nucleation

THEORY-799: Under geometric coarse-graining of the S¹ hidden-metric network model the average degree flows as r^ν, with ν = 2/(γ − 1) − 1 when the degree exponent γ sets it and 2/β − 1 when β does, so the coarse-grained networks run toward a complete graph for γ < 3 or β < 2 and toward a ring otherwise

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Properties5
StatusProposed
Tagsnetwork-science, complex-systems
SourceLIT-869 — Multiscale unfolding of real networks by geometric renormalization
ExtendsTHEORY-190 — Under geometric coarse-graining of the S¹ hidden-metric network model, which merges blocks of r angular neighbours into supernodes linked if any of their members were, the supernodes' link probability approaches the S¹ form with β kept, κ′ = (Σκ^β)^{1/β}, μ′ = μ/r and R′ = R/r, its ratio to that form tending to 1 as their separation grows large against the block width and the coupling μκ′κ″; between nearby supernodes it exceeds that form
AnswersQUESTION-203 — How does the average degree of the S¹ hidden-metric model flow under geometric coarse-graining when the closure fails between nearby supernodes, and does an exact ν(γ, β, r) confirm or move the phase boundaries γ = 3 and β = 2?

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Source

García-Pérez, Boguñá and Serrano (2017), LIT-869: the flow ⟨k⟩′ = r^ν⟨k⟩ with ν = 2/(γ − 1) − 1 or 2/β − 1, and the phases it implies, derived from the renormalized S¹ parameters under a power-law approximation.

What is asserted, and by whom

That the average degree of the model obeys this flow, with phase boundaries at γ = 3 and β = 2: small-world networks that densify toward a complete graph below them, and sparsening toward a ring above. The authors derive it; it is a statement about the model.

What the record holds toward it

  • THEORY-190 (Active): the coarse-grained link probability tends to the S¹ form with the stated maps as the supernodes’ separation grows large, with relative error of order β times block width over separation.
  • Against the derivation as given: between neighbouring supernodes the union of member links exceeds the S¹ form, by 32–86% in the worked example with two-node blocks (THEORY-190’s note of 2026-10-20). Most of a node’s links lie there, so the flow inherits the excess.

What would confirm or reject it

An exact ν(γ, β, r) that accounts for the excess and keeps the boundaries confirms it; one that moves them refutes the stated form. QUESTION-203 asks the same thing as a question.

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