### abstract

- How much information can be carried over a wireless network with a multiplicity of nodes, and how should the nodes cooperate to transfer information? To study these questions, we formulate a model of wireless networks that particularly takes into account the distances between nodes, and the resulting attenuation of radio signals, and study a performance measure that weights information by the distance over which it is transported. Consider a network with the following features. i) n nodes located on a plane, with minimum separation distance ρmin > 0. ii) A simplistic model of signal attenuation eγρ/ρδ over a distance ρ, where γ ≥ 0 is the absorption constant (usually positive, unless over a vacuum), and δ > 0 is the path loss exponent. iii) All receptions subject to additive Gaussian noise of variance σ2. The performance measure we mainly, but not exclusively, study is the transport capacity CT := sup Σℓ=1m Rℓ · ρℓ, where the supremum is taken over m, and vectors (R1, R2, ..., Rm) of feasible rates for m source-destination pairs, and ρℓ is the distance between the ℓth source and its destination. It is the supremum distance-weighted sum of rates that the wireless network can deliver. We show that there is a dichotomy between the cases of relatively high and relatively low attenuation. When γ > 0 or δ > 3, the relatively high attenuation case, the transport capacity is bounded by a constant multiple of the sum of the transmit powers of the nodes in the network. This shows that there is a positive lower bound on the energy price in joules per bit-meter of information transport. If the nodes are individually power limited, the transport capacity consequently scales as O(n). This order is, in fact, sharp, i.e., the transport capacity is Θ(n), for regular planar networks where the nodes are situated at integer lattice sites in a square. Consider now the "multihop" strategy where packets are routed over possibly multiple paths, and, along each path, packets are relayed from node to node with full decoding of each packet at each hop, treating all interference as noise, i.e., employing only point-to-point coding. This strategy is an order optimal strategy when the relaying burden can be balanced across the nodes, with no hop being too long. Or, in a randomly picked scenario, if nodes in a regular planar network randomly choose destination nodes, then the maximum common throughput that can be furnished to all nodes by multihop transport is nearly order optimal with respect to the transport capacity, differing only by a factor 1/√log n. Hence, up to order, there is no need for network coding or multiuser estimation. Thus, information theory can shed some light on what is an order-optimal architecture for wireless networks in situations where the load can be nearly balanced across nodes. In particular, the order optimality or near order optimality of multihop transport in such scenarios is of interest because much protocol development activity currently is actually aimed at realizing this strategy. However, when γ = 0 and δ < 2/3, the low-attenuation case, we show that there exist networks that can provide unbounded transport capacity for fixed total power, yielding zero energy priced communication. When nodes lie on a straight line and δ < 1 (a physical impossibility in the three-dimensional world, but perhaps the examples can be generalized to a plane with larger values of δ), there are networks which can even attain superlinear scaling Θ(nθ) for θ < 2. Both these results are achieved by a strategy of coherent multistage relaying with interference subtraction. These examples show that nodes can profitably cooperate over large distances using coherence and multiuser estimation when the attenuation is low. These results are established by developing a coding scheme and an achievable rate for Gaussian multiple-relay channels, a result that may be of interest in its own right.