Superconcentration on a Pair of Butterflies

Abstract: 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.