  By discretizing an argument of Kislyakov, Naor and Schechtman proved that the
1-Wasserstein metric over the planar grid $\{0,1,\dots n\}^2$ has
$L_1$-distortion bounded below by a constant multiple of $\sqrt{\log n}$. We
provide a new "dimensionality" interpretation of Kislyakov's argument, showing
that, if $\{G_n\}_{n=1}^\infty$ is a sequence of graphs whose isoperimetric
dimension and Lipschitz-spectral dimension equal a common number $\delta \in
[2,\infty)$, then the 1-Wasserstein metric over $G_n$ has $L_1$-distortion
bounded below by a constant multiple of $(\log |G_n|)^{\frac{1}{\delta}}$. We
proceed to compute these dimensions for $\oslash$-powers of certain graphs. In
particular, we get that the sequence of diamond graphs
$\{\mathsf{D}_n\}_{n=1}^\infty$ has isoperimetric dimension and
Lipschitz-spectral dimension equal to 2, obtaining as a corollary that the
1-Wasserstein metric over $\mathsf{D}_n$ has $L_1$-distortion bounded below by
a constant multiple of $\sqrt{\log| \mathsf{D}_n|}$. This answers a question of
Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of
$L_1$-embeddable graphs whose sequence of 1-Wasserstein metrics is not
$L_1$-embeddable.
