  In a geometric graph, $G$, the \emph{stretch factor} between two vertices,
$u$ and $w$, is the ratio between the Euclidean length of the shortest path
from $u$ to $w$ in $G$ and the Euclidean distance between $u$ and $w$. The
\emph{average stretch factor} of $G$ is the average stretch factor taken over
all pairs of vertices in $G$. We show that, for any constant dimension, $d$,
and any set, $V$, of $n$ points in $\mathbb{R}^d$, there exists a geometric
graph with vertex set $V$, that has $O(n)$ edges, and that has average stretch
factor $1+ o_n(1)$. More precisely, the average stretch factor of this graph is
$1+O((\log n/n)^{1/(2d+1)})$. We complement this upper-bound with a lower
bound: There exist $n$-point sets in $\mathbb{R}^2$ for which any graph with
$O(n)$ edges has average stretch factor $1+\Omega(1/\sqrt{n})$. Bounds of this
type are not possible for the more commonly studied worst-case stretch factor.
In particular, there exists point sets, $V$, such that any graph with
worst-case stretch factor $1+o_n(1)$ has a superlinear number of edges.
