  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.
