Some quantity sits on the nodes of a network and moves along its edges. The same network appears in all three parts below; pick a different one at the top and everything rebuilds. The circle around each node shows how much of the quantity it holds: red for positive amounts, blue for negative ones.
1 Spreading with the adjacency matrix
Multiplying by A sends each node's whole amount to every neighbor at once: vn+1 = Avn, so An is n-step spreading. Nothing stops the total from growing.
Click a node to start with all of the quantity on it.
2 Diffusion with the Laplacian
With L = D − A (degrees on the diagonal), each node loses in proportion to its degree and gains from its neighbors. Diffusion runs continuously in time: dv/dt = −cLv, where c sets the speed.
Click a node to drop one more unit on it and highlight its equation.
Every node's equation
3 Eigenmodes: build a state and watch it settle
Every state can be written as a mix of the eigenvectors of L (Lxi = λixi). Under diffusion each ingredient fades on its own clock: v(t) = Σi ci e−λit xi. Mix your own starting state with the sliders, then drag or play time forward. Here c = 1 and the sign is −L.
Each curve is cie−λit; the vertical line is the current time.