标题： 有向网络中互惠关系的回归分析 显示英文标题

标题： Regression Analysis of Reciprocity in Directed Networks

摘要： 互惠性——个体形成相互联系的倾向——是许多有向网络的基本结构特征。 尽管互惠性普遍存在，但它在统计网络模型中仍未能得到充分整合，尤其是在与协变量信息相关方面。 在本文中，我们引入了$R^{2}$模型，这是一种新颖且灵活的框架，能够显式地建模互惠性，同时结合协变量效应。 基于广义的$p_1$模型，我们的框架既能适应网络稀疏性，又能处理节点异质性，提供了迄今为止最全面的互惠性参数化方法——不仅捕捉其基线水平，还捕捉其如何系统地随观测协变量变化。 为应对高维性和干扰参数带来的挑战，我们开发了一种条件似然估计器，能够隔离并一致地估计互惠效应。 我们建立了其理论保证，包括在广泛稀疏性条件下的一致性、渐近正态性和最小最大最优性。 大量模拟和实际应用展示了$R^{2}$模型的灵活性、可解释性和强大的有限样本性能，突显了其在揭示有向网络中由协变量驱动的互惠模式方面的实际价值。 显示英文摘要

摘要： Reciprocity--the tendency of individuals to form mutual ties--is a fundamental structural feature of many directed networks. Despite its ubiquity, reciprocity remains insufficiently integrated into statistical network models, particularly in relation to covariate information. In this paper, we introduce the $R^{2}$-Model, a novel and flexible framework that explicitly models reciprocity while incorporating covariate effects. Built upon a generalized $p_1$ model, our framework accommodates both network sparsity and node heterogeneity, offering the most comprehensive parametrization of reciprocity to date--capturing not only its baseline level but also how it systematically varies with observed covariates. To address the challenges posed by high dimensionality and nuisance parameters, we develop a conditional likelihood estimator that isolates and consistently estimates the reciprocity effects. We establish its theoretical guarantees, including consistency, asymptotic normality, and minimax optimality under broad sparsity regimes. Extensive simulations and real-world applications demonstrate the $R^{2}$-Model's flexibility, interpretability, and strong finite-sample performance, highlighting its practical utility for uncovering covariate-driven patterns of reciprocity in directed networks.