We apply combinatorial tools, including P´olya’s theorem, to enumerate all possible
networks for which (1) the network contains distinguishable input and output nodes
as well as partially distinguishable intermediate nodes; (2) all connections are directed
and for each pair of nodes, there are at most two connections, that is, at most one
connection per direction; (3) input nodes send connections but don’t receive any, while
output nodes receive connections but don’t send any; (4) every intermediate node
receives a path from an input node and sends a path to at least one output node; and
(5) input nodes don’t send direct connections to output nodes. We first obtain the
generating function for the number of such networks, and then use it to obtain precise
estimates for the number of networks. Finally, we develop a computer algorithm that
allows us to generate these networks. This work could become useful in the field of
neuroscience, in which the problem of deciphering the structure of hidden networks
is of the utmost importance, since there are several instances in which the activity
of input and output neurons can be directly measured, while no direct access to the
intermediate network is possible. Our results can also be used to count the number of
finite automata in which each cell plays a relevant role
