Exact Capacity Scaling of Extended Wireless Networks

n source and destination pairs randomly located in an area extending linearly with n want to communicate with each other. Signals transmitted from one user to another at distance r apart are subject to a power attenuation of r
^-α
and random phase changes. Classical multihop architectures that decode and forward packets can deliver a √n-scaling of the aggregate throughput, while recently proposed hierarchical cooperation achieves n
^2-α/2
-scaling, which is superior to multi-hop for α < 3. The study of information theoretic upper bounds has revealed the optimality of multi-hop for α > 4, while the moderate-attenuation regime (2 ≤ α ≤ 4) remains uncharacterized. We close this gap by deriving a tight upper bound on the scaling of the aggregate throughput, valid for all α ≥ 2. Our result shows that the mentioned schemes are scaling-optimal, namely that no other scheme can beat hierarchical cooperation when α < 3, nor can it beat classical multi-hop when α ≥ 3. The key ingredient is a careful evaluation of the scaling of the cut-set bound.