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^~alpha and random phase changes. Classical multihop architectures that decode and forward packets can deliver a radicn-scaling of the aggregate throughput, while recently proposed hierarchical cooperation achieves n^2-alpha/2-scaling, which is superior to multi-hop for alpha < 3. The study of information theoretic upper bounds has revealed the optimality of multi-hop for alpha > 4, while the moderate- attenuation regime (2 les alpha les 4) remains uncharacterized. We close this gap by deriving a tight upper bound on the scaling of the aggregate throughput, valid for all alpha ges 2. Our result shows that the mentioned schemes are scaling-optimal, namely that no other scheme can beat hierarchical cooperation when alpha 3, nor can it beat classical multi-hop when alpha ges3. The key ingredient is a careful evaluation of the scaling of the cut-set bound.