Regression Analysis of Reciprocity in Directed Networks

TL;DR

This paper introduces the $R^{2}$-Model, a new statistical framework for analyzing reciprocity in directed networks, incorporating covariate effects and addressing high-dimensional challenges with theoretical guarantees.

Contribution

It presents the $R^{2}$-Model, the most comprehensive model to date for reciprocity, integrating covariate effects and providing consistent estimators with strong theoretical properties.

Findings

The $R^{2}$-Model accurately captures reciprocity variations with covariates.

The estimator is consistent, asymptotically normal, and minimax optimal.

Simulations and real data validate the model's effectiveness.

Abstract

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.