Multicasting in Large Random Wireless Networks: Bounds on the Minimum Energy per Bit Aman Jain Department of Electrical Engineering Princeton University Princeton, NJ 08544, USA Email: amanjain@princeton.edu Sanjeev R. Kulkarni Department of Electrical Engineering Princeton University Princeton, NJ 08544, USA Email: kulkarni@princeton.edu Sergio Verd´ u Department of Electrical Engineering Princeton University Princeton, NJ 08544, USA Email: verdu@princeton.edu Abstract—We consider scaling laws for maximal energy effi- ciency of communicating a message to all the nodes in a random wireless network, as the number of nodes in the network becomes large. Two cases of large wireless networks are studied — dense random networks and constant density (extended) random networks. We first establish an information-theoretic lower bound on the minimum energy per bit for multicasting that holds for arbitrary wireless networks when the channel state information is not available at the transmitters. These lower bounds are then evaluated for two cases of random networks. Upper bounds are also obtained by constructing a simple flooding scheme that requires no information at the receivers about the channel states or the locations and identities of the nodes. The gap between the upper and lower bounds is only a constant factor for dense random networks and differs by a poly-logarithmic factor for extended random networks. Furthermore, the proposed upper and lower bounds hold almost surely in the node locations as the number of nodes approaches infinity. I. I NTRODUCTION A. Prior Work Determining energy efficiency of a point-to-point channel is a fundamental information-theoretic problem. This problem is considerably more complicated for networks. Even when just one helper (relay) node is added to the two terminal AWGN channel, the minimum energy per bit is still unknown despite many efforts ([4], [11] and references therein). As the number of relays k in a network grows, we can ask whether the energy efficiency improves and at what rate. It was shown in [3] that a two hop distributed beamforming scheme gives very good energy efficiency in dense random networks with the energy requirement falling as Θ(1/ √ k). It is not clear, however, how to extend this idea to noncoherent or to multicasting scenarios. Cooperation between nodes (also known as cooperative diversity) leads to capacity or reliability gains even with simple schemes. A simple cooperation idea in a multicast setting involves letting many nodes transmit the same signal (at lower power levels), so that each receiver can combine several low reliability signals to construct progressively better estimates. The works of [6], [7], [5] presented such multi- stage decode and forward schemes to reduce the transmission This research was supported in part by the Office of Naval Research under contract numbers W911NF-07-1-0185 and N00014-07-1-0555. energy. In [7], achievability schemes were presented for dense networks, whereas our interest is in order of growth of energy requirement for simpler power allocation (uniform) for both dense and extended networks. A major difference in our setup from the previous works is our emphasis on minimal network and channel state information. This implies, among other things, that no centrally optimized transmission or power policies can be implemented. The problem of communicating the same message to a set of nodes (multicasting) in a network with minimum energy consumption has drawn a lot of research interest. For wireless networks, there is an inherent wireless multicast advantage [10] that allows all the nodes within the coverage range to receive the message at no additional cost. Even the nodes out of the coverage range can overhear the transmissions made over the wireless medium. Such an advantage has been termed Cooperative Wireless Advantage (CWA) in [5]. A more fundamental approach to the modeling and analysis of wireless networks may yield better results based on exploiting the broadcast nature of wireless communications. B. Summary of Results In this work, our aim is to determine the maximum possible energy efficiency for multicasting in wireless relay networks. Besides developing converse bounds, we also show how cooperative communication is instrumental to approach them. We first present, in Theorem 1, a lower bound on the energy requirement for multicasting in arbitrary wireless networks when there is no constraint on the available bandwidth. This lower bound is inversely proportional to the effective radius of the network, which is a fundamental property of the network and depends on the channel gains between the nodes. This bound is applicable when channel state information is not available at the transmitters, regardless of whether the channel state information is available at the receivers. For the achievability part, we propose a simple wideband flooding algorithm that does not require knowledge of the node locations, identities or channel states. These converse and achievability bounds are then evaluated for two cases of large random networks when the same message needs to be communicated to all the nodes. Following recent trends, our focus is on the order of scaling of the energy efficiency. For ISIT 2009, Seoul, Korea, June 28 - July 3, 2009 978-1-4244-4313-0/09/$25.00 ©2009 IEEE 2627