Source
Yang, Zhou, Ying, Chen & King (2023), LIT-871 (ICML 2023), read in full in NOTE-669.
What was actually shown
The assumption tested. Hyperbolic embedding methods train on losses that never mention the hierarchy: a softmax over hyperbolic distances, a cross-entropy for node classes, a Fermi–Dirac link likelihood. A hierarchy’s root near the origin and its levels further out is expected to arise by itself, after Krioukov et al.’s placement of level r at radius proportional to r. The paper measures it with the hyperbolic distance of each node to the origin (HDO), read as level.
The trees. Two synthetic 3-ary trees, eight levels, 1,093 nodes, with labels following the tree two ways (by top-level subtree, homophily 0.998; by level, homophily 0.018).
Position (§3.2, Fig. 2). HGCN for link prediction or node classification puts the root at HDO 1.8–3.3 while the minimum is 1.2–2.4. The HDO distribution is roughly normal, where a tree’s level counts would put most nodes far out.
Level order (Table 9). Of 5,000 random node pairs, the fraction whose HDO order matches their true level order is 72.5% and 68.9% at dimension 16 and 75.3% and 73.7% at 256. With recentring on the hyperbolic centroid and a loss rewarding a large HDO-weighted mean HDO, it is 75.4–83.5%.
What could have come out otherwise. The root could have sat at the minimum and the order could have been near complete. Neither held.
What this does not say
- Why the root is off-centre. A translation of the whole embedding changes no pairwise distance, so under a loss built only on pairwise hyperbolic distances the origin is arbitrary (NOTE-669, Key insight). How much of the diagnosis survives recentring alone is not separated. The ablation’s alignment-only column hints that on the most tree-like data recentring is most of it.
- That the fix recovers hierarchy. The outward push uses each node’s own current HDO as its weight, and the root is the centroid, so it amplifies whatever radial order training produced. Its largest benchmark gains are on Citeseer and Cora, the least tree-like graphs, which the authors credit to class separation. On the real graphs levels are never measured.
- That other hyperbolic models behave alike. Asserted, not shown; Fig. 2 is HGCN only.
- That radius is the only encoding of hierarchy. Angular containment and entailment cones (THEORY-189) are not considered.
- Anything about attribute lattices. The hierarchies are graph trees, not implication among attributes.
Connections
- QUESTION-025. Not an answer. The caution it carries applies to any answer: a space suited to a hierarchy does not show that a trained representation in it holds the hierarchy, which has to be measured on data with known levels.
- Ganea et al. (LIT-879, THEORY-566) is among the works this paper says assumed the radial layout would arise by itself.
- THEORY-196 gives the near-isometric tree embedding the geometry makes possible. This account says task training realises it only partly.
Note of 2026-10-20: narrowed and promoted
The promotion audit (batch 073) narrowed the title to what LIT-871 measured and promoted it on that one study, under ADR-047 v3. NOTE-669 read it in full; the review re-checked Figure 2’s inset statistics (ROOT against MIN: 3.1/2.0, 1.8/1.2, 3.3/2.1, 2.7/2.4) and Table 9 (72.5%, 68.9%, 75.3%, 73.7%) against the arXiv PDF. The old title opened with hyperbolic graph embeddings in general; the paper shows HGCN only and asserts the rest (“other hyperbolic models have demonstrated comparable results”). Under ADR-047 v4 that general claim is not dropped: it is filed as the Proposed THEORY-825, which extends this account and inherits its former promote_when. The pairwise-distance clause is the record’s own and was re-derived step by step: a loss that depends only on pairwise hyperbolic distances is invariant under every isometry, the isometries act transitively, so the loss does not fix where the origin lies. It covers the link-prediction runs; HGCN’s classifier reads the tangent space at the origin and is not pairwise-only.