  Given a finite Lie incidence geometry which is either a polar space of rank
at least $3$ or a strong parapolar space of symplectic rank at least $4$ and
diameter at most $4$, or the parapolar space arising from the line Grassmannian
of a projective space of dimension at least $4$, we show that its point graph
is determined by its local structure. This follows from a more general result
which classifies graphs whose local structure can vary over all local
structures of the point graphs of the aforementioned geometries. In particular,
this characterises the strongly regular graphs arising from the line
Grassmannian of a finite projective space, from the half spin geometry related
to the quadric $Q^+(10,q)$ and from the exceptional group of type
$\mathsf{E_6}(q)$ by their local structure.
