Ultrametric Convergence — Variance Collapse Under Coarse-Graining
Watch diversity collapse into uniformity — not by intervention, but because ultrametric geometry makes it inevitable
Tree Parameters
Prime (p)
p = 2
p = 3
p = 5
Tree Depth (d)
3
4
5
6
Coarse-Graining Level
k = 0 (leaves)
▶ Run Collapse
↺ Reseed
Reset
✓ Verify
Results
Variance at level k
—
Variance at k = 0 (leaves)
—
Collapse Ratio
—
Var(X) = E[Var(X|Y)] + Var(E[X|Y])
Law of total variance: coarse-graining reduces variance because it eliminates within-group variation. The second term
Var(E[X|Y])
≤ Var(X).
In an ultrametric tree, each level
k
groups leaves by their ancestor at depth
d−k
. Variance monotonically decreases with k.
Seed:
Apply
Ultrametric Tree (k =
0
)
Leaf Value Distribution
Variance vs. Coarse-Graining Level
Welcome!
Next →
Skip