  We report the results of an in-depth study of the role of graph topology on
quantum transport efficiency in random removal and Watts-Strogatz networks. By
using four different environmental models -- noiseless, driving by classical
random telegraph noise (RTN), thermal quantum bath, and bath+RTN -- we compare
the role of the environment and of the change in network topology in
determining the quantum transport efficiency. We find that small and specific
changes in network topology is more effective in causing large change in
efficiency compared to that achievable by environmental manipulations for both
network classes. Furthermore, we have found that noise dependence of transport
efficiency in these networks can be categorized into six classes. In general,
our results highlight the interplay that network topology and environment
models play in quantum transport, and pave the way for transport studies for
networks of increasing size and complexity -- when going beyond so far often
used few-site transport systems.
