You can find definitions and properties in Fan Chung's paper,
"Discrete Isoperimetric Inequalities,"
*Surveys in Differential Geometry IX*, International Press, 2004, 53--82
(PDF download link).
She says, "In a way, a graph can be viewed as a
discretization of a Riemannian manifold in $\mathbb{R}^n$ where $n$ is roughly equal to
[the isoperimetric dimension] $\delta$." Here is the precise definition of $\delta$:

We say that a graph $G$ has isoperimetric dimension $\delta$ with an isoperimetric constant $c_\delta$ if for all subsets
$X$ of $V(G)$, the number of edges between $X$ and the complement $\bar{X}$ of $X$,
denoted by $e(X,\bar{X})$, satisifies
$$e(X,\bar{X}) \ge c_\delta \; \mathrm{vol} (X)^{\frac{\delta-1}{\delta}}$$
where $\mathrm{vol} (X) \le \mathrm{vol} (\bar{X})$
and $c_\delta$ is a constant depending only on $\delta$.

Earlier:

For a graph $G$ and a subset $X$ of vertices in $G$, the volume vol$(X)$ is deﬁned by
$$\mathrm{vol} (X) = \sum_{v \in X} d_v$$
where $d_v$ is the degree of $v$.