Let $H_\mu, 0 < \mu < < 1$ be a small perturbation of size $\mu$ of an initially
hyperbolic Hamiltonian system. We prove that Graff tori satisfy the transitivity
property : if $T_1, T_2$ and $T_3$ are three Graff tori such that $W^+(T_1)$ (resp. $W^+(T_2))$
and $W^-$ $(T_2)$ (resp. $W^-$ $(T_3))$ intersect transversally in a given energy level $H$ with
an angle of order $\mu$, then $W^+(T_1)$ and $W^-$ $(T_3)$ intersect transversally in $H$ with
an angle of order $\mu- c(\mu)$, with $c(\mu)$ exponentially small in $\mu$. The proof is based
on a quantitative version of the $\lambda$-lemma for Graff tori called the transfer lemma.
This result allows us to compute the Arnold diffusion time along transition chains
for initially hyperbolic Hamiltonian systems. We prove that this time is polynomial
in the inverse of the perturbing parameter.