Wireless networks with a minimum inter-node separation distance are studied where the signal attenuation grows in magnitude as 1/ρ δ  with distance ρ. Two performance measures of wireless networks are analyzed. The transport capacity is the supremum of the total distance-rate products that can be supported by the network. The energy cost of information transport is the infimum of the ratio of the transmission energies used by all the nodes to the number of bit-meters of information thereby transported.If the phases of the attenuations between node pairs are uniformly and independently distributed, it is shown that the expected transport capacity is upper-bounded by a multiple of the total of the transmission powers of all the nodes, whenever δ > 2 for two-dimensional networks or δ > 5/4 for one-dimensional networks, even if all the nodes have full knowledge of all the phases, i.e., full channel state information. If all nodes have an individual power constraint, the expected transport capacity grows at most linearly in the number of nodes due to the linear growth of the total power. This establishes the best case order of expected transport capacity for these ranges of path-loss exponents since linear scaling is also feasible.If the phases of the attenuations are arbitrary, it is shown that the transport capacity is upper-bounded by a multiple of the total transmission power whenever δ > 5/2 for two-dimensional networks or δ > 3/2 for one-dimensional networks, even if all the nodes have full channel state information. This shows that there is indeed a positive energy cost which is no less than the reciprocal of the above multiplicative constant. It narrows the transition regime where the behavior is still open, since it is known that when δ < 3/2 for two-dimensional networks, or δ < 1 for one-dimensional networks, the transport capacity cannot generally be bounded by any multiple of the total transmit power.
['Liang-Liang Xie', 'P. R. Kumar']
On the path-loss attenuation regime for positive cost and linear scaling of transport capacity in wireless networks