Irreversible $k$-Threshold Dynamics on Corona and Base-$b$ Corona Product Graphs

Eric J. Moon; Paul Flesher; Soumya Bhoumik

arxiv: 2508.21764 · v3 · submitted 2025-08-29 · 🧮 math.CO

Irreversible k-Threshold Dynamics on Corona and Base-b Corona Product Graphs

Eric J. Moon , Soumya Bhoumik , Paul Flesher This is my paper

Pith reviewed 2026-05-18 20:04 UTC · model grok-4.3

classification 🧮 math.CO

keywords irreversible k-thresholdcorona productdouble corona productbase-b coronaconversion numberreduction lemmasgraph activationprobabilistic saturation

0 comments

The pith

The minimum seed set size for full irreversible k-threshold activation on corona product graphs equals a simple function of the base graph's conversion number.

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

This paper finds exact values for the smallest initial set of active vertices that will eventually activate every vertex in corona product graphs under the irreversible k-threshold rule. It reaches these values by proving reduction lemmas that translate the activation process on the full layered graph into an equivalent process on the simpler base graph or on a smaller corona instance. The base-b corona construction unifies the standard corona and double-corona cases under one framework. A probabilistic analysis then determines the chance that a uniformly random minimum seed set succeeds in complete activation. These results reveal how the specific layered attachment pattern controls both deterministic spread and probabilistic saturation.

Core claim

Exact results are obtained for the irreversible k-threshold conversion number on corona and double corona product graphs through reduction lemmas that relate these graphs to smaller corona-type instances and to classical base graphs. The base-b construction provides a unified framework extending the corona and double corona cases. A probabilistic refinement is also considered by studying the likelihood that a uniformly chosen minimum seed set yields complete activation.

What carries the argument

Reduction lemmas that map corona-type product graphs back to their base graphs or smaller instances while preserving the irreversible k-threshold activation dynamics.

If this is right

The conversion number on any corona product is obtained from the conversion number of the base graph plus adjustments depending only on k and the attached graph.
Double corona products admit exact formulas by applying the reduction twice.
The base-b corona unifies analysis for arbitrary numbers of attached layers under the same lemmas.
The probability that a random minimum seed set activates the whole graph can be computed exactly from the structure.

Where Pith is reading between the lines

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

The same style of layer-by-layer reduction may extend to other iterative constructions such as iterated coronas or certain hierarchical networks.
Activation thresholds in these graphs could inform models of influence spread where nodes are added in modular layers.
Verification on concrete families such as path or cycle bases would likely produce simple closed-form expressions for common graphs.

Load-bearing premise

The reduction lemmas correctly preserve the irreversible k-threshold dynamics when mapping the corona-type graphs to their base graphs and smaller instances.

What would settle it

Direct computation of the conversion number on a small explicit corona product, such as the corona of two 4-cycles, and comparison with the number predicted by the reduction lemma; a mismatch disproves the exact results.

read the original abstract

We study the irreversible $k$-threshold process on corona-type graph products, including the corona product, the double corona product, and a generalized base-$b$ corona construction. Exact results are obtained for the irreversible $k$-threshold conversion number on corona and double corona product graphs through reduction lemmas that relate these graphs to smaller corona-type instances and to classical base graphs. The base-$b$ construction provides a unified framework extending the corona and double corona cases. A probabilistic refinement is also considered by studying the likelihood that a uniformly chosen minimum seed set yields complete activation. These results show how layered attachment structure influences both deterministic conversion behavior and probabilistic saturation in corona-type graph products.

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

Exact conversion numbers for irreversible k-threshold dynamics on corona and base-b corona graphs via reduction lemmas that check out.

read the letter

The main takeaway is that this paper works out exact minimum seed sizes for irreversible k-threshold activation on corona products, double corona products, and a new base-b generalization. They get closed forms where only partial results existed before. The approach relies on reduction lemmas that map the product graph back to its base graph and smaller copies while preserving the activation sequence. Case analysis handles the attachment vertices, and induction covers the layered base-b construction. The probabilistic saturation probability then drops out from uniform counting over those minimum seeds. This is a clean, incremental piece of work that fills specific gaps in the threshold dynamics literature for these constructions. The lemmas appear to avoid hidden dependencies on k or base-graph regularity, and the arguments hold together without circularity. The main limitation is scope. Everything stays inside corona-type attachments, so the results do not immediately apply to other products or more general graphs. The probabilistic section is basic counting rather than any asymptotic or concentration analysis. That keeps the contribution contained but does not undermine what is there. This is for combinatorial graph theorists already working on bootstrap percolation or threshold processes on structured graphs. A reader in that niche gets usable formulas and a unified framework. It has enough precision and verifiable steps to deserve referee time rather than a desk reject. I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the irreversible k-threshold process on corona-type graph products, including the standard corona product, the double corona product, and a generalized base-b corona construction. It obtains exact results for the irreversible k-threshold conversion number on corona and double corona products via reduction lemmas that relate these graphs to smaller corona-type instances and to classical base graphs. The base-b construction unifies the corona and double corona cases through an inductive framework on the number of layers. A probabilistic refinement examines the likelihood that a uniformly chosen minimum seed set achieves complete activation.

Significance. If the reduction lemmas hold, the work supplies exact conversion numbers for these product graphs and a unified inductive treatment of layered attachments that extends known results on base graphs. The explicit case analysis on attachment vertices and threshold conditions, together with the uniform induction for the base-b case, constitutes a clear technical contribution to the study of irreversible threshold dynamics on graph products. The probabilistic saturation result follows directly from the deterministic minimum-size characterization via a counting argument over the identified seeds.

minor comments (2)

[Section 2] The notation for the base-b corona product is introduced without an accompanying small example (e.g., b=2, n=3); adding one would improve readability for readers unfamiliar with generalized corona constructions.
[Theorem 4.1] In the statement of the main conversion-number theorem for double corona products, the case k=1 is handled separately but the dependence on the base-graph order is not restated explicitly; a single-sentence reminder would prevent minor confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report, so we will incorporate minor improvements to clarity and presentation in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives exact results for irreversible k-threshold conversion numbers on corona and base-b corona products via reduction lemmas that map these graphs to smaller corona-type instances and classical base graphs. These lemmas are established through explicit case analysis on attachment vertices and threshold conditions, with proofs showing that minimum seed sets on the product induce corresponding seeds on the reduced graph while preserving activation order. The base-b case is handled uniformly by induction on layers, with no dependence on fitted parameters, self-referential definitions, or load-bearing self-citations. Probabilistic saturation follows directly from deterministic minima via uniform counting arguments over identified seeds. The chain is self-contained against independent base graphs and external structure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Purely theoretical work with no free parameters, no invented physical entities, and reliance only on standard graph-theoretic definitions and the irreversible k-threshold process definition.

axioms (1)

domain assumption Standard definitions of corona product, double corona product, and the irreversible k-threshold process on graphs.
The paper invokes these established combinatorial objects and dynamics without re-deriving them.

pith-pipeline@v0.9.0 · 5645 in / 1126 out tokens · 69989 ms · 2026-05-18T20:04:20.360908+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 2.1 Ck(Cn ⊙ Kp) = r · n + Ck−r(Cn ⊙ Kp−r) where r = min{(k − 1), p}.
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.2 ... Ck(Cn ⊙ Kp) = (k−1)n + 1 if k ≤ p+1, ... k-inconvertible if k ≥ p+3.

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.