Hodge decomposition

中文導讀

Hodge decomposition 把 manifold 上的 vector field 唯一拆成三塊:gradient(來自 scalar potential)、divergence-free rotational(curl)、harmonic(residual)。在 single-cell dynamics 裡(ddHodge),這三塊各自對應 biology:gradient→potential landscape / ordering、rotational→cycle(cell cycle)、divergence→ stability / differentiation potency。重點 insight:如果一個系統 gradient 佔絕大多數(embryogenesis ~88%),它本質上就是 Waddington 那種 potential-landscape;如果 rotational 很大(FUCCI ~49%),potential 就 給不出可靠的 temporal ordering。

Definition

For a vector field ω on a manifold, the Hodge decomposition is the unique split

ω = grad α + curl·β + γ
     (gradient) (rotational) (harmonic residual)

• gradient — derives from a scalar potential; integrable, “downhill” flow.
• rotational (divergence-free) — circulation / cycles; carries no potential.
• harmonic — the residual consistent with the manifold’s topology.

The divergence of the field measures source/sink behavior; the curl measures rotation.

Why it matters for cell-state dynamics

ddHodge applies this on the single-cell data manifold and reads each component biologically:

• gradient potential → temporal ordering (latent-time / pseudotime) and the potential-landscape (Waddington).
• divergence → cell-state stability / differentiation potency (negative = canalized/stable; positive = plastic/diverging).
• curl / harmonic → oscillation / cycle (e.g. cell cycle).

The proportion of gradient vs rotational is itself a finding: embryogenesis is ~88% gradient (a genuine potential-landscape system), whereas the FUCCI cell cycle is ~49% rotational (cyclic — where the potential gives no faithful ordering).

Relation to physical time

Hodge decomposition is a geometric analysis of an already-estimated field; it does not anchor absolute time (see physical-time-grounding). It enriches the structure of inferred time — separating directed (gradient) from cyclic (rotational) flow and exposing acceleration — but the potential ordering remains ordinal and scale-inheriting.

Related

ddHodge · potential-landscape · manifold-consistent-velocity · latent-time · physical-time-grounding · RNA velocity · optimal-transport