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Testing For Global Covariate Effects in Dynamic Interaction Event Networks

Alexander Kreiss, Enno Mammen, Wolfgang Polonik

Abstract

In statistical network analysis it is common to observe so called interaction data. Such data is characterized by actors forming the vertices and interacting along edges of the network, where edges are randomly formed and dissolved over the observation horizon. In addition covariates are observed and the goal is to model the impact of the covariates on the interactions. We distinguish two types of covariates: global, system-wide covariates (i.e. covariates taking the same value for all individuals, such as seasonality) and local, dyadic covariates modeling interactions between two individuals in the network. Existing continuous time network models are extended to allow for comparing a completely parametric model and a model that is parametric only in the local covariates but has a global non-parametric time component. This allows, for instance, to test whether global time dynamics can be explained by simple global covariates like weather, seasonality etc. The procedure is applied to a bike-sharing network by using weather and weekdays as global covariates and distances between the bike stations as local covariates.

Testing For Global Covariate Effects in Dynamic Interaction Event Networks

Abstract

In statistical network analysis it is common to observe so called interaction data. Such data is characterized by actors forming the vertices and interacting along edges of the network, where edges are randomly formed and dissolved over the observation horizon. In addition covariates are observed and the goal is to model the impact of the covariates on the interactions. We distinguish two types of covariates: global, system-wide covariates (i.e. covariates taking the same value for all individuals, such as seasonality) and local, dyadic covariates modeling interactions between two individuals in the network. Existing continuous time network models are extended to allow for comparing a completely parametric model and a model that is parametric only in the local covariates but has a global non-parametric time component. This allows, for instance, to test whether global time dynamics can be explained by simple global covariates like weather, seasonality etc. The procedure is applied to a bike-sharing network by using weather and weekdays as global covariates and distances between the bike stations as local covariates.

Paper Structure

This paper contains 22 sections, 16 theorems, 224 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Model Specification and Test Strategy
  3. Main Results
  4. Main Result
  5. Assumptions
  6. Proof of Theorem \ref{['thm:T1']}
  7. Numerical Results
  8. Implementation
  9. Empirical Application
  10. Simulation
  11. Conclusion
  12. Relation to Kreiss (2021) and Mixing
  13. Discussion of Assumptions
  14. Motivation for Assumption (mDep)
  15. Hubs
  16. ...and 7 more sections

Key Result

Theorem 3.1

Suppose that Assumption (SP) and all the assumptions from Section subsec:assumptions hold. Further, assume that model eq:mod holds with $\alpha_0(t)=\alpha(\theta_0,t)+c_n\Delta_n(t),$ where $c_n=(N\sqrt{h})^{-1/2},$ and $\Delta_n$ is uniformly bounded and continuously differentiable with uniformly where with $f_n(r,s):=\int_0^ThK_{h,t}(s)K_{h,t}(r)w(t)dt,$$\gamma=\int_0^T\int_0^TK_{h,t}(s)\frac{

Figures (7)

  • Figure 1: Kernel density estimate for bike rides over a period of two weeks in 2018 (May 13 - May 26).
  • Figure 2: Temperature and Precipitation in Washington D.C.
  • Figure 3: Parametric (P) and Non-parametric (NP) estimates of the baseline intensity for the second week.
  • Figure 4: Choices of baseline intensities used for the simulations (x-axis in hours)
  • Figure 5: Percentage of rejections of a test of level $\alpha=0.05$.
  • ...and 2 more figures

Theorems & Definitions (43)

  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 6.1
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 33 more