Triple-dyad ratio estimation for the $p_1$ model

2026-01-10

AI Summary

This work addresses the long-standing theoretical challenge in the $p_1$ model concerning the lack of scalable and statistically guaranteed maximum likelihood estimation (MLE). We propose an explicit parameter estimation method based on ratios of triadic binary indicators, which, for the first time, yields a consistent and asymptotically normal estimator for the $p_1$ model. This approach overcomes the computational and analytical bottlenecks of conventional MLE in large-scale networks. By incorporating an analytic bias correction and a complementary hypothesis testing framework, our method enables efficient inference of reciprocity effects in directed networks with up to millions of nodes. Both theoretical analysis and numerical experiments demonstrate that the proposed estimator achieves statistical performance comparable to MLE in large samples while offering superior computational efficiency and robust statistical guarantees.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Preferences

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

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.

Problem

• p1 model
• asymptotic theory
• parameter estimation
• consistency
• maximum likelihood estimator

Innovation

• triple-dyad ratio estimator
• p1 model
• asymptotic normality
• reciprocity effect
• bias correction