Example Flow

Consider a network with $4$ nodes: source $s$, nodes $a$ and $b$, and sink $t$. Edges: $(s,a)$ capacity $10$, $(s,b)$ capacity $5$, $(a,b)$ capacity $15$, $(a,t)$ capacity $10$, $(b,t)$ capacity $10$.

Maximum flow is $15$: send $10$ through $s \to a \to t$ and $5$ through $s \to b \to t$. The bottleneck is the total capacity leaving $s$ ($10$ + $5$ = $15$).

This small example illustrates flow conservation: what enters a node must leave it (except source and sink). Maximum flow finds the best way to route without exceeding capacities.