A class of graphs is bridge-addable if given a graph
$G$
in the class, any graph obtained by adding an edge between two connected components of
$G$
is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if
${\mathcal{G}}$
is bridge-addable and
$G_{n}$
is a uniform
$n$
-vertex graph from
${\mathcal{G}}$
, then
$G_{n}$
is connected with probability at least
$(1+o_{n}(1))e^{-1/2}$
. The constant
$e^{-1/2}$
is best possible, since it is reached for the class of all forests.
In this paper, we prove a form of uniqueness in this statement: if
${\mathcal{G}}$
is a bridge-addable class and the random graph
$G_{n}$
is connected with probability close to
$e^{-1/2}$
, then
$G_{n}$
is asymptotically close to a uniform
$n$
-vertex random forest in a local sense. For example, if the probability converges to
$e^{-1/2}$
, then
$G_{n}$
converges in the sense of Benjamini–Schramm to the uniformly infinite random forest
$F_{\infty }$
. This result is reminiscent of so-called “stability results” in extremal graph theory, the difference being that here the stable extremum is not a graph but a graph class.