Source
Bianconi and Rahmede (2017), LIT-867, read in NOTE-678. The thin-triangle bound below is the record’s own. It is written out in record/promotion-audit/decisions/THEORY-193.yaml.
What was actually shown
The model: start from one d-simplex. At each step a (d − 1)-face α is chosen with probability ∝ 1 + s n_α, where the flavor s is −1, 0 or 1 and n_α is the number of simplices on α minus one. A new d-simplex is glued to α through one new node.
- The gluing is a tree. Each new simplex meets the old complex in one face, and has as parent the earliest simplex holding that face. With s = −1 a face holds at most two simplices, so the face-adjacency graph (the source’s dual) is itself a tree. With s = 0 or 1 several simplices can be glued on one face, which makes them a clique in that graph. The parent structure is a tree for every flavor.
- The 1-skeleton is Gromov hyperbolic, with δ ≤ 2 at every size (the record’s derivation). The simplices, with parent links, give a tree decomposition of the 1-skeleton G whose bags are cliques: each vertex’s simplices form a subtree, because a simplex holding an old vertex holds it in the face it was glued to. If a bag lies on the tree path between bags of u and v, every u–v path in G meets it. Now take a geodesic triangle x, y, z and a point p on [x, y]. Let q be the node of the tripod spanned by bags of x, y and z that is nearest a bag of p. Then q’s bag meets [x, p] and [p, y] on either side of p, so p is within 1 of it. It also meets a second side of the triangle. Since a bag is a clique, p is within 2 of that side. The bound does not depend on N, d or s. In graph-theoretic terms, G is chordal with tree-length 1.
- The drawing. With every link of equal length, the source draws each simplex as an ideal simplex of the Poincaré ball. The initial nodes sum to zero, and each new node sits at the normalised sum of its parent face’s positions. For d = 2 and s = −1, each new node lands on the open arc cut off by its parent edge. The drawn triangles then have disjoint interiors and fill the disc as every free edge is glued: a tiling of H² by ideal triangles. For s = 0 or 1, siblings coincide (NOTE-678). For d ≥ 3, a (d − 2)-face of an ideal simplex has at least two ideal vertices, so it passes through H^d. In a face-to-face tiling, the tiles around such a face form a finite cycle of face-adjacent simplices, whose dihedral angles sum to 2π. The grown complex has no such cycle. So for d ≥ 3 the drawing is not a tiling: around each such face the simplices form a path, and they overlap once their angles pass 2π or never close up. The source asserts the tiling and its uniqueness without proof, and NOTE-678 grades that part weak.
- The statistics, derived. Degrees follow from a master equation: geometric for d + s = 1, and a power law with γ = 2 + 1/(d + s − 1) ≤ 3 for d + s > 1, matched by simulation. Simulation also gives high clustering and modularity for d ≥ 2. None of this depends on the drawing, and it could have failed.
What could have come out otherwise is the degree law, the clustering, and the thin-triangle bound, whose proof needs the gluing to be a tree and would not carry over to a rule that closes cycles of simplices. The drawing’s constant curvature could not: the equal-length convention defines it.
What this does not say
- Not that the drawing’s curvature comes from the growth. The source says the constant-curvature geometry follows from a “proto-geometry” of equal links. With links of different lengths, the same complex can be embedded in a d-sphere. What the growth fixes is the coarse geometry of the graph metric, which is hyperbolic whatever drawing is chosen.
- Not that real networks grow this way, or that their hyperbolicity (THEORY-188) has this origin. No real network is fitted.
- Not much beyond what trees already give. A tree embeds in hyperbolic space with low distortion (Sarkar–Sala, THEORY-196), and a tree is 0-hyperbolic. This account extends the point to complexes whose gluing is a tree, with δ ≤ 2, and adds the growth statistics. It says nothing about complexes with cycles of simplices, and it gives no low-distortion embedding in H^d with finite link lengths.
- Not a statement about learned embeddings. The source’s placement of a node at the normalised sum of its ancestors is a rule it chooses. It does not bear on whether a co-occurrence embedding carries a hierarchy that way, which QUESTION-025 asks.