$U$-statistics on bipartite exchangeable networks

$U$-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. Since row-column exchangeable matrices are an actual representation for exchangeable bipartite networks, we apply these results to statistical inference in network analysis.