  The models of $k$-core percolation and interdependent networks (IN) have been
extensively studied in their respective fields. A recent study has revealed
that they share several common critical exponents. However, several newly
discovered exponents in IN have not been explored in $k$-core percolation, and
the origin of the similarity still remains unclear. Here, we investigate k-core
percolation in random networks. We find that for k-core percolation,the
fractality of the giant component fluctuations is manifested by a fractal
fluctuation dimension, $\widetilde d_f = 3/4$, within a correlation \emph{size}
$N'$ that scales as $N' \propto (p-p_c)^{-\widetilde\nu}$, with $\widetilde\nu
= 2$, same as found in IN. Indeed, here, $\widetilde\nu \equiv d\cdot \nu'$ and
$\widetilde{d}_f \equiv d'_f/d$, where $\nu'$ and $d'_f$ are respectively the
same as the correlation \emph{length} exponent and the fractal fluctuation
dimension observed in $d$-dimensional IN spatial networks. These two new
exponents found here for $k$-core percolation demonstrate the same scaling
behaviors as found for IN with the same critical exponents, reinforcing the
similarity between the two models. Furthermore, we suggest that these two
models are similar since both have two types of interactions: short-range (SR)
connectivity and long-range (LR) influences. In IN the LR are the influences of
dependency links while in k-core we find here that for $k=1$ and $k=2$ the
influences are short range while for $k\geq3$ the influence is long range. In
addition, analytical arguments for a universal hyper-scaling relation for the
fractal fluctuation dimension of the $k$-core giant component and for IN as
well as for any mixed-order transition are established.Our analysis enhances
the comprehension of k-core percolation and supports the generalization of the
concept of fractal fluctuations in mixed-order phase transitions.
