Luca Gugelmann, Konstantinos Panagiotou and Ueli Peter (2012), ICALP 2012, Lecture Notes in Computer Science — ARXIV-1205.1470
Key takeaways
- Model. n points in a hyperbolic disk of radius R = 2 log n + C, angles uniform, radii with density α sinh(αr)/(cosh(αR) − 1), α > 1/2; two points joined if their hyperbolic distance is at most R. This is Krioukov et al.’s model at zero temperature with curvature fixed at −1, so α here is the ratio α/ζ there.
- Clustering (Theorem 2.1). The average of the local clustering coefficients is Θ(1) in expectation and with high probability (1 + o(1)) times its expectation. A lower bound, not a limit.
- Degrees (Theorems 2.2–2.4). The fraction of vertices of degree k is (1 + o(1)) times an explicit Gamma-function expression for k up to n^δ′, and the fraction of degree at least k is (1 + o(1)) times a constant times k^(−2α) up to n^(1/(2α)) up to logarithms: a power law with exponent 2α + 1 up to the maximum degree. The average degree tends to an explicit constant and the maximum degree is n
- Scope. Only the threshold model. The soft model with a distance-dependent connection probability is named and not treated.
Standing in the record
Filed on 2026-10-20 by the promotion audit (batch 057) for THEORY-188, whose promote_when asks for a proof, not a simulation, of the model’s degree law and clustering. It covers the T = 0 half of that request. Its reading is NOTE-1010, skimmed: the theorems and the geometric lemmas read, the proofs followed in outline only.