THEORY-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

Source

García-Pérez, Boguñá and Serrano (2017; Nature Physics 2018), LIT-869, read in NOTE-673.

What was actually shown

The model is S¹: nodes on a circle of radius N/2π with angles θ and hidden degrees κ, linked with probability 1/(1 + (RΔθ/(μκκ′))^β). Its hyperbolic form is the ℍ² model of THEORY-188. The transformation merges blocks of r consecutive nodes on the circle into supernodes, linked if any of their members were.

• Closure. The probability that two supernodes are linked is one minus the product of the r² member pairs’ non-link probabilities. Cut after its first-order term, and with every member pair treated as at the blocks’ separation, it is again of the S¹ form, provided κ′ = (Σκ_j^β)^{1/β}, μ′ = μ/r, R′ = R/r and β′ = β. The supernode angle is any order-preserving mean; the source picks a power mean that composes. This could have failed in simulation. On a synthetic network of about 225,000 nodes (γ = 2.5, β = 1.5), the empirical connection curve stays on the theoretical one through eight layers, and mean clustering stays flat (Fig. 3A).
• Degree exponent. A sum of r variables κ^β stays power-law, and γ is asymptotically kept, when (γ−1)/2 < β. Otherwise scale-freeness is lost along the flow.
• Average degree. ⟨k⟩′ = r^ν⟨k⟩. If the renormalized hidden degrees stay a pure power law, which the source says is not true in general, then ν = 2/(γ−1) − 1 where (γ−1)/β < 1 and ν = 2/β − 1 where 1 < (γ−1)/β < 2. So ν > 0, and the flow goes to a complete graph, if γ < 3 or β < 2. ⟨k⟩ is kept on γ = 3 (β ≥ 2) or β = 2 (γ ≥ 3), an unstable fixed line between small-world and non-small-world. ν < 0, and the flow goes to a ring that loses the small world, if γ > 3 and β > 2. An exact series for r = 2 agrees with this for large γ or β.
• Links kept are long-range. Links that survive to a coarser layer join nodes at larger angular separation.

What this does not say

• Not that real networks have hidden metric spaces. The source finds six embedded real networks self-similar under the flow and calls this new evidence that hidden metric spaces underlie real networks. The account here is about the model. In the data, the collapse is judged by eye over two to five layers, with no null ordering of the circle to show that geometric blocks are what produce it.
• Not exact. Closure holds to first order with equidistant blocks, and that is weakest for adjacent supernodes. The closed forms for ν rest on a further approximation. No error bound is given.
• Not a hierarchy found in the data. Each node sits in one supernode per layer, so the layers nest, but the nesting is set by the procedure (contiguous blocks on the circle), not discovered. The ℍ² radius is popularity, as in THEORY-188. Neither is a hierarchy in which one kind implies another, which is what QUESTION-025 asks about.
• Not unique. Other choices of block (equal arcs, connected pairs only) are said to give the same closure. The supernode angle is a convention, and its power-mean form depends on where the angle origin is placed.

Note of 2026-10-20: the closure at large separation, and where it fails

Worked for the promotion audit (batch 048).

The bound. In S¹, members i and j at arc distance d, with a_ij = μκ_iκ_j, link with p = 1/(1 + (d/a_ij)^β), so (a/d)^β(1 − (a/d)^β) ≤ p ≤ (a/d)^β. Let two blocks of r members each span an arc of at most w, with supernode angles inside their blocks, at arc distance D = RΔθ′. Every member pair then has d_ij in [D − w, D + w]. The supernodes link with the union probability U, and by the Bonferroni inequalities Σp_ij − Σ_{pairs} p p′ ≤ U ≤ Σp_ij. Since Σ_ij a_ij^β = μ^β(Σκ_i^β)(Σκ_j^β) = (μκ′κ″)^β with κ′ = (Σκ^β)

U = (μκ′κ″/D)^β (1 + ε), |ε| ≤ βw/D (1 + o(1)) + O(r² max(a_ij/D)

After coarse-graining, μ′ = μ/r and R′ = R/r give R′Δθ′/μ′ = D/μ, so the S¹ form is p′ = (μκ′κ″/D)^β(1 + O((μκ′κ″/D)^β)). Hence U/p′ → 1 as D grows large against w and μκ′κ″, with β unchanged.

Where it fails. Two-node blocks of equal hidden degree with μκ² = 1, members at unit spacing (positions 0, 1 and s, s + 1), supernode separation s. Then U = 1 − Π(1 − 1/(1 + d^β)) over the four member pairs and p′ = 1/(1 + (s/2^{2/β})^β). The ratio U/p′ at s = 2, 4, 8, 16, 64 is 1.32, 1.18, 1.07, 1.03, 1.003 for β = 1.5; 1.42, 1.17, 1.05, 1.01, 1.001 for β = 2; and 1.86, 1.23, 1.05, 1.01, 1.001 for β = 3. The excess is largest between neighbouring supernodes, which is where most of a node’s links lie, so a degree flow derived from the closed form inherits it.

Note of 2026-10-20: the wider claim, filed Proposed

Under ADR-047 v4 the narrowing-loss sweep (batch 069) filed THEORY-799 and THEORY-803 as Proposed, extending this entry. They carry the two claims this entry set aside when batch 048 narrowed it: the source’s degree flow ⟨k⟩′ = r^ν⟨k⟩ with its phase boundaries, and the source’s finding that real networks are self-similar under the renormalization, as evidence of hidden metric spaces. This entry is their evidence and is not changed by the filing.