By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid
{
0
,
1
,
…
,
n
}
2
\{0,1,\dots , n\}^2
has
L
1
L_1
-distortion bounded below by a constant multiple of
log
n
\sqrt {\log n}
. We provide a new “dimensionality” interpretation of Kislyakov’s argument, showing that if
{
G
n
}
n
=
1
∞
\{G_n\}_{n=1}^\infty
is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number
δ
∈
[
2
,
∞
)
\delta \in [2,\infty )
, then the 1-Wasserstein metric over
G
n
G_n
has
L
1
L_1
-distortion bounded below by a constant multiple of
(
log
|
G
n
|
)
1
δ
(\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
{
D
n
}
n
=
1
∞
\{\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
D
n
\mathsf {D}_n
has
L
1
L_1
-distortion bounded below by a constant multiple of
log
|
D
n
|
\sqrt {\log | \mathsf {D}_n|}
. This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of
L
1
L_1
-embeddable graphs whose sequence of 1-Wasserstein metrics is not
L
1
L_1
-embeddable.