On the capacity of wireless networks: the relay case

Gupta and Kumar (see IEEE Transactions an Information Theory, vol.46, no.2, p.388-404, 2000) determined the capacity of wireless networks under certain assumptions, among them point-to-point coding, which excludes for example multi-access and broadcast codes. We consider essentially the same physical model of a wireless network under a different traffic pattern, namely the relay traffic pattern, but we allow for arbitrarily complex network coding. In our model, there is only one active source/destination pair, while all other nodes assist this transmission. We show code constructions leading to achievable rates and derive upper bounds from the max-flow min-cut theorem. It is shown that lower and upper bounds meet asymptotically as the number of nodes in the network goes to infinity, thus proving that the capacity of the wireless network with n nodes under the relay traffic pattern behaves like log n bits per second. This demonstrates also that network coding is essential: under the point-to-point coding assumption considered by Gupta et al., the achievable rate is constant, independent of the number of nodes. Moreover, the result of this paper has implications' and extensions to fading channels and to sensor networks.