  We study the environment-assisted enhancement of the excitation-transport
efficiency across a network of interacting quantum particles or sites. Our
study reveals a crucial influence of the network configuration - and especially
its degree of connectivity - on the amount of environment-supported
enhancement. In particular, we find a significant interplay of direct and
indirect connections between excitation sending and receiving sites. On the
other hand, the non-Markovianity induced by memory-bearing, finite sized
environments does not seem to provide a critical resource for the enhanced
excitation-transport mechanism.
