Poisson migration stream without a return path ($j\to k\text{ stream has rate }{\alpha }_j{p}_{jk}$)

If no directed routing path returns from $k$ to $j$, the set of colonies reachable from $k$ has no outgoing edge to its complement. The complement is an autonomous open migration process, and the marked $j\to k$ transition is an exit from it. In time reversal of an open migration process, that exit is a state-independent Poisson immigration channel of rate ${\alpha }_j{p}_{jk}$. Reversing a stationary two-sided Poisson process preserves its law, proving the forward stream is Poisson. This argument does not assert Poisson internal streams for networks with feedback.