Some examples of very optimized pathfinding can be found in the node A Better Word Morph Builder. I think that the general idea for solving your problem is giving each machine an unique number (which is already done if the network map is implemented as you described), buiding a datastructure like

%network_map = (
12567 =>            # the machine id
[335, 456, 338],    # the id's of machines it is directly connected to
335 => [12657, 558, 667, 333]
# etc
)
[download]

and browsing this datastructure with some tree search algorithm. I myself prefer bidirectional search, the working example is given e.q. in this node, but i think that your network is small enough for using the breadth-first search without any significant performance loss.

Sorry for my English :)