We consider single-source single-sink (ss-ss) multi-hop relay networks, with slow-fading links and single-antenna half-duplex relay nodes. While two-hop cooperative relay networks have been studied in great detail in terms of the diversity-multiplexing tradeoff (DMT), few results are available for more general networks. In this paper, we identify two families of networks that are multi-hop generalizations of the two-hop network: K-Parallel-Path (KPP) networks and layered networks. KPP networks can be viewed as the union of K node-disjoint parallel relaying paths, each of length greater than one. KPP networks are then generalized to KPP(I) networks, which permit interference between paths and to KPP(D) networks, which possess a direct link from source to sink. We characterize the DMT of these families of networks completely for K > 3. Layered networks are networks comprising of layers of relays with edges existing only between adjacent layers, with more than one relay in each layer. We prove that a linear DMT between the maximum diversity dmax and the maximum multiplexing gain of 1 is achievable for single-antenna fully-connected layered networks. This is shown to be equal to the optimal DMT if the number of relaying layers is less than 4. For multiple-antenna KPP and layered networks, we provide an achievable DMT, which is significantly better than known lower bounds for half duplex networks. For arbitrary multi-terminal wireless networks with multiple source-sink pairs, the maximum achievable diversity is shown to be equal to the min-cut between the corresponding source and the sink, irrespective of whether the network has half-duplex or full-duplex relays. For arbitrary ss-ss single-antenna directed acyclic networks with full-duplex relays, we prove that a linear tradeoff between maximum diversity and maximum multiplexing gain is achievable. Along the way, we derive the optimal DMT of a generalized parallel channel and derive lower bounds for the DMT of triangular channel matrices, which are useful in DMT computation of various protocols. All protocols in this paper are explicit and use only amplify-and-forward (AF) relaying. We also construct codes with short block-lengths based on cyclic division algebras that achieve the optimal DMT for all the proposed schemes. Two key implications of the results in the paper are that the half-duplex constraint does not entail any rate loss for a large class of cooperative networks and that simple AF protocols are often sufficient to attain the optimal DMT.

L’objectif est de distinguer les îlots boisés relictuels des néoformés dans une matrice agricole, de les recenser avant de les caractériser et d’évaluer leur biodiversité et pour finir d’en tirer un modèle. La méthode associe le recours aux archives et aux photographies aériennes à l’enquête de terrain (1996-2007). L’approche cartographique s’est faite par analyse régressive (2004-1620). Le seuil temporel permet de reconnaître 152 îlots (<50ha) dont l’existence est attestée avant 1620. L’analyse régressive s’appuie sur le réseau viaire, élément fixe et structurant du paysage, pour cartographier les petits bois.On constate une forte relation entre la localisation des îlots relictuels, les tracés antiques et les chemins médiévaux. La caractérisation et l’évaluation de la biodiversité des relictuels s’effectuent selon une démarche spatio-temporelle pluriscalaire, et à partir d’une comparaison avec les îlots néoformés. L’analyse à partir de la canopée informe sur la texture de la couverture boisée et la dynamique du manteau forestier, sur les coupes, les lisières et le réseau viaire intraforestier. L’analyse de la structure interne porte sur la diversité taxinomique et la diversité écologique des espèces végétales et des espèces aviaires indicatrices. Ces analyses font apparaître la spécificité des relictuels, dont est proposé un modèle. Il s’agit soit de fragments de forêts anciennes, soit en plus grand nombre de formations boisées secondaires mais anciennes, gallo-romaines, mérovingiennes ou médiévales. / The purpose is to distinguish the relictual forest islands from the recent ones in an agricultural matrix; to inventory and characterize them and to assess their biodiversity, and eventually to develop a model. The method combines the resort to archives and aerial photographs with field studies (1996-2007). The mapping approach has been based on regression analysis (2004-1620). The time threshold has made it possible to identify 152 woodlots (<50ha) which presence was attested before 1620. The regression analysis to map the wood patches has been based on the path network, which is a fixed and structuring element of the landscape. There is a strong relation between the location of the relictual forest islands, the antique tracks and the medieval paths. The characterization and assessment of the relictual forest islands biodiversity have been carried out following a time and space pluri-scalar approach and from a comparison with recent forest islands. The analysis was based on the shapeless canopy, the texture and dynamics of the forest cover, the cuttings, the edges and the internal forest path network. The analysis of the internal structure focuses on the taxonomic diversity and the ecological diversity of the plant species and the indicator bird species. These analyses have stressed the specificity of the relictual forest islands, a model of which has been developed. They are either fragment of ancient forests, or, in most cases, secondary but old forest forms from the Gallo-Roman, Merovingian or medieval time.