  Recent observations have indicated a strong connection between compact ($a
\lesssim 0.5$ au) super-Earth and mini-Neptune systems and their outer ($a
\gtrsim$ a few au) giant planet companions. We study the dynamical evolution of
such inner systems subject to the gravitational effect of an unstable system of
outer giant planets, focussing on systems whose end configurations feature only
a single remaining outer giant. In contrast to similar studies which used on
N-body simulations with specific (and limited) parameters or scenarios, we
implement a novel hybrid algorithm which combines N-body simulations with
secular dynamics with aims of obtaining analytical understanding and scaling
relations. We find that the dynamical evolution of the inner planet system
depends crucially on $N_{\mathrm{ej}}$, the number of mutual close encounters
between the outer planets prior to eventual ejection/merger. When
$N_{\mathrm{ej}}$ is small, the eventual evolution of the inner planets can be
well described by secular dynamics. For larger values of $N_{\mathrm{ej}}$, the
inner planets gain orbital inclination, inclination and eccentricity in a
stochastic fashion analogous to Brownian motion. We develop a theoretical
model, and compute scaling laws for the final orbital parameters of the inner
system. We show that our model can account for the observed eccentric
super-Earths/mini-Neptunes with inclined cold Jupiter companions, such as
HAT-P-11, Gliese 777 and $\pi$ Men.
