  We investigate the interacting dark energy models by using the diagnostics of
statefinder hierarchy and growth rate of structure. We wish to explore the
deviations from $\Lambda$CDM and to differentiate possible degeneracies in the
interacting dark energy models with the geometrical and structure growth
diagnostics. We consider two interacting forms for the models, i.e., $Q_1=\beta
H\rho_c$ and $Q_2=\beta H\rho_{de}$, with $\beta$ being the dimensionless
coupling parameter. Our focus is the I$\Lambda$CDM model that is a
one-parameter extension to $\Lambda$CDM by considering a direct coupling
between the vacuum energy ($\Lambda$) and cold dark matter (CDM), with the only
additional parameter $\beta$. But we begin with a more general case by
considering the I$w$CDM model in which dark energy has a constant $w$
(equation-of-state parameter). For calculating the growth rate of structure, we
employ the "parametrized post-Friedmann" theoretical framework for interacting
dark energy to numerically obtain the $\epsilon(z)$ values for the models. We
show that in both geometrical and structural diagnostics the impact of $w$ is
much stronger than that of $\beta$ in the I$w$CDM model. We thus wish to have a
closer look at the I$\Lambda$CDM model by combining the geometrical and
structural diagnostics. We find that the evolutionary trajectories in the
$S^{(1)}_3$--$\epsilon$ plane exhibit distinctive features and the departures
from $\Lambda$CDM could be well evaluated, theoretically, indicating that the
composite null diagnostic $\{S^{(1)}_3, \epsilon\}$ is a promising tool for
investigating the interacting dark energy models.
