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THEORY-803: Real complex networks embedded in the hyperbolic plane are self-similar under geometric renormalization, their coarse-grained layers keeping the degree distribution, clustering and degree–degree correlations of the original, and this self-similarity is evidence that a hidden metric space underlies them

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Properties4
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

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Source

García-Pérez, Boguñá and Serrano (2017), LIT-869: six real networks, read in NOTE-673 (the Internet, airports, a metabolic network, a proteome, a music network and a word network), embedded in ℍ², renormalized, and their layers compared with the original (Fig. 7).

What is asserted, and by whom

That real networks show the self-similarity the S¹ model has under geometric coarse-graining, and that this is new evidence that hidden metric spaces underlie them. The authors assert both.

What the record holds toward it

  • THEORY-190 (Active): what the model does under the transformation, asymptotically. It says nothing of real networks; its former promote_when said explicitly that a collapse in data cannot show the model is why.
  • Against the inference: the layers nest because the procedure takes contiguous blocks on the circle, so some collapse may come from the blocking itself. No null ordering was run, and the collapse was judged by eye.

What would confirm or reject it

The comparison against a null ordering that the promote_when describes. If null layers collapse as well, the self-similarity is not evidence of the geometry.

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