  [Abridged] Filaments and hubs have received special attention recently thanks
to studies showing their role in star formation. While the column density and
velocity structures of both filaments and hubs have been studied, their
magnetic fields (B-field) are not yet characterized. We aim to understand the
role of the B-field in the dynamical evolution of the NGC 6334 hub-filament
network. We present new observations of the dust polarized emission at
850$\mu$m towards NGC 6334 obtained with the JCMT/POL-2. We study the
distribution and dispersion of the polarized intensity ($PI$), the polarization
fraction ($PF$), and the B-field angle ($\theta_{B}$). We derive the power
spectrum of the intensity and $\theta_{B}$ along the ridge crest. Our analyses
show a complex B-field structure when observed over the whole region ($\sim10$
pc), however, at smaller scales ($\sim1$ pc), $\theta_{B}$ varies coherently
along the filaments. The observed power spectrum of $\theta_{B}$ can be well
represented with a power law function with a slope $-1.33\pm0.23$, which is
$\sim20\%$ shallower than that of $I$. This result is compatible with the
properties of simulated filaments and may indicate the processes at play in the
formation of filaments. $\theta_{B}$ rotates from being mostly perpendicular to
the filament crests to mostly parallel as they merge with the hubs. This
variation of $\theta_{B}$ may be tracing local velocity flows of matter
in-falling onto the hubs. Our analysis suggests a variation of the energy
balance along the crests of these filaments, from magnetically
critical/supercritical at their far ends to magnetically subcritical near the
hubs. We detect an increase of $PF$ towards the high-column density star
cluster-forming hubs that may result from the increase of grain alignment
efficiency due to stellar radiation from the newborn stars.
