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A functional tensor model for dynamic multilayer networks with common invariant subspaces and the RKHS estimation

Runshi Tang, Runbing Zheng, Anru R. Zhang, Carey E. Priebe

Abstract

Dynamic multilayer networks are frequently used to describe the structure and temporal evolution of multiple relationships among common entities, with applications in fields such as sociology, economics, and neuroscience. However, exploration of analytical methods for these complex data structures remains limited. We propose a functional tensor-based model for dynamic multilayer networks, with the key feature of capturing the shared structure among common vertices across all layers, while simultaneously accommodating smoothly varying temporal dynamics and layer-specific heterogeneity. The proposed model and its embeddings can be applied to various downstream network inference tasks, including dimensionality reduction, vertex community detection, analysis of network evolution periodicity, visualization of dynamic network evolution patterns, and evaluation of inter-layer similarity. We provide an estimation algorithm based on functional tensor Tucker decomposition and the reproducing kernel Hilbert space framework, with an effective initialization strategy to improve computational efficiency. The estimation procedure can be extended to address more generalized functional tensor problems, as well as to handle missing data or unaligned observations. We validate our method on simulated data and two real-world cases: the dynamic Citi Bike trip network and an international food trade dynamic multilayer network, with each layer corresponding to a different product.

A functional tensor model for dynamic multilayer networks with common invariant subspaces and the RKHS estimation

Abstract

Dynamic multilayer networks are frequently used to describe the structure and temporal evolution of multiple relationships among common entities, with applications in fields such as sociology, economics, and neuroscience. However, exploration of analytical methods for these complex data structures remains limited. We propose a functional tensor-based model for dynamic multilayer networks, with the key feature of capturing the shared structure among common vertices across all layers, while simultaneously accommodating smoothly varying temporal dynamics and layer-specific heterogeneity. The proposed model and its embeddings can be applied to various downstream network inference tasks, including dimensionality reduction, vertex community detection, analysis of network evolution periodicity, visualization of dynamic network evolution patterns, and evaluation of inter-layer similarity. We provide an estimation algorithm based on functional tensor Tucker decomposition and the reproducing kernel Hilbert space framework, with an effective initialization strategy to improve computational efficiency. The estimation procedure can be extended to address more generalized functional tensor problems, as well as to handle missing data or unaligned observations. We validate our method on simulated data and two real-world cases: the dynamic Citi Bike trip network and an international food trade dynamic multilayer network, with each layer corresponding to a different product.

Paper Structure

This paper contains 35 sections, 4 theorems, 60 equations, 12 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Model
  4. Functional tensor model for single dynamic networks
  5. Multilayer functional tensor dynamic networks
  6. Identifiability
  7. Estimation via Functional Tensor Tucker Decomposition and RKHS
  8. Likelihood formulation and RKHS-constrained estimation
  9. Optimization algorithm and implementation
  10. Initialization via spectral methods
  11. Simulation Experiments
  12. Estimation accuracy
  13. Dynamic network clustering
  14. Vertex community detection
  15. Real Data Experiments
  16. ...and 20 more sections

Key Result

Proposition 1

(MFTDN identifiability) Suppose an MFTDN can be represented by two sets of parameters, $(\mathbf{X}, \mathbf{Y}, \mathbf{R}^{[t]}_s)$ and $(\mathbf{X}', \mathbf{Y}', (\mathbf{R}')^{[t]}_s)$, such that they yield the same probability matrices $\{\mathbf{P}^{[t]}_s\}_{t \in \mathcal{T}, s \in [K]}$. I where the matrix on the left is the horizontal concatenation of all $\{\mathbf{R}_s^{[t]}\}_{s\in[

Figures (12)

  • Figure 1: Empirical $\operatorname{Err}(\widehat{\mathbf{X}},\widehat{\mathbf{Y}})$ and $\operatorname{Acc}(\widehat{\mathbf{R}})$ for the MFTDN model estimation under the following settings: (1) varying $n \in \{30, 50, 70, 90, 110, 130\}$ while fixing $m=20$, $K=4$, and $d=3$ fixed, (2) varying $m \in \{8, 12, 16, 20, 24, 28, 32\}$ while fixing $n=100$, $K=5$, and $d=2$, and (3) varying $K \in \{3, 4, 5, 6, 7\}$ while fixing $n=100$, $m=20$, and $d=3$. Additional details of the settings are provided in Section \ref{['sec:simu_error']}. The lines represent the means of $64$ independent Monte Carlo replicates, with error bars indicating the $20$th and $80$th quantiles. The total runtime for all settings is 1.5 hours using 64 parallel computations. The gray dashed lines with error bars show the results of the COSIE model for comparison.
  • Figure 2: Empirical accuracy, measured as the proportion of layers clustered correctly, using the MFTDN model estimation under the following settings: (1) varying $n \in \{10, 20, 30, 40, 50\}$ while fixing $m=15$ and $K=10$, (2) varying $m \in \{10, 14, 18, 22, 26\}$ while fixing $n=30$ and $K=10$, and (3) varying $K \in \{20, 25, 30, 35, 40\}$ while fixing $n=20$ and $m=10$. Additional details of the settings are provided in Section \ref{['sec:multi']}. The results are means of $256$ independent Monte Carlo replicates. The gray and black dashed lines show the results of COSIE and MFTDN without temporal smoothing, respectively, for comparison.
  • Figure 3: Empirical overall accuracy, measured as the average of the proportions of vertices correctly clustered for outgoing and incoming edges, using the FTDN and other methods under the following settings: (1) varying $n \in \{40, 60, 80, 100, 120, 150, 200\}$ while fixing $m = 20$, and (2) varying $m \in \{5, 8, 12, 15, 20\}$ while fixing $n = 100$. Additional details of the settings are provided in Section \ref{['sec:dynamic SBM']}. The results are medians of $50$ independent Monte Carlo replicates.
  • Figure 4: The $k$-means clustering results for the $2240$ Citi Bike stations in New York City, grouped into 46 clusters. The colored crosses represent the cluster centers, with the four boroughs, Bronx, Brooklyn, Manhattan, and Queens, represented by different colors. The smaller grey dots represent the original stations.
  • Figure 5: Kernel and kernel parameter selection by BIC.
  • ...and 7 more figures

Theorems & Definitions (11)

  • Definition 1: Functional tensor dynamic network
  • Definition 2: Multilayer functional tensor dynamic network
  • Proposition 1
  • Proposition 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Proposition A.1
  • ...and 1 more