  The homogeneous Poisson point process (PPP) is widely used to model spatial
distribution of base stations and mobile terminals. The same process can be
used to model underlay device-to-device (D2D) network, however, neglecting
homophilic relation for D2D pairing presents underestimated system insights. In
this paper, we model both spatial and social distributions of interfering D2D
nodes as proximity based independently marked homogeneous Poisson point
process. The proximity considers physical distance between D2D nodes whereas
social relationship is modeled as Zipf based marks. We apply these two
paradigms to analyze the effect of interference on coverage probability of
distance-proportional power-controlled cellular user. Effectively, we apply two
type of functional mappings (physical distance, social marks) to Laplace
functional of PPP. The resulting coverage probability has no closed-form
expression, however for a subset of social marks, the mark summation converges
to digamma and polygamma functions. This subset constitutes the upper and lower
bounds on coverage probability. We present numerical evaluation of these bounds
on coverage probability by varying number of different parameters. The results
show that by imparting simple power control on cellular user, ultra-dense
underlay D2D network can be realized without compromising the coverage
probability of cellular user.
