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Register a new userA sparse $p_0$ model with covariates for directed networks
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Added by
Qiuping Wang
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
Authors
Qiuping Wang
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We are concerned here with unrestricted maximum likelihood estimation in a sparse $p_0$ model with covariates for directed networks. The model has a density parameter $ u$, a $2n$-dimensional node parameter $bs{eta}$ and a fixed dimensional regression coefficient $bs{gamma}$ of covariates. Previous studies focus on the restricted likelihood inference. When the number of nodes $n$ goes to infinity, we derive the $ell_infty$-error between the maximum likelihood estimator (MLE) $(widehat{bs{eta}}, widehat{bs{gamma}})$ and its true value $(bs{eta}, bs{gamma})$. They are $O_p( (log n/n)^{1/2} )$ for $widehat{bs{eta}}$ and $O_p( log n/n)$ for $widehat{bs{gamma}}$, up to an additional factor. This explains the asymptotic bias phenomenon in the asymptotic normality of $widehat{bs{gamma}}$ in cite{Yan-Jiang-Fienberg-Leng2018}. Further, we derive the asymptotic normality of the MLE. Numerical studies and a data analysis demonstrate our theoretical findings.
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The edges in networks are not only binary, either present or absent, but also take weighted values in many scenarios (e.g., the number of emails between two users). The covariate-$p_0$ model has been proposed to model binary directed networks with the degree heterogeneity and covariates. However, it may cause information loss when it is applied in weighted networks. In this paper, we propose to use the Poisson distribution to model weighted directed networks, which admits the sparsity of networks, the degree heterogeneity and the homophily caused by covariates of nodes. We call it the emph{network Poisson model}. The model contains a density parameter $mu$, a $2n$-dimensional node parameter ${theta}$ and a fixed dimensional regression coefficient ${gamma}$ of covariates. Since the number of parameters increases with $n$, asymptotic theory is nonstandard. When the number $n$ of nodes goes to infinity, we establish the $ell_infty$-errors for the maximum likelihood estimators (MLEs), $hat{theta}$ and $hat{{gamma}}$, which are $O_p( (log n/n)^{1/2} )$ for $hat{theta}$ and $O_p( log n/n)$ for $hat{{gamma}}$, up to an additional factor. We also obtain the asymptotic normality of the MLE. Numerical studies and a data analysis demonstrate our theoretical findings. ) for b{theta} and Op(log n/n) for b{gamma}, up to an additional factor. We also obtain the asymptotic normality of the MLE. Numerical studies and a data analysis demonstrate our theoretical findings.
The $p_0$ model is an exponential random graph model for directed networks with the bi-degree sequence as the exclusively sufficient statistic. It captures the network feature of degree heterogeneity. The consistency and asymptotic normality of a differentially private estimator of the parameter in the private $p_0$ model has been established. However, the $p_0$ model only focuses on binary edges. In many realistic networks, edges could be weighted, taking a set of finite discrete values. In this paper, we further show that the moment estimators of the parameters based on the differentially private bi-degree sequence in the weighted $p_0$ model are consistent and asymptotically normal. Numerical studies demonstrate our theoretical findings.
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