Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Triple-dyad ratio estimation for the $p_1$ model

Qunqiang Feng, Yaru Tian, Ting Yan

TL;DR

This work addresses the lack of asymptotic theory for the $p_1$ model by introducing the triple-dyad ratio estimator, which yields explicit, scalable parameter estimates for $\rho$, $\theta$, $\alpha_i$, and $\beta_j$. The authors prove consistency and asymptotic normality under suitable conditions, derive bias-corrected inference for the density and reciprocity parameters, and develop a reciprocity test based on plug-in variance estimates. Through simulations and a real-world Sina Weibo analysis, TRE demonstrates comparable accuracy to the MLE in large networks while offering substantial computational advantages, highlighting its practical utility for massive directed networks. The results provide a practical framework for likelihood-free inference in network models with reciprocity and enable robust testing and comparison of network structure across datasets.

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.

Triple-dyad ratio estimation for the $p_1$ model

TL;DR

This work addresses the lack of asymptotic theory for the model by introducing the triple-dyad ratio estimator, which yields explicit, scalable parameter estimates for , , , and . The authors prove consistency and asymptotic normality under suitable conditions, derive bias-corrected inference for the density and reciprocity parameters, and develop a reciprocity test based on plug-in variance estimates. Through simulations and a real-world Sina Weibo analysis, TRE demonstrates comparable accuracy to the MLE in large networks while offering substantial computational advantages, highlighting its practical utility for massive directed networks. The results provide a practical framework for likelihood-free inference in network models with reciprocity and enable robust testing and comparison of network structure across datasets.

Abstract

Although the 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.
Paper Structure (8 sections, 4 theorems, 39 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 8 sections, 4 theorems, 39 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Method of triple-dyad ratio estimation
  3. Asymptotic properties of estimators
  4. Testing the reciprocal effect
  5. Numerical experiments
  6. Simulations
  7. Real data analysis
  8. Discussion

Key Result

Theorem 1

(1) Suppose that $\|\Theta\|_{\infty} \le C$ for a fixed constant $C>0$. Then the triple-dyad ratio estimator $\widehat{\Theta}$ satisfies with probability at least $1- O(n^{-1})$. (2) Define If and hold, then $\widehat{\Theta}$ satisfies with probability at least $1- O(n^{-1})$.

Figures (3)

  • Figure 1: QQ-plots for $(\hat{\theta}-\theta-\theta^*)/\sigma_{\theta}$ and $(\hat{\rho}-\rho-\rho^*)/\sigma_{\rho}$. The horizontal and vertical axes are the theoretical and sample quantiles. The red color indicates the diagonal line.
  • Figure 2: QQ-plots for $(\hat{\alpha}_i-\alpha_i)/\sigma_{\alpha_i}$ and $(\hat{\beta}_i-\beta_i)/\sigma_{\beta_i}$, $i=1,n/2,n$. The horizontal and vertical axes are the theoretical and sample quantiles. The red color indicates the diagonal line.
  • Figure 3: Histogram of $\hat{\alpha}$ and $\hat{\beta}$. The red color indicates the density estimator.

Theorems & Definitions (10)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • Corollary 1
  • Remark 5
  • Remark 6