Capacity Scaling of Wireless Ad Hoc Networks: Shannon Meets Maxwell

In this paper, we characterize the information-theoretic capacity scaling of wireless ad hoc networks with $n$ randomly distributed nodes. By using an exact channel model from Maxwell's equations, we successfully resolve the conflict in the literature between the linear capacity scaling by \"{O}zg\"{u}r et al. and the degrees of freedom limit given as the ratio of the network diameter and the wavelength $\lambda$ by Franceschetti et al. In dense networks where the network area is fixed, the capacity scaling is given as the minimum of $n$ and the degrees of freedom limit $\lambda^{-1}$ to within an arbitrarily small exponent. In extended networks where the network area is linear in $n$, the capacity scaling is given as the minimum of $n$ and the degrees of freedom limit $\sqrt{n}\lambda^{-1}$ to within an arbitrarily small exponent. Hence, we recover the linear capacity scaling by \"{O}zg\"{u}r et al. if $\lambda=O(n^{-1})$ in dense networks and if $\lambda=O(n^{-1/2})$ in extended networks. Otherwise, the capacity scaling is given as the degrees of freedom limit characterized by Franceschetti et al. For achievability, a modified hierarchical cooperation is proposed based on a lower bound on the capacity of multiple-input multiple-output channel between two node clusters using our channel model.