The Flavor Complex FC

585 ingredients from 1,901,884 RecipeNLG recipes, with copies and same-formula variants merged. A set is a simplex when its ingredients co-occur more than chance predicts.

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Filtration by lift

4.00lift threshold t: pairs seen together ≥ 4× chance

168421.5

vertices: 585
edges: –
triangles: –
held back: –
β₀: –
β₁: –

β₁ (solid): triangles fill as soon as their edges exist, unless the trio's own count rules it out. Dashed: a triangle fills only once its count proves lift ≥ t. Click the chart to jump.

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Context fiber

Highlights pairs certified good within one course or dish type. FC is the union of these fibers.

Not flag: 46 empty triangles

Each pair is good, but the three together appear less often than chance (certified at 1% FDR). Their pairs are weak: in 45 of 46 the weakest pair has lift below 1.5. So they sit below the slider's range and are drawn dashed. At each t the slider's own non-flag triangles are the held-back ones, drawn solid.

How this is built

Lift. For a pair, lift = (n_ab+5)/(E_ab+5), where E_ab = n_an_b/N is the co-occurrence count expected if they were independent. The +5 shrinks rare pairs toward 1.

Filtration. K_t contains every edge with lift ≥ t. A triangle enters with its last edge, unless the trio's own count (95% upper bound) shows its lift is below t. 730 triangles enter later than their edges; at t = 1.5, 47 of 203,188 are still held back. Higher simplices are flag-filled. With this optimistic reading, persistence nearly matches the flag complex. The conservative reading has 1,571 H₁ bars against 1,199, most of them born at low lift.

Empty triangles are strict witnesses: all three pairs are significantly favored, and the trio is significantly below independence. 44 of the 46 stay significant after stratifying by website, course and dish. None stay significant inside a single dish fiber, so most non-flagness comes from mixing recipe contexts.

Cleaning.

H₂. Classes come from the persistence pairing of the same filtration (tetrahedra flag-filled). A representative is the class's birth triangle plus triangles that entered before it, which no older class can cancel; the small spheres shown were found by suspension search or local reduction and then shrunk.

Layout. UMAP on each ingredient's row of log-lifts (cosine), in three or two dimensions. The 3D view orbits with a quaternion trackball, so there is no gimbal lock; the axis that best separates sweet from savory starts out horizontal on wide screens. Color is the share of an ingredient's recipes that are desserts, breads or drinks.

Ingredient co-occurrence in RecipeNLG (Bień et al., INLG 2020), 1,823,093 recipes after dropping sites with stripped ingredient lists and merging copies and same-formula variants (recipes whose proportions agree within 10%). Pairs are tested with hypergeometric tests, triples with an exact no-three-way-interaction test, all at 1% FDR. Part of e-infinity.space/viz.