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Network regression and supervised centrality estimation

Junhui Cai, Dan Yang, Ran Chen, Wu Zhu, Haipeng Shen, Linda Zhao

TL;DR

This work addresses the problem of learning network effects when the observed network is noisy by introducing a unified framework that jointly models network generation and network regression. The key contribution is SuperCENT, a supervised centrality estimation method that uses the regression data to inform centrality estimation, yielding faster convergence and valid confidence intervals compared to the traditional two-stage approach. Theoretical results show SuperCENT dominates the two-stage procedure under broad conditions, with optimal tuning $ ext{lambda}$ improving both centrality estimation and regression inference, as demonstrated in extensive simulations and a case study predicting currency risk premia from the global trade network. Practically, this leads to more reliable inference and economically meaningful insights, including better investment strategies based on centrality-informed currency risk predictors.

Abstract

The centrality in a network is often used to measure nodes' importance and model network effects on a certain outcome. Empirical studies widely adopt a two-stage procedure, which first estimates the centrality from the observed noisy network and then infers the network effect from the estimated centrality, even though it lacks theoretical understanding. We propose a unified modeling framework to study the properties of centrality estimation and inference and the subsequent network regression analysis with noisy network observations. Furthermore, we propose a supervised centrality estimation methodology, which aims to simultaneously estimate both centrality and network effect. We showcase the advantages of our method compared with the two-stage method both theoretically and numerically via extensive simulations and a case study in predicting currency risk premiums from the global trade network.

Network regression and supervised centrality estimation

TL;DR

This work addresses the problem of learning network effects when the observed network is noisy by introducing a unified framework that jointly models network generation and network regression. The key contribution is SuperCENT, a supervised centrality estimation method that uses the regression data to inform centrality estimation, yielding faster convergence and valid confidence intervals compared to the traditional two-stage approach. Theoretical results show SuperCENT dominates the two-stage procedure under broad conditions, with optimal tuning improving both centrality estimation and regression inference, as demonstrated in extensive simulations and a case study predicting currency risk premia from the global trade network. Practically, this leads to more reliable inference and economically meaningful insights, including better investment strategies based on centrality-informed currency risk predictors.

Abstract

The centrality in a network is often used to measure nodes' importance and model network effects on a certain outcome. Empirical studies widely adopt a two-stage procedure, which first estimates the centrality from the observed noisy network and then infers the network effect from the estimated centrality, even though it lacks theoretical understanding. We propose a unified modeling framework to study the properties of centrality estimation and inference and the subsequent network regression analysis with noisy network observations. Furthermore, we propose a supervised centrality estimation methodology, which aims to simultaneously estimate both centrality and network effect. We showcase the advantages of our method compared with the two-stage method both theoretically and numerically via extensive simulations and a case study in predicting currency risk premiums from the global trade network.
Paper Structure (52 sections, 16 theorems, 235 equations, 19 figures, 4 tables, 6 algorithms)

This paper contains 52 sections, 16 theorems, 235 equations, 19 figures, 4 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. A unified framework
  3. Set-up and background of network
  4. A unified framework
  5. Methodology
  6. The two-stage procedure
  7. SuperCENT methodology
  8. Theoretical properties
  9. Simulation
  10. Simulation setup
  11. Simulation results
  12. Global trade network and currency risk premium
  13. Centrality estimation.
  14. Inference of regression.
  15. Conclusion and discussion
  16. ...and 37 more sections

Key Result

Theorem 1

Under the unified framework eq:model-supercent and Assumptions assump:normal-assump:consistent, suppose ${\color{black} \sqrt{\kappa}}\frac{\sigma_y}{\sqrt{\beta_{u}^{2}+\beta_{v}^{2}}} = o(1)$, $\sigma_y = o\left(\sqrt{\frac{n}{\log n}}\right)$, and $\left|\frac{\beta_{u}}{\beta_{v}}\right| \in[\u where $\left( \right) \sim N( {\bf 0}_{(2n+2+p)\times 1}, {\hbox{\boldmath $C$}}\left( \right){{

Figures (19)

  • Figure 1: Boxplot of $\log_{10}(l(\hat{\hbox{\boldmath $u$}},{\hbox{\boldmath $u$}}))$ for the four estimators across different ${\sigma_{a}}$, ${\sigma_{y}}$ and ${\beta_u}$ with fixed $d = 1,~{\beta_v} = 1$. The super-imposed red symbols show the theoretical rates of the two-stage and SuperCENT calculated from Theorems \ref{['thm:two-stage-normality']} and \ref{['thm:supercent-normality']} respectively in Figure \ref{['fig:u-2']} and the median of ${\hat{\beta}_u} - \beta_u$ in Figure \ref{['fig:betau-bias-2']} respectively.
  • Figure 2: Empirical coverage and $\log_{10}$ of the width of $CI_{{\beta_u}}$ across different ${\sigma_{a}}$, ${\sigma_{y}}$ and ${\beta_u}$ with $d = 1$ and ${\beta_v} = 1$. ${{\text{SuperCENT}}}$ variants are labelled as circles ($\circ\;\bullet$) and the two-stage variants are labelled as triangles ($\vartriangle\mathlarger{\mathlarger{\mathlarger{\blacktriangledown}}}\;\blacktriangle$). The hollow ones are for oracles and the solid ones are for non-oracles.
  • Figure 3: Time series of ranking of risk premium in descending order and ranking of hub centrality estimated by two-stage and SuperCENT in ascending order from 2003 to 2012. The vertical dashed line indicates 2008, the year of the financial crisis.
  • Figure 4: Time series of the next-year return from 2004 to 2013 based on a strategy that takes a long position on the currencies with the lowest 3 centralities and a short position on the currencies with the highest 3 centralities estimated from 2003 to 2012 respectively.
  • Figure S5: A toy network to illustrate the hub and authority centrality.
  • ...and 14 more figures

Theorems & Definitions (53)

  • Remark 1
  • Remark 2: Identifiability of ${\hat{\beta}_u},{\hat{\beta}_v},\widehat{{\hbox{\boldmath $u$}}},\widehat{{\hbox{\boldmath $v$}}}$
  • Remark 3: Full-rankness of $({\mathbf{X}}, {\hbox{\boldmath $u$}}^{(t)}, {\hbox{\boldmath $v$}}^{(t)})$
  • Remark 4: Algorithmic convergence of Algorithm \ref{['algo:supercent']}
  • Remark 5: Verification of the full-rank assumption of $({\hbox{\boldmath $X$}}, {{\hbox{\boldmath $u$}}}, {{\hbox{\boldmath $v$}}})$
  • Remark 6: Practical implication of the full-rank assumption of $({\hbox{\boldmath $X$}}, {{\hbox{\boldmath $u$}}}, {{\hbox{\boldmath $v$}}})$
  • Theorem 1
  • Remark 7: Asymptotic results for SuperCENT estimators from Algorithm \ref{['algo:supercent']}
  • Proposition 1
  • Remark 8
  • ...and 43 more