$U$-statistics on bipartite exchangeable networks

TL;DR

This paper develops asymptotic theory for quadruplet U-statistics on bipartite exchangeable networks, enabling statistical inference such as parameter estimation and motif counting in network models.

Contribution

It introduces new asymptotic results for quadruplet U-statistics on bipartite exchangeable matrices and applies them to network inference in BEDD models.

Findings

Established weak convergence for quadruplet U-statistics.

Proved a central limit theorem for dissociated matrices.

Demonstrated applications in network parameter estimation and motif counting.

Abstract

Bipartite networks with exchangeable nodes can be represented by row-column exchangeable matrices. A quadruplet is a submatrix of size $2 \times 2$. A quadruplet $U$-statistic is the average of a function on a quadruplet over all the quadruplets of a matrix. We prove several asymptotic results for quadruplet $U$-statistics on row-column exchangeable matrices, including a weak convergence result in the general case and a central limit theorem when the matrix is also dissociated. These results are applied to statistical inference in network analysis. We suggest a method to perform parameter estimation, network comparison and motifs count for a particular family of row-column exchangeable network models: the bipartite expected degree distribution (BEDD) models. These applications are illustrated by simulations.