  In covert communication, Alice tries to communicate with Bob without being
detected by a warden Willie. When the distance between Alice and Bob becomes
large compared to the distance between Alice and Willie(s), the performance of
covert communication will be degraded. In this case, multi-hop message
transmission via intermediate relays can help to improve performance. Hence, in
this work multi-hop covert communication over a moderate size network and in
the presence of multiple collaborating Willies is considered. The relays can
transmit covertly using either a single key for all relays, or different
independent keys at the relays. For each case, we develop efficient algorithms
to find optimal paths with maximum throughput and minimum end-to-end delay
between Alice and Bob. As expected, employing multiple hops significantly
improves the ability to communicate covertly versus the case of a single-hop
transmission. Furthermore, at the expense of more shared key bits, analytical
results and numerical simulations demonstrate that multi-hop covert
communication with different independent keys at the relays has better
performance than multi-hop covert communication with a single key.
