Cross-Validation in Bipartite Networks

TL;DR

This paper introduces Bipartite Cross-Validation (BCV), a novel data-driven method for accurately estimating community numbers in bipartite networks, addressing theoretical gaps and demonstrating strong empirical performance.

Contribution

It develops the first model selection consistency framework for bipartite networks, accommodating scaling community numbers and addressing asymmetry challenges.

Findings

BCV achieves consistent community number estimation in bipartite networks.

Simulations show BCV outperforms existing methods in finite samples.

Real-data applications validate BCV's practical effectiveness.

Abstract

Bipartite networks, which encode interactions between two distinct types of entities, arise widely in applications and exhibit inherent asymmetry across node sets. Despite a growing literature on bipartite community detection, estimating community numbers $(K_1, K_2)$, a critical issue for bipartite network analysis, remains theoretically underdeveloped without any model selection consistency established, to our knowledge. Indeed, the inherent asymmetry and the two-dimensional parameter space with possibly drastically different $K_1$ and $K_2$ pose unique challenges that differ from unipartite cases. In particular, the candidate models may simultaneously overfit one node set while underfitting the other. To address these challenges, we propose Bipartite Cross-Validation (BCV), a penalized cross-validation framework that jointly selects $(K_1,K_2)$ in a fully data-driven manner. We establish the first model selection consistency for bipartite networks, notably accommodating the regime where the numbers of communities scale with the network size, revealing the intricate interplay between sparsity and model complexity. Simulations and real-data applications demonstrate strong finite-sample performance of BCV.