  Supergranules in the quiet Sun are outlined by a web-like structure of
enhanced magnetic field strength, the so-called magnetic network. We aim to map
the magnetic network field around the average supergranule near disk center. We
use observations of the line-of-sight component of the magnetic field from the
Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory
(SDO). The average supergranule is constructed by coaligning and averaging over
3000 individual supergranules. We determine the positions of the supergranules
with an image segmentation algorithm that we apply on maps of the horizontal
flow divergence measured using time-distance helioseismology. In the center of
the average supergranule the magnetic (intranetwork) field is weaker by about
2.2 Gauss than the background value (3.5 Gauss), whereas it is enhanced in the
surrounding ring of horizontal inflows (by about 0.6 Gauss on average). We find
that this network field is significantly stronger west (prograde) of the
average supergranule than in the east (by about 0.3 Gauss). With time-distance
helioseismology, we find a similar anisotropy. The observed anisotropy of the
magnetic field adds to the mysterious dynamical properties of solar
supergranulation.
