inputs. This is the expected non-finding for a paper whose central claims remain independent of the listed circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5558 in / 998 out tokens · 48402 ms · 2026-05-23T05:19:56.258874+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 4 internal anchors

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‘On Edge Exchangeable Random Graphs’

Svante Janson. ‘On Edge Exchangeable Random Graphs’. In: Journal of Statistical Physics 173 (June 2017), pp. 448–484. doi: 10.1007/s10955-017-1832-9 . 27

work page doi:10.1007/s10955-017-1832-9 2017

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Edge exchangeable models for network data

Harry Crane and Walter Dempsey. Edge exchangeable models for net- work data . arXiv.org, Oct. 2016. doi: 10.48550/arXiv.1603.04571. url: https://arxiv.org/abs/1603.04571

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1603.04571 2016

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‘Edge Exchangeable Models for In- teraction Networks’

Harry Crane and Walter Dempsey. ‘Edge Exchangeable Models for In- teraction Networks’. In: Journal of the American Statistical Association 113 (June 2018), pp. 1311–1326. doi: 10.1080/01621459.2017.1341413. (Visited on 16/01/2025)

work page doi:10.1080/01621459.2017.1341413 2018

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Edge-exchangeable graphs and sparsity

Diana Cai, Trevor Campbell and Tamara Broderick. Edge-exchangeable graphs and sparsity. Neural Information Processing Systems, 2016. url: https://papers.nips.cc/paper_files/paper/2016/hash/1a0a283bfe7c549dee6c638a05 (visited on 05/08/2024)

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work page 2023

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work page doi:10.1214/21-ejs1916 2021

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work page doi:10.1214/07-aop350 2008

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