  One of the most frequently used indicators to characterize the magnetic
field's influence on star formation is the relation between magnetic field
strength and gas density ($B-\rho$ relation), usually expressed as $B \propto
\rho^{\kappa}$. The value of $\kappa$ is an indication of the dynamical
importance of the magnetic field during gas compression. Investigating the
global magnetic field's impact on this relation and its evolution, we conduct
MHD simulations of Milky-Way-like galaxies including gravity, star formation,
and supernova feedback along with non-equilibrium chemistry up to $H_2$
formation fueling star formation. Two initial magnetic field morphologies are
studied: one completely ordered (toroidal) and the other completely random. In
these models, we study the dynamical importance of the magnetic field through
the plasma $\beta$ and the $B-\rho$ relation. For both magnetic morphologies,
low-density regions are thermally supported, while high-density regions are
magnetically dominated. Equipartition is reached earlier and at lower densities
in the toroidal model. However, the $B-\rho$ relation is not unique even within
the same galaxy, as it consistently includes two different branches for a given
density, with $\kappa$ ranging from about 0.2 to 0.8. The mean value of
$\kappa$ for each model also displays significant variations over time, which
supersede the differences between the two models. While our findings suggest
that the magnetic field morphology does influence the galactic $B-\rho$
relation, its impact is transient, since time-averaged differences between the
models fall within the large temporal scatter. The context and time-dependent
nature of the $B-\rho$ relation underscore the need for comprehensive research
and observations to understand the intricate role of magnetic fields in star
formation processes across diverse galactic environments.
