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Optimal Nonparametric Inference on Network Effects with Dependent Edges

Wenqin Du, Yuan Zhang, Wen Zhou

TL;DR

This paper develops a fully nonparametric framework for testing network effects in directed, weighted networks with dependent edges by leveraging the Aldous–Hoover exchangeable representation. It addresses the challenge of indeterminate degeneracy in network U-statistics through a novel U-statistics reduction, and provides Berry–Esseen type bounds for finite-sample type I error control, along with minimax-type results for test power. The methodology covers four network effects, offering degeneracy diagnostics and separate non-degenerate and degenerate inference procedures with reduced-network moments to ensure computational scalability and robustness. An empirical evaluation on simulated data and real faculty hiring networks demonstrates substantial speed, accuracy, and robustness advantages over SRM-based methods, with practically meaningful findings such as strong same-sender effects and reciprocity in real networks.

Abstract

Testing network effects in weighted directed networks is a foundational problem in econometrics, sociology, and psychology. Yet, the prevalent edge dependency poses a significant methodological challenge. Most existing methods are model-based and come with stringent assumptions, limiting their applicability. In response, we introduce a novel, fully nonparametric framework that requires only minimal regularity assumptions. While inspired by recent developments in $U$-statistic literature (arXiv:1712.00771, arXiv:2004.06615), our approach notably broadens their scopes. Specifically, we identified and carefully addressed the challenge of indeterminate degeneracy in the test statistics $-$ a problem that aforementioned tools do not handle. We established Berry-Esseen type bound for the accuracy of type-I error rate control. Using original analysis, we also proved the minimax optimality of our test's power. Simulations underscore the superiority of our method in computation speed, accuracy, and numerical robustness compared to competing methods. We also applied our method to the U.S. faculty hiring network data and discovered intriguing findings.

Optimal Nonparametric Inference on Network Effects with Dependent Edges

TL;DR

This paper develops a fully nonparametric framework for testing network effects in directed, weighted networks with dependent edges by leveraging the Aldous–Hoover exchangeable representation. It addresses the challenge of indeterminate degeneracy in network U-statistics through a novel U-statistics reduction, and provides Berry–Esseen type bounds for finite-sample type I error control, along with minimax-type results for test power. The methodology covers four network effects, offering degeneracy diagnostics and separate non-degenerate and degenerate inference procedures with reduced-network moments to ensure computational scalability and robustness. An empirical evaluation on simulated data and real faculty hiring networks demonstrates substantial speed, accuracy, and robustness advantages over SRM-based methods, with practically meaningful findings such as strong same-sender effects and reciprocity in real networks.

Abstract

Testing network effects in weighted directed networks is a foundational problem in econometrics, sociology, and psychology. Yet, the prevalent edge dependency poses a significant methodological challenge. Most existing methods are model-based and come with stringent assumptions, limiting their applicability. In response, we introduce a novel, fully nonparametric framework that requires only minimal regularity assumptions. While inspired by recent developments in -statistic literature (arXiv:1712.00771, arXiv:2004.06615), our approach notably broadens their scopes. Specifically, we identified and carefully addressed the challenge of indeterminate degeneracy in the test statistics a problem that aforementioned tools do not handle. We established Berry-Esseen type bound for the accuracy of type-I error rate control. Using original analysis, we also proved the minimax optimality of our test's power. Simulations underscore the superiority of our method in computation speed, accuracy, and numerical robustness compared to competing methods. We also applied our method to the U.S. faculty hiring network data and discovered intriguing findings.
Paper Structure (27 sections, 20 theorems, 50 equations, 3 figures, 6 tables)

This paper contains 27 sections, 20 theorems, 50 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Literature review
  4. Our contributions
  5. Reference chart for main results
  6. Problem set up: network effects in exchangeable networks
  7. Inference for same-sender effect eta3 and same-receiver effect eta4
  8. Inference for $\eta_3$
  9. Degeneracy status of $\widehat{\eta}_{3,n}$
  10. Testing $\eta_{3}$ using the U-statistics reduction for $\widehat{\eta}_{3,n}$
  11. Theoretical guarantees
  12. Inference for $\eta_4$
  13. Inference for sender-receiver effect eta5
  14. Diagnosis of degeneracy
  15. The non-degenerate case
  16. ...and 12 more sections

Key Result

Theorem 2.2

A network $\{e_{i,j}\}_{1\leq \{i,j\}\leq n}$ satisfying Definition def::WEdef admits the universal representation where $F$ is a latent, potentially asymmetric function that encodes all network structures, and latent variables are generated as $\{X_i\}_{1\leq i\leq n} \cup \{X_{(i,j)} = X_{(j,i)}\}_{1\leq i<j\leq n}\stackrel{\rm i.i.d.}\sim \mathrm{Uniform}[0,1]$.

Figures (3)

  • Figure 1: Q-Q plots of the null distribution of test statistics under normal configurations across settings \ref{['test_eta2_add']} to \ref{['test_comp_d']}.
  • Figure 2: Local network effects for the business school hiring network. Left: Density plots of local network effects. Right: Scatter plot of the local reciprocity effect versus the local same-sender effect. Institutes with larger local reciprocity and same-sender effects are highlighted in red.
  • Figure 3: Radar plots of local network effects for institutions marked in red in Figure \ref{['fig::local-network-effects']}, ordered by the U.S. News & World Report rankings in 2012 for business schools.

Theorems & Definitions (25)

  • Definition 2.1: Exchangeable networks, hoff2007modeling
  • Theorem 2.2: hoover1979relationsaldous1981representations
  • Proposition 3.1
  • Lemma 3.1
  • Proposition 3.2
  • Example 3.1: Indeterminate degeneracy of $\widehat{\eta}_{3,n}$ under $H_0^{(3)}$
  • Theorem 3.1
  • Theorem 3.2
  • Example 4.1: Indeterminate degeneracy of $\widehat{\eta}_{5,n}$ under $H_0^{(5)}$
  • Proposition 4.1
  • ...and 15 more