Elisabetta Candellero and Nikolaos Fountoulakis (2015), Internet Mathematics — ARXIV-1309.0459, DOI-10.1080/15427951.2015.1067848
Key takeaways
- Two models. The disc model joins vertices at distance at most R; the binomial model joins them with probability 1/(exp(β(ζ/2)(d(u, v) − R)) + 1), β the inverse temperature, and has the disc model as its β → ∞ limit. N = ν e^(ζR/2) vertices.
- Global clustering (Theorem 1.2). For β > 1, C2 → L∞(β, ζ, α) > 0 in probability when 0 < ζ/α < 1, and C2 → 0 when 1 ≤ ζ/α < 2. For β ≤ 1, C2 → 0. The limit does not depend on ν, so clustering and average degree can be tuned separately.
- Why it vanishes for heavy tails. When ζ/α ≥ 1 (degree exponent at most 3), vertices of type above R/2 appear; their degrees add many paths of length two and few triangles. Removing them, the clustering of the subgraph of vertices of type at most t stays positive for every ζ/α < 2 when β > 1 (Theorem 1.3).
- Not the local average. The result is for C2. The average of local coefficients, the quantity Krioukov et al. simulated and Gugelmann et al. bounded, is said to need “significant modification” of the analysis.
Standing in the record
Filed on 2026-10-20 by the promotion audit (batch 057) for THEORY-188’s clustering clause. It confirms clustering is tuned by the temperature with a transition at T = 1, but only for the global coefficient, and only when the degree exponent exceeds 3; below that the global coefficient vanishes at every temperature. Its reading is NOTE-1011, skimmed: §1 read in full, the theorems and the heuristic of §1.3 followed, the proofs of §§2–8 not read.