A network Poisson model for weighted directed networks with covariates

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-p0 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 network Poisson model. The model contains a density parameter μ, a 2n-dimensional node parameter θ and a fixed dimensional regression coefficient γ of covariates. Since the number of parameters increases with n, asymptotic theory is non standard. When the number n of nodes goes to infinity, we establish the ℓ∞-errors for the maximum likelihood estimators (MLEs), θ̂ and γ̂, which are Op(( log n/n)1/2) for θ̂ and Op( log n/n) for γ̂, up to an additional factor. We also obtain the asymptotic normality of the MLE. Numerical studies and a data analysis demonstrate our theoretical findings.</p

Scheme optimization process.

Scheme optimization process.</p

Weight vectors.

Weight vectors.</p

The highway route scheme optimization algorithm.

The highway route scheme optimization algorithm.</p

Evaluation results of the relative degree of closeness.

Evaluation results of the relative degree of closeness.</p

Theorem 1: If (<i>j = 1</i>, <i>2</i>, <i>…</i>, <i>n</i>), then the optimization formula (15) has a unique solution.

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Evaluation index system for a highway route scheme.

Evaluation index system for a highway route scheme.</p

Evaluation indexes for the program.

Evaluation indexes for the program.</p

General form of the judgment matrix.

General form of the judgment matrix.</p

Highway route plan.

Highway route plan.</p