Modelling directed networks with reciprocity

Summary Asymmetric relational data are becoming increasingly prevalent in diverse fields, underscoring the need for developing directed network models to address the complex challenges posed by the unique structure of such data. Unlike undirected models, directed models can capture reciprocity, the tendency of nodes to form mutual links. This work addresses a fundamental question: what is the effective sample size for modelling reciprocity? We examine this question by analysing the Bernoulli model with reciprocity, allowing for varying sparsity levels between non-reciprocal and reciprocal effects. We then extend this framework to a model that incorporates node-specific heterogeneity and link-specific reciprocity using covariates. Our findings reveal the intriguing interplay between non-reciprocal and reciprocal effects in sparse networks. We propose a straightforward inference procedure based on maximum likelihood estimation that operates without prior knowledge of sparsity levels, whether covariates are included or not.