  Virialized Ultra-Light Fields (VULFs) are viable cold dark matter candidates
and include scalar and pseudo-scalar bosonic fields, such as axions and
dilatons. Direct searches for VULFs rely on low-energy precision measurement
tools. While the previous proposals have focused on detecting coherent
oscillations of the VULF signals at the VULF Compton frequencies at individual
devices, here I consider a network of such devices. VULFs are essentially dark
matter {\em waves} and as such they carry both temporal and spatial phase
information. Thereby, the discovery reach can be improved by using networks of
precision measurement tools. To formalize this idea, I derive a spatio-temporal
two-point correlation function for the ultralight dark matter fields in the
framework of the standard halo model. Due to VULFs being Gaussian random
fields, the derived two-point correlation function fully determines $N$-point
correlation functions. For a network of $N_{d}$ devices within the coherence
length of the field, the sensitivity compared to a single device can be
improved by a factor of $\sqrt{N_{d}}$. Further, I derive a VULF dark matter
signal profile for an individual device. The resulting line shape is strongly
asymmetric due to the parabolic dispersion relation for massive
non-relativistic bosons. I discuss the aliasing effect that extends the
discovery reach to VULF frequencies higher than the experimental sampling rate.
I present sensitivity estimates and develop a stochastic field SNR statistic.
Finally, I consider an application of the developed formalism to atomic clocks
and their networks.
