  We consider a broad class of random bipartite networks, the distribution of
which is invariant under permutation within each type of nodes. We are
interested in $U$-statistics defined on the adjacency matrix of such a network,
for which we define a new type of Hoeffding decomposition. This decomposition
enables us to characterize non-degenerate $U$-statistics -- which are then
asymptotically normal -- and provides us with a natural and easy-to-implement
estimator of their asymptotic variance. \\ We illustrate the use of this
general approach on some typical random graph models and use it to estimate or
test some quantities characterizing the topology of the associated network. We
also assess the accuracy and the power of the proposed estimates or tests, via
a simulation study.
