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Inference in covariate-adjusted bipartite network models

Wu Zuhui, Wang Qiuping, Yan Ting

Abstract

In this paper, we introduce a general model for jointly modelling the nodal heterogeneity and covariates in weighted or unweighted bipartite networks, which contains two different types of nodes. The model has a degree heterogeneity parameter for each node and a fixed-dimensional regression coefficient for the covariates. We use the method of moments to estimate the unknown parameters. When the model belongs to the exponential family of distributions, the moment estimator is identical to the maximum likelihood estimator. We show the uniform consistency of the moment estimator, when the number of actors and the number of events both go to infinity under some conditions. Further, we derive an asymptotic representation of the moment estimator, which leads to their asymptotic normal distributions under some conditions. We present two applications to illustrate the unified results. Numerical simulations and a real-data analysis demonstrates our theoretical findings.

Inference in covariate-adjusted bipartite network models

Abstract

In this paper, we introduce a general model for jointly modelling the nodal heterogeneity and covariates in weighted or unweighted bipartite networks, which contains two different types of nodes. The model has a degree heterogeneity parameter for each node and a fixed-dimensional regression coefficient for the covariates. We use the method of moments to estimate the unknown parameters. When the model belongs to the exponential family of distributions, the moment estimator is identical to the maximum likelihood estimator. We show the uniform consistency of the moment estimator, when the number of actors and the number of events both go to infinity under some conditions. Further, we derive an asymptotic representation of the moment estimator, which leads to their asymptotic normal distributions under some conditions. We present two applications to illustrate the unified results. Numerical simulations and a real-data analysis demonstrates our theoretical findings.

Paper Structure

This paper contains 14 sections, 7 theorems, 65 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A covariate-adjusted bipartite network model
  3. Moment estimation
  4. Asymptotic properties
  5. Preliminaries
  6. Consistency
  7. Asymptotic normality
  8. Applications
  9. The logistic model
  10. The Poisson model
  11. Numerical Analysis
  12. Simulation
  13. A real Data example
  14. Summary and discussion

Key Result

Theorem 1

Let $V=\partial F(\theta^*, \gamma^*)/\partial \theta^\top$ and $\sigma_{m,n}^2 = (m+n-1)^2 \| (V^{-1}-S) \mathrm{Cov}(g) (V^{-1}-S) \|_{\max}$, where $S$ is defined in eq:Smatrix. Suppose Conditions condition-0--assumption-3 hold, and Then the moment estimator $(\widehat{\beta},\widehat{\gamma})$ exists with high probability, and we further have $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 6.1: QQ plot of $\hat{\zeta}_i$ for $m=300$, $n=100$
  • Figure 6.2: Plots of the MLEs for the degree parameters.
  • Figure 6.3: Histogram density estimation of the fitted parameter estimates.

Theorems & Definitions (10)

  • Example 1
  • Example 2
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Corollary 2
  • Corollary 3
  • Corollary 4