  In several works, Mendel and Naor have introduced and developed theory
surrounding a nonlinear expansion constant similar to the spectral gap for
sequences of graphs, in which one considers embeddings of a graph $G$ into a
metric space $X$ \cite{mendel2010towards, mendel2013nonlinear,
mendel2014expanders}. Here, we investigate the open question of whether the
random regular graph $G_{n, d}$ is an expander when embedded into the metric
space of a random regular graph $G_{m, d}$ a.a.s., where $m\leq n$. We show
that if $m$ is fixed, the answer is affirmative. In addition, when $m\to
\infty$, we provide partial solutions to the problem in the case that $d$ is
fixed or that $d\to \infty$ under the constraint $d=o(m^{1/2})$.
