I'll be as intuitive and hand-wavy as possible so as to get to what I am looking for. Suppose we have a graph $G=(V,E)$. One notion of expansion, namely vertex expansion, can be intuitively described as how much a subset $A$ (of size at most $\|V\|/2$) grows within the graph. Here growth is described using edge connections of the graph, that is
$inf_{A} \frac{\|N_1(A)\|}{\|A\|}$
where $N_1(A)$ is the set of all vertices that can be reached from a vertex in $A$ using atmost one step (edge). This notion of expansion is useful and well-studied.
Is there a generalization of this using two steps, say? That is, let $N_2(A)$ be the vertices reachable from $A$ using at most two steps (or a length 2 path of edges). Then we can define a $2$-expansion $inf_{A} \frac{\|N_2(A)\|}{\|N_1(A)\|}$ In a sense, this describes how much better two steps would be compared to one step. Now for instance if the graph expands and also 2-expands, its like the growth is pretty rapid. We can of course, generalize this to $i$-steps, and if the graph $i$-expands for $i=1,2,3,\dots$, then its a kind of "super" expander where sets grow like crazy.
Now all of this is just an intuition, so I'm curious if any of this makes sense and has been studied. Or its possible its all trivial and useless, reducing to the standard expansion. Please do let me know any references or directions in this regard.