Let $G = (L,R,E)$ be a bi-regular bipartite graph, with $|L|=n$ and $|R| = C cdot n$, where $C$ is a large constant. Let $d$ be its (constant) right-degree.

We know $G$ is a good spectral expander. Say, for concreteness, that its normalized second eigenvalue is roughly $frac{1}{sqrt{d}}$, so in particular we have good vertex expansion from both sides.

I wish to pick a random subset $S subseteq R$ of cardinality, say, $frac{n}{4}$. Let’s think of $d >4$, so we are at a situtation in which with high probability, $Gamma(S)$ covers nearly all of $L$. Let $G’ = (Gamma(S),S,E’)$ be the induced subgraph, where we keep only the edges that come out of $S$. Note that $G’$ is still $d$ right-regular, but is no longer left-regular.

The question is: What can we say about the expansion propetries of a typical $S$? Specifically, what is the expected vertex expansion from left to right? I don’t expect it to be as good, but is it only mildly good, say that each small enough set $X subseteq Gamma(S)$ is mapped to a set of size $frac{|X|}{2}$? (note that it’s actually a good expansion, as the “imbalanceness” is roughly $frac{1}{4}$). A simple union-bound does not seem to give anything meaningful.