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Quantitative BiologyEnglishPublished

High‑order hyperbolic embeddings group scientists and proteins more coherently than pairwise methods

November 3, 2026arXiv: 2610.41684v1

A new embedding called hyperbolic higher‑order (HHO) inserts higher‑order connections—groups of three or more nodes—directly into a hyperbolic space. In the Cora co‑authorship network, HHO shrank 61 % of the triangles formed by three authors that had non‑zero area in the pairwise embedding, with a median area ratio of 0.577. The ordering of filled and empty triangles is only weakly separated, with a fold of 1.05 and a probability that a filled triangle is smaller than an empty triangle of 0.533.

Figure 3 panels c–e illustrating area compression and distance preservation for Cora: histogram of log10(AHHO/Apairwise) (median 0.577), density of log10​areas (fold 1.05, P≈0.533), and scatter of embedding distance vs. shortest‑path distance (Spearman ρ = 0.417).
Figure 3. Figure 3 panel c–e display the distribution of area ratios for filled triangles, the density of triangle areas, and the Spearman correlation between hyperbolic and shortest‑path distances in the Cora co‑authorship complex. Median ratio 0.577 and fold 1.05, P(Afilled < Aempty) = 0.533 are shown.Guillemaud, Martin, Latora, Vito, Chavez, Mario · arXiv:2610.41684v1 · Figure 3: ·
CC BY 4.0

For the Escherichia coli protein interaction map, HHO brought the members of known protein complexes into highly compact regions: 99.7 % of the 5 366 filled triangles were smaller than under the pairwise embedding and the median ratio was 7.9 × 10⁻⁴. The probability that a filled triangle is smaller than an empty one rose from 0.704 in the pairwise graph to 0.993 under HHO.

Figure 4 panels c–e depicting area compression and distance preservation for E. coli: histogram of log10(AHHO/Apairwise) (99.7 % smaller), density of log10​areas (fold ≈ 1, P≈0.993), and scatter of embedding distance vs. shortest‑path distance (Spearman ρ = 0.683).
Figure 4. Figure 4 panels c–e show the distribution of area ratios for filled triangles, the density of triangle areas, and the Spearman correlation between hyperbolic and shortest‑path distances in the E. coli protein‑interaction network. 99.7 % area ratio ≈ 7.9×10⁻⁴ and P(Afilled < Aempty) = 0.993 are presented.Guillemaud, Martin, Latora, Vito, Chavez, Mario · arXiv:2610.41684v1 · Figure 4: · CC BY 4.0

Across both studies the correlation improved for Cora (–0.0784 to 0.417) but declined for E. coli (0.712 to 0.683). These gains suggest that incorporating higher‑order interactions gives a more faithful geometric representation of the underlying simplicial complexes.

The visualizations (Figure 3 and Figure 4) illustrate the area compression and distance preservation achieved by HHO. The same quantitative trends that appear in the tabulated and plotted results are emphasized in the figures, confirming that higher‑order structure reorganises angular coordinates and normalises gaps between triangles.

Source paper

Hyperbolic embedding of higher-order simplicial networks

Quantitative BiologyPhysics
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High‑order hyperbolic embeddings group scientists and proteins more coherently than pairwise methods | arXiv News