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We consider a broad class of random bipartite networks, the distribution ofwhich is invariant under permutation within each type of nodes. We areinterested in $U$-statistics defined on the adjacency matrix of such a network,for which we define a new type of Hoeffding decomposition. This decompositionenables us to characterize non-degenerate $U$-statistics -- which are thenasymptotically normal -- and provides us with a natural and easy-to-implementestimator of their asymptotic variance. \\ We illustrate the use of thisgeneral approach on some typical random graph models and use it to estimate ortest some quantities characterizing the topology of the associated network. Wealso assess the accuracy and the power of the proposed estimates or tests, viaa simulation study.

Hoeffding-type decomposition for $U$-statistics on bipartite networks