Cascade Detection via Fixed-Point Iteration

Committing a flight to the live network raises the load on every sector along its route. If any sector exceeds capacity, the excess must be redirected — but redirection can push the next sector over capacity, propagating the disruption. Cascade detection asks whether one round of redirection settles the load surface or triggers another round.

We model propagation with a cascade operator $T$ acting on the load vector $\mathbf{L}=(L_1,\ldots,L_n)$ over sectors arranged in a downstream chain with capacities $C_1,\ldots,C_n$. Define the excess at sector $i$ as

$$e_i = \max(0,\, L_i - C_i).$$

The cascade operator subtracts each excess and pushes it one step downstream:

$$T(\mathbf{L})_i = L_i - e_i + e_{i-1}, \qquad e_0 = 0,$$

with $e_n$ leaving the chain (absorbed at the boundary).

A cascade is present at $\mathbf{L}$ when $T(\mathbf{L}) \neq \mathbf{L}$, equivalently when some $e_i > 0$.

For instance, with $\mathbf{C}=(3,2,2)$ and $\mathbf{L}=(4,1,0)$:

$$\begin{align*} e_1 &= \max(0,4-3) = 1, \quad e_2 = 0, \quad e_3 = 0, \\[3pt] T(\mathbf{L}) &= (4-1+0,\ 1-0+1,\ 0-0+0) \\[3pt] &= (3,2,0). \end{align*}$$

Since $T(\mathbf{L}) \neq \mathbf{L}$, a cascade is present.