Network Simplification for Secure AF Relaying
  We consider a class of Gaussian layered networks where a source communicates with a destination through L intermediate relay layers with N nodes in each layer in the presence of a single eavesdropper which can overhear the transmissions of the nodes in the last layer. For such networks we address the question: what fraction of maximum secure achievable rate can be maintained if only a fraction of available relay nodes are used in each layer? In particular, we provide upper bounds on additive and multiplicative gaps between the optimal secure AF when all N relays in each layer are used and when only k, 1 <= k < N, relays are used in each layer. We show that asymptotically (in source power), the additive gap increases at most logarithmically with ratio N/k and L, and the corresponding multiplicative gap increases at most quadratically with ratio N/k and L. To the best of our knowledge, this work offers the first characterization of the performance of network simplification in layered amplify-and-forward relay networks in the presence of an eavesdropper. 
Tulika Agrawal and Samar Agnihotri
cs.IT math.IT