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Mesoscale two-sample testing for networks

Peter W. MacDonald, Elizaveta Levina, Ji Zhu

Abstract

Networks arise naturally in many scientific fields as a representation of pairwise connections. Statistical network analysis has most often considered a single large network, but it is common in a number of applications to observe multiple networks on a shared node set. When these networks are grouped by case-control status or another categorical covariate, the classical statistical question of two-sample comparison arises. In this work, we address the problem of testing for statistically significant differences in a given arbitrary subset of connections. This general framework allows an analyst to focus on a single node, a specific region of interest, or compare whole networks. Our ability to conduct ``mesoscale'' testing on a meaningful group of edges is particularly relevant for applications such as neuroimaging and distinguishes our approach from prior work, which tends to focus either on a single node or the whole network. In this mesoscale setting, we develop statistically sound projection-based tests for two-sample comparison in both weighted and binary edge networks. The key to our approach is to leverage network information from outside the set of interest to learn informative low-rank projections which leads to more powerful tests.

Mesoscale two-sample testing for networks

Abstract

Networks arise naturally in many scientific fields as a representation of pairwise connections. Statistical network analysis has most often considered a single large network, but it is common in a number of applications to observe multiple networks on a shared node set. When these networks are grouped by case-control status or another categorical covariate, the classical statistical question of two-sample comparison arises. In this work, we address the problem of testing for statistically significant differences in a given arbitrary subset of connections. This general framework allows an analyst to focus on a single node, a specific region of interest, or compare whole networks. Our ability to conduct ``mesoscale'' testing on a meaningful group of edges is particularly relevant for applications such as neuroimaging and distinguishes our approach from prior work, which tends to focus either on a single node or the whole network. In this mesoscale setting, we develop statistically sound projection-based tests for two-sample comparison in both weighted and binary edge networks. The key to our approach is to leverage network information from outside the set of interest to learn informative low-rank projections which leads to more powerful tests.

Paper Structure

This paper contains 42 sections, 16 theorems, 225 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mesoscale testing methodology
  3. Notation
  4. An exponential family edge model
  5. Mesoscale null hypotheses and projection-based tests
  6. Classes of projected tests (Stage II)
  7. Exponential family model with known mean-variance relationship
  8. Exponential family edge model with unknown dispersion
  9. Gaussian edge model
  10. Gaussian edge model with projected data
  11. Learning projections from held-out edges
  12. Row and column hypothesis sets
  13. Theoretical guarantees
  14. Test properties with fixed projections
  15. The general exponential family edge model
  16. ...and 27 more sections

Key Result

Proposition 1

Suppose $U_{-[r]}$ and $V_{-[c]}$ have linearly independent columns. Then $\mathbb{T}$ has rank $d_{\mathcal{S}}$, and the column and row spaces of $\mathbb{T}$ coincide with the column and row spaces of $\Theta^{(1)}_{\mathcal{S}} - \Theta^{(2)}_{\mathcal{S}}$.

Figures (9)

  • Figure 1: Rejection rate for Gaussian edge networks with inner product latent space model. Dashed lines correspond to $\alpha=0.05$ and $1$. Point colors correspond to $2p$ or $d$, with basic tests displayed in black and oracle projection tests in orange.
  • Figure 2: Rejection rate for Gaussian edge networks with Euclidean distance latent space model. Dashed lines correspond to $\alpha=0.05$ and $1$. Point colors correspond to $2p$ or $d$, with basic tests displayed in black and oracle projection tests in orange.
  • Figure 3: Rejection rates for binary networks modeled with logistic link and the inner product latent space model. Dashed lines correspond to $\alpha=0.05$ and $1$. Point colors correspond to $2p$ or $d$, with basic tests displayed in black and oracle projection tests in orange.
  • Figure 4: Rejection rates for binary edge logistic link networks with inner product latent space model and overdispersion parameter $\eta=2$. Dashed lines correspond to $\alpha=0.05$ and $1$. Point colors correspond to $2p$ or $d$, with basic tests displayed in black and oracle projection tests in orange.
  • Figure 5: Median $-\log(p)$ for testing the CBM/CBM hypothesis in fRMI data as a function of sample size $m$. For $m < 20$, vertical bars span from the 25th to 75th empirical quantiles over $200$ replications. The horizontal dotted line corresponds to $\alpha=0.05$.
  • ...and 4 more figures

Theorems & Definitions (36)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Remark 3
  • Lemma 1
  • Corollary 1
  • Proposition 5
  • ...and 26 more