The equivalences of community detection methods for bipartite networks

Investigating the community structures of bipartite networks is a frequent topic of discussion in the ecological and social fields. The most widely used methods, as proposed by numerous academics from varying perspectives, include spectral graph partitioning, modularity, nonnegative matrix factorization, and stochastic block model. In this paper, we demonstrate three equivalences among these four methods. One, both Dhillon spectral graph partitioning and Barber modularity clustering are equivalent to solving for the matrix's left and right singular vectors after relaxing the discrete constraints. Two, the nonnegative matrix factorization clustering is equivalent to the Dhillon spectral graph partitioning. Three, The bipartite stochastic block model is equivalent to the constraint-based NMF that uses K-L divergence as its cost function. These equivalences, obtained through rigorous mathematical derivations, will aid in the future development of efficient algorithms for community detection in bipartite networks.