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Hypothesis testing for community structure in temporal networks using e-values

Eric Yanchenko, Jonathan P. Williams, Ryan Martin

TL;DR

This work proposes a simple yet powerful test using e-values, an alternative to p-values that is more flexible in certain ways, and applies the proposed test to synthetic and real-world networks, demonstrating various features inherited from the e-value formulation and exposing some of the inherent difficulties of testing on temporal networks.

Abstract

Community structure in networks naturally arises in various applications. But while the topic has received significant attention for static networks, the literature on community structure in temporally evolving networks is more scarce. In particular, there are currently no statistical methods available to test for the presence of community structure in a sequence of networks evolving over time. In this work, we propose a simple yet powerful test using e-values, an alternative to p-values that is more flexible in certain ways. Specifically, an e-value framework retains valid testing properties even after combining dependent information, a relevant feature in the context of testing temporal networks. We apply the proposed test to synthetic and real-world networks, demonstrating various features inherited from the e-value formulation and exposing some of the inherent difficulties of testing on temporal networks.

Hypothesis testing for community structure in temporal networks using e-values

TL;DR

This work proposes a simple yet powerful test using e-values, an alternative to p-values that is more flexible in certain ways, and applies the proposed test to synthetic and real-world networks, demonstrating various features inherited from the e-value formulation and exposing some of the inherent difficulties of testing on temporal networks.

Abstract

Community structure in networks naturally arises in various applications. But while the topic has received significant attention for static networks, the literature on community structure in temporally evolving networks is more scarce. In particular, there are currently no statistical methods available to test for the presence of community structure in a sequence of networks evolving over time. In this work, we propose a simple yet powerful test using e-values, an alternative to p-values that is more flexible in certain ways. Specifically, an e-value framework retains valid testing properties even after combining dependent information, a relevant feature in the context of testing temporal networks. We apply the proposed test to synthetic and real-world networks, demonstrating various features inherited from the e-value formulation and exposing some of the inherent difficulties of testing on temporal networks.

Paper Structure

This paper contains 34 sections, 34 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Challenges in temporal network testing
  5. E-values
  6. Basics
  7. Combination
  8. Methodology
  9. Proposed test
  10. Step 1: Compute p-values:
  11. Step 2: Convert p-values to e-values:
  12. Step 3: Average e-values:
  13. Step 4: Rejection decision:
  14. Properties
  15. Simulation Study
  16. ...and 19 more sections

Figures (10)

  • Figure 1: Median e-value over 100 MC simulations for correlated SBM networks with $\rho=0.25$. Grey line at $E=20$ corresponding to $\alpha=0.05$ rejection threshold.
  • Figure 2: Median e-value over 100 MC simulations for dynamic SBM networks with $\boldsymbol \pi_1$. Grey line at $E=20$ corresponding to $\alpha=0.05$ rejection threshold.
  • Figure 3: Median e-value over 100 MC simulations for dynamic DCBM networks with $\varepsilon=0.6$. Grey line at $E=20$ corresponding to $\alpha=0.05$ rejection threshold.
  • Figure 4: Simulation rejection rates for correlated SBM networks with $\rho=0.25$ and rejection threshold $\bar{E}_T>20$. Dashed black line corresponds to $\alpha=0.05$.
  • Figure 5: Simulation rejection rates for dynamic SBM networks with $\boldsymbol{\pi}_1$ and rejection threshold $\bar{E}_T>20$. Dashed black line corresponds to $\alpha=0.05$.
  • ...and 5 more figures