there exist positive numbers $c$ and $R$ such that for every positive number $S$, $$\begin{aligned}
\liminf_{m \rightarrow \infty} \left(
\inf \left\{ \frac{ \sharp(N_R(Y)) }
{ \sharp(Y) } \ \colon\ 
\emptyset \neq Y \subseteq X_m, \mathrm{diam}(Y) \le S
\right\} \right) > 1 + c.\end{aligned}$$

To get a better understanding about a weak expander sequence, take an expanding sequence of subsets $Y, N_R(Y), N_R(N_R(Y)), \cdots \subseteq X_m$. Let us think that the sequence describes how information spreads. The numbers $\sharp(Y)$, $\sharp(N_R(Y))$, $\cdots$ indicate how many points the information reaches. The inequality $$\begin{aligned}
\inf \left\{ \sharp(N_R(Y)) / \sharp(Y)  \ \colon\ 
\emptyset \neq Y \subseteq X_m, \mathrm{diam}(Y) \le S
\right\} > 1 + c.\end{aligned}$$ means that the number of the points increases exponentially ($\ge O((1 + c)^n))$), until the diameter of the area exceeds $S$. We can make the constant $S$ bigger, by choosing a larger component of $\bigsqcup X_m$.

The notion of weak expander spaces is a generalization of expander graphs. We recall the definition of expander graphs. For a connected graph $(X, E)$ with a vertex set $X$ and an edge set $E$, we define a metric on $X$ by $$\begin{aligned}
d(x, y) = \mathrm{min} \{d \in \mathbb{N} \ 
|\ x = 