  Suppose we concatenate two directed graphs, each isomorphic to a $d$
dimensional butterfly (but not necessarily identical to each other). Select any
set of $2^k$ input and $2^k$ output nodes on the resulting graph. Then there
exist node disjoint paths from the input nodes to the output nodes. If we take
two standard butterflies and permute the order of the layers, then the result
holds on sets of any size, not just powers of two.
