Source
Krioukov, Papadopoulos, Kitsak, Vahdat and Boguñá (2010), LIT-865, read in NOTE-680.
What was actually shown
The model: N nodes in a disk of radius R ~ (2/ζ) ln N in the hyperbolic plane of curvature −ζ², angles uniform, radial density ∝ sinh(αr); each pair linked independently with probability 1/(e^{β(ζ/2)(x−R)} + 1), x their hyperbolic distance, T = 1/β.
- Degrees. For large R, a node at radius r has expected degree ∝ e^{−ζr/2}, and since radii are exponentially distributed the degree distribution is a power law with γ = 2α/ζ + 1 when α/ζ ≥ ½ (γ = 2 below), for T < 1; for T > 1, γ = 2αT/ζ + 1. Only the ratio of the density exponent to the curvature matters. Fig. 4 checks the degree formula against simulation; this part could have failed and did not.
- Clustering. Below T = 1 average clustering is positive and decreases almost linearly from its maximum at T = 0 to zero at T = 1, where the chemical potential R diverges (a phase transition in the ensemble); above T = 1 it is zero in the large-network limit. The cold-regime curve is numerical integration against simulation (Fig. 6), exact only at β = 2, γ = 3.
- Equivalence. The S¹ model, nodes on a circle with power-law hidden degrees κ and connection a function of distance over κκ′, maps onto this one under κ = κ₀e^{ζ(R−r)/2}: degree becomes radial depth. The mapping is exact for the approximate distance x ≈ r + r′ + (2/ζ) ln sin(Δθ/2).
- Degenerate limits. Sending ζ and T to infinity at fixed ratio removes the angular term, so x = r_i + r_j and link probability depends only on the product of expected degrees: the configuration model, with no clustering. Heating at fixed α, ζ gives Erdős–Rényi graphs.
Statistical mechanics frames it: edges are fermions with energy the hyperbolic distance, and the exponential random graph’s link fields are linear in it.
What this does not say
- Not that real networks are hyperbolic. The source fits the Internet’s degree distribution, neighbour degree and clustering with three parameters (α = 0.55, ζ = 1, β = 2). That shows the model can reproduce those statistics, not that the Internet’s links were generated by distance in a hyperbolic space. Inferring coordinates for a real network is left to other work.
- Not that every heterogeneous network has a hyperbolic geometry. The source’s “effective hyperbolic geometry” is the S¹ ⇄ H² equivalence: it assumes the network already has a similarity metric and power-law hidden degrees, and shows the two together can be read as a hyperbolic space. The extension to other boundaries is argued by analogy.
- Not that hierarchy causes heterogeneity. The source motivates hyperbolicity by approximately tree-like hierarchies of node similarity and reads e^α as their branching factor. In the model, the “hierarchy” is the radial ordering by expected degree. Nothing in the derivation needs a hierarchy of groups, and nothing tests that one is present.
- Not that hyperbolic geometry is the only route to the two signatures. Other mechanisms produce power laws and clustering. The account says that this geometry produces both from one assumption, and that removing the metric term removes the clustering.
- Not a statement about embeddings of data in machine learning. The contrast with THEORY-185 is the record’s, not the source’s: there, log co-occurrence additive across independent factors gives linear structure; here, the additive per-node part alone is the degenerate configuration model, and the structure sits in the non-additive angular term. QUESTION-025 is where that contrast is open.
Note of 2026-10-20: the rigorous literature
Promotion audit, batches 048 and 057. The promote_when asks for proofs in the limit N → ∞. Two held sources now state them in part.
- Degrees, T = 0. Gugelmann, Panagiotou and Peter (LIT-1282, Theorems 2.2–2.4) prove for the threshold model at curvature −1 that the degree sequence is a power law with exponent 2α + 1 up to the maximum degree, which is 2α/ζ + 1 at ζ = 1, with the large-distance approximation’s error bounded (their Lemmas 3.1–3.2). The degree law at T > 0 is attributed by Candellero and Fountoulakis to a paper of Fountoulakis’s that the record does not hold.
- Clustering, T = 0. The same paper proves the average local clustering coefficient is Θ(1) with high probability (Theorem 2.1): a lower bound, not a limit.
- Clustering, T > 0. Candellero and Fountoulakis (LIT-1283, Theorem 1.2) prove, for the global clustering coefficient C2 with 0 < ζ/α < 2: below T = 1, C2 tends to a positive constant when ζ/α < 1 and to zero when 1 ≤ ζ/α < 2; at and above T = 1, C2 tends to zero. So “bounded away from zero below T = 1” holds for C2 only when the degree exponent exceeds 3. For exponents between 2 and 3 the hubs drive C2 to zero at every temperature, while the clustering among vertices of bounded type stays positive (Theorem 1.3). Neither source treats the average local coefficient at 0 < T < 1, the quantity LIT-865 simulates.
- The configuration-model limit needs no source: with x = r_i + r_j the link probability is a function of e^(−ζr_i/2)e^(−ζr_j/2), the product of the two expected degrees, by inspection.
What keeps the account Proposed: both readings are skims (NOTE-1010, NOTE-1011), the theorems taken as stated and the proofs followed only in outline, which is what held THEORY-270 in batch 038; and the T < 1 local-clustering clause has no held source. If promoted, the title should name the clustering coefficient and the ζ/α < 1 condition for the global one.