The rational homotopy groups of the group Diff($S^4$) of
self-diffeomorphisms of $S^4$ with the $C^\infty$-topology. We present a
method to prove that there are many `exotic' non-trivial elements in the
rational homotopy groups of Diff($S^4$) parametrized by trivalent graphs.
The proof utilizes Kontsevich's characteristic classes for smooth disk
bundles and a version of clasper surgery for families. In fact, these
are analogues of Chern--Simons perturbation theory in 3-dimension and
clasper theory due to Goussarov and Habiro.