Triple-dyad ratio estimation for the $p_1$ model

TL;DR

This paper introduces a novel triple-dyad ratio estimator for the $p_1$ model, providing the first asymptotic theory, bias correction, and scalable inference for large networks, addressing a long-standing open problem.

Contribution

It proposes a new ratio-based estimator for the $p_1$ model with proven consistency, asymptotic normality, bias correction, and scalability to large networks.

Findings

Estimator is consistent and asymptotically normal.

Bias correction formulas improve inference accuracy.

Performs comparably to MLE in large networks.

Abstract

Although the $p_1$ model was proposed 40 years ago, little progress has been made to address asymptotic theories in this model, that is, neither consistency of the maximum likelihood estimator (MLE) nor other parameter estimation with statistical guarantees is understood. This problem has been acknowledged as a long-standing open problem. To address it, we propose a novel parametric estimation method based on the ratios of the sum of a sequence of triple-dyad indicators to another one, where a triple-dyad indicator means the product of three dyad indicators. Our proposed estimators, called \emph{triple-dyad ratio estimator}, have explicit expressions and can be scaled to very large networks with millions of nodes. We establish the consistency and asymptotic normality of the triple-dyad ratio estimator when the number of nodes reaches infinity. Based on the asymptotic results, we develop a test statistic for evaluating whether is a reciprocity effect in directed networks. The estimators for the density and reciprocity parameters contain bias terms, where analytical bias correction formulas are proposed to make valid inference. Numerical studies demonstrate the findings of our theories and show that the estimator is comparable to the MLE in large networks.