Asymmetric relational data is increasingly prevalent across diverse fields, underscoring the need for directed network models to address the complex challenges posed by their unique structures. Unlike undirected models, directed models can capture reciprocity, the tendency of nodes to form mutual links. In this work, we address a fundamental question: what is the effective sample size for modeling reciprocity? We examine this by analyzing 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 intriguing interplays 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.

翻译：非对称关系数据在多个领域日益普遍，这凸显了有向网络模型对于应对其独特结构所带来的复杂挑战的必要性。与无向模型不同，有向模型能够捕捉互惠性，即节点形成双向链接的倾向。在本研究中，我们探讨一个基本问题：建模互惠性的有效样本量是多少？我们通过分析具有互惠性的伯努利模型来研究此问题，该模型允许非互惠效应与互惠效应之间存在不同的稀疏度水平。随后，我们将此框架扩展到一个模型，该模型利用协变量纳入了节点特异性异质性和链接特异性互惠性。我们的研究结果揭示了稀疏网络中非互惠效应与互惠效应之间有趣的相互作用。我们提出了一种基于最大似然估计的简单推断程序，该程序在无论是否包含协变量的情况下，都无需预先知晓稀疏度水平即可运行。