arXiv subjects

Yaru Tian

Publications and source records attributed to Yaru Tian.

4 recordsLinked to original sources

arXiv2026

Subgraph counting estimation for the $β$-model in sparse networks

The $β$-model is popular for characterizing the commonly observed degree heterogeneity phenomenon in real-world networks. In this study, we develop a cycle counting approach to estimate $n$ node-specific parameters in the $β$-model for moderate or extremely sparse networks. Our proposed estimators, called \emph{Cycle Counting Ratio (CCR) Estimator}, are based on the log-ratios of two network cycle counting statistics with explicit expressions and therefore easy to compute. We focus on conditions to guarantee statistical properties of the single estimator for each node. Under the very weak conditions that $\max_t θ_t \to 0$ and $θ_t \|θ\|_1 \to \infty$, we show that the CCR estimator is consistent and achieves the minimax rate in terms of the mean squared error, which is the squared signal-to-noise ratio for $\hatβ_t$ up to a constant factor. Here, $\hatβ_t$ is the CCR estimator of the node-specific parameter $β_t$, $θ_t = \exp(β_t)$ and $θ=(θ_1, \ldots, θ_n)$. Even if the whole network density is close to the Erdős-Rényi lower bound $\log n/n$, the CCR estimator for the single parameter $β_t$ is still consistent as long as $θ_t \|θ\|_1 \to \infty$. To the best of our knowledge, this is the first time to derive the minimax rate and consistency result under such weak conditions. Under a slight stronger condition, we further establish its uniform consistency and asymptotic normality, whose asymptotic variance is $θ_t \|θ\|_1$. Numerical studies and an application to a sparse network data set demonstrate our theoretical findings.

stat.ME↗

arXiv2026

Triple-dyad ratio estimation for the $p_1$ model

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.

stat.ME↗

arXiv2026

Optimal estimators and tests for reciprocal effects

The $p_1$ model plays a fundamental role in modeling directed networks, where the reciprocal effect parameter $\rho$ is of special interest in practice. However, due to nonlinear factors in this model, how to estimate $\rho$ efficiently is a long-standing open problem. We tackle the problem by the cycle count approach. The challenge is, due to the nonlinear factors in the model, for any given type of generalized cycles, the expected count is a complicated function of many parameters in the model, so it is unclear how to use cycle counts to estimate $\rho$. However, somewhat surprisingly, we discover that, among many types of generalized cycles with the same length, we can carefully pick a pair of them such that in the ratio between the expected cycle counts of the two types, the non-linear factors cancel out nicely with each other, and as a result, the ratio equals to $\mathrm{exp}(\rho)$ exactly. Therefore, though the expected count of cycles of any type is not tractable, the ratio between the expected cycle counts of a (carefully chosen) pair of generalized cycles may have an utterly simple form. We study to what extent such pairs exist, and use our discovery to derive both an estimate for $\rho$ and a testing procedure for testing $\rho = \rho_0$. In a setting where we allow a wide range of reciprocal effects and a wide variety of network sparsity and degree heterogeneity, we show that our estimator achieves the optimal rate and our test achieves the optimal phase transition. Technically, first, motivated by what we observe on real networks, we do not want to impose strong conditions on reciprocal effects, network sparsity, and degree heterogeneity. Second, our proposed statistic is a type of $U$-statistic, the analysis of which involves complex combinatorics and is error-prone. For these reasons, our analysis is long and delicate.

math.ST↗

arXiv2025

Limit laws for the generalized Zagreb indices of random graphs

In this paper, we study the limiting behavior of the generalized Zagreb indices of the classical Erdős-Rényi (ER) random graph $G(n,p)$, as $n\to\infty$. For any integer $k\ge1$, we first give an expression for the $k$-th order generalized Zagreb index in terms of the number of star graphs of various sizes in any simple graph. The explicit formulas for the first two moments of the generalized Zagreb indices of an ER random graph are then obtained by this expression. Based on the asymptotic normality of the numbers of star graphs of various sizes, several joint limit laws are established for a finite number of generalized Zagreb indices with a phase transition for $p$ in different regimes. Finally, we provide a necessary and sufficient condition for any single generalized Zagreb index of $G(n,p)$ to be asymptotic normal.

math.PR↗