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Triadic structures in multislice networks

Kevin Ren, Tara Trauthwein, Gesine Reinert

TL;DR

This work extends motif-based analysis to multislice networks by developing a multislice Erdős-Rényi (MSER) model and proving a multivariate Poisson approximation for triangle counts across layers. The authors define 1D, 2D, and 3D triangles, derive means and bounds for covariances, and provide a total-variation bound for approximating the joint triangle distribution with a product of Poisson laws, enabling statistical goodness-of-fit tests. Through two real datasets (Florentine and Lazega), they demonstrate the approach: Florentine shows no rejection of the MSER null, while Lazega shows strong rejection, illustrating the framework's utility for assessing multislice network structure and motif distributions. The results highlight the utility and limitations of Poisson approximations in multislice contexts and point to future extensions to more complex multiplex models and other motifs.

Abstract

Networks provide a popular representation of complex data. Often, different types of relational measurements are taken on the same subjects. Such data can be represented as a \textit{multislice network}, a collection of networks on the same set of nodes, with connections between the different layers to be determined. For the analysis of multislice networks, we take inspiration from the analysis of simple networks, for which small subgraphs (motifs) have proven to be useful; motifs are even seen as building blocks of complex networks. A particular instance of a motif is a triangle, and while triangle counts are well understood for simple network models such as Erdős-Rényi random graphs, with i.i.d. distributed edges, even for simple multislice network models little is known about triangle counts. Here we address this issue by extending the analysis of triadic structures to multislice Erdős-Rényi networks. Again taking inspiration from the analysis of sparse Erdős-Rényi random graphs, we show that the distribution of triangles across multiple layers in a multislice Erdős-Rényi network can be well approximated by an appropriate Poisson distribution. This theoretical result opens the door to statistical goodness of fit tests for multislice networks.

Triadic structures in multislice networks

TL;DR

This work extends motif-based analysis to multislice networks by developing a multislice Erdős-Rényi (MSER) model and proving a multivariate Poisson approximation for triangle counts across layers. The authors define 1D, 2D, and 3D triangles, derive means and bounds for covariances, and provide a total-variation bound for approximating the joint triangle distribution with a product of Poisson laws, enabling statistical goodness-of-fit tests. Through two real datasets (Florentine and Lazega), they demonstrate the approach: Florentine shows no rejection of the MSER null, while Lazega shows strong rejection, illustrating the framework's utility for assessing multislice network structure and motif distributions. The results highlight the utility and limitations of Poisson approximations in multislice contexts and point to future extensions to more complex multiplex models and other motifs.

Abstract

Networks provide a popular representation of complex data. Often, different types of relational measurements are taken on the same subjects. Such data can be represented as a \textit{multislice network}, a collection of networks on the same set of nodes, with connections between the different layers to be determined. For the analysis of multislice networks, we take inspiration from the analysis of simple networks, for which small subgraphs (motifs) have proven to be useful; motifs are even seen as building blocks of complex networks. A particular instance of a motif is a triangle, and while triangle counts are well understood for simple network models such as Erdős-Rényi random graphs, with i.i.d. distributed edges, even for simple multislice network models little is known about triangle counts. Here we address this issue by extending the analysis of triadic structures to multislice Erdős-Rényi networks. Again taking inspiration from the analysis of sparse Erdős-Rényi random graphs, we show that the distribution of triangles across multiple layers in a multislice Erdős-Rényi network can be well approximated by an appropriate Poisson distribution. This theoretical result opens the door to statistical goodness of fit tests for multislice networks.

Paper Structure

This paper contains 10 sections, 3 theorems, 68 equations, 23 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. A multislice Erdős–Rényi model for multislice networks
  4. A Poisson approximation
  5. Illustration
  6. Florentine family data
  7. Lazega's lawyer data
  8. Conclusion
  9. Stein's method for Multivariate Poisson approximation
  10. Proofs

Key Result

Proposition 1

We have the following bounds: for covariances involving 1D triangles, and For covariances involving 2D triangles but not 1D triangles, For the covariances of 3D triangles,

Figures (23)

  • Figure 1: Example of multislice network with two layers. Each layer consists of a Erdős–Rényi random graph with 10 nodes and edge probabilities of $0.2$ and $0.3$ respectively. Here all nodes are connected between layers.
  • Figure 2: Examples of 1D, 2D and 3D triangles, across one, two or three layers
  • Figure 3: 1D and 2D counts from the Monte Carlo simulations, for the Florentine family multislice network
  • Figure 4: Dependent triangles for Case 1
  • Figure 5: Dependent triangles in Case 2, sharing the layer with two edges
  • ...and 18 more figures

Theorems & Definitions (5)

  • Proposition 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • proof