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.

翻译：尽管$p_1$模型在四十年前已被提出，但针对该模型的渐近理论进展甚微，即最大似然估计量（MLE）的一致性或其他具有统计保证的参数估计均未得到理解。该问题已被公认为一个长期存在的开放性问题。为解决此问题，我们提出了一种基于三元组指标序列和之比率的新型参数估计方法，其中三元组指标指三个二元组指标的乘积。我们提出的估计量，称为\emph{三元组比率估计量}，具有显式表达式，并可扩展至具有数百万节点的超大规模网络。我们建立了当节点数趋于无穷时三元组比率估计量的一致性与渐近正态性。基于这些渐近结果，我们构建了一个用于检验有向网络中是否存在互惠效应的检验统计量。针对密度参数与互惠参数的估计量包含偏差项，我们提出了解析的偏差校正公式以进行有效推断。数值研究验证了我们理论的结果，并表明该估计量在大型网络中与MLE具有可比性。