Backpropagation on a computational graph

In topological order the adjacency matrix is strictly upper-triangular — every edge points from a lower to a higher index. The forward pass fills v and the local Jacobian D (edge labels). The backward pass seeds ∂f/∂f = 1 and, in reverse order, sets ∂f/∂i = Σ_j (∂f/∂j)·Dij over children j — the chain rule. Node x feeds two paths, so ∂f/∂x is a sum of both contributions. Gradient step: (x,y) ← (x,y) − η·∇f (∇f is norm-clipped to 6 for a stable animation).