Contribution
Puts Krioukov et al.’s claim that temperature tunes clustering, with a transition at T = 1, on a rigorous footing for the global clustering coefficient, and finds a second transition the simulations did not report: in the regime of degree exponent at most 3 the global coefficient vanishes at every temperature.
Key results
- Theorem 1.2. In the binomial model G(N; ζ, α, β, ν) with 0 < ζ/α < 2: if β > 1, C2 → L∞(β, ζ, α) ∈ (0, 1] in probability when ζ/α < 1, and C2 → 0 when 1 ≤ ζ/α < 2; if β ≤ 1, C2 → 0. L∞ is an explicit triple integral with C_β = 2/(β sin(π/β)), and does not depend on ν.
- Theorem 1.3. For fixed t > 0 and β > 1, the global clustering of the subgraph induced by vertices of type at most t converges to an explicit positive L(t; β, ζ, α) for every 0 < ζ/α < 2.
- Cited, not read here. The degree law with exponent 2α/ζ + 1 for the binomial model at β > 1 is attributed to Fountoulakis (a paper the record does not hold), and the density transition at β = 1 (average degree growing at least logarithmically for β ≤ 1) likewise.
Method (from §1.3, not checked)
A vertex’s type is R minus its radius. Two vertices of types t_u, t_v are adjacent with probability bounded away from zero roughly when their relative angle is below e^((ζ/2)(t_u + t_v − R)). Triangles among vertices of type below R/2 number order N; paths of length two number order N when α > ζ, but order RN when α = ζ and more when α < ζ, because vertices of type near R/2 with degree about N^(1/2) appear. The ratio then tends to zero.
Limitations
- Global coefficient only. The average of local coefficients, which LIT-865 simulates and LIT-1282 bounds, is not treated; the authors say their analysis would need significant modification for it.
- Proofs not read. The record holds the statements.
Bearing on the record
Bears on THEORY-188’s clustering clause. “Clustering stays bounded away from zero below T = 1 and vanishes above it” holds for the global coefficient only when ζ/α < 1, that is degree exponent above 3; for exponents between 2 and 3 the global coefficient vanishes below T = 1 as well. The title as written is therefore too broad for the global coefficient, and the held sources say nothing about the local average at 0 < T < 1.