Overlapping and nonoverlapping models

TL;DR

This paper introduces a new model for directed networks with overlapping row nodes and nonoverlapping column nodes, providing identifiability conditions, an extension for degree variation, and spectral algorithms with theoretical guarantees.

Contribution

It proposes an overlapping and nonoverlapping model for directed networks with specific identifiability conditions and develops spectral algorithms with proven consistency.

Findings

Model is identifiable when $K_{r} \,\leq\, K_{c}$.

Spectral algorithms achieve consistent estimation.

Numerical studies illustrate the algorithms' effectiveness.

Abstract

Consider a directed network with $K_{r}$ row communities and $K_{c}$ column communities. Previous works found that modeling directed networks in which all nodes have overlapping property requires $K_{r}=K_{c}$ for identifiability. In this paper, we propose an overlapping and nonoverlapping model to study directed networks in which row nodes have overlapping property while column nodes do not. The proposed model is identifiable when $K_{r}\leq K_{c}$. Meanwhile, we provide one identifiable model as extension of ONM to model directed networks with variation in node degree. Two spectral algorithms with theoretical guarantee on consistent estimations are designed to fit the models. A small scale of numerical studies are used to illustrate the algorithms.