  $U$-statistics are used to estimate a population parameter by averaging a
function on a subsample over all the subsamples of the population. In this
paper, the population we are interested in is formed by the entries of a
row-column exchangeable matrix. We consider $U$-statistics derived from
functions of quadruplets, i.e. submatrices of size $2 \times 2$. We prove a
weak convergence result for these $U$-statistics in the general case and we
establish a Central Limit Theorem when the matrix is also dissociated. We shed
further light on these results using the Aldous-Hoover representation theorem
for row-column exchangeable random variables. Finally, to illustrate these
results, we give examples of hypothesis testing for bipartite networks.
