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Dynamic networks clustering via mirror distance

Runbing Zheng, Avanti Athreya, Marta Zlatic, Michael Clayton, Carey E. Priebe

TL;DR

This work introduces Dynamic Network Clustering through Mirror Distance (DNCMD), a method that clusters multiple dynamic networks by comparing their Euclidean mirrors—low-dimensional representations of network evolution derived from latent-position changes. It provides rigorous guarantees of exact recovery under deterministic and random latent-position models, with error rates that scale favorably with network sparsity and time horizon. The approach combines adjacency spectral embedding, CMDS, Procrustes alignment, and hierarchical clustering, and it is validated through simulations and real data analyses of Drosophila connectomes and international trade networks. The results demonstrate that mirror-based clustering captures meaningful evolutionary patterns, enabling discovery of key structural differences across dynamic networks and offering scalable, distributed computation advantages for large-scale data.

Abstract

The classification of different patterns of network evolution, for example in brain connectomes or social networks, is a key problem in network inference and modern data science. Building on the notion of a network's Euclidean mirror, which captures its evolution as a curve in Euclidean space, we develop the Dynamic Network Clustering through Mirror Distance (DNCMD), an algorithm for clustering dynamic networks based on a distance measure between their associated mirrors. We provide theoretical guarantees for DNCMD to achieve exact recovery of distinct evolutionary patterns for latent position random networks both when underlying vertex features change deterministically and when they follow a stochastic process. We validate our theoretical results through numerical simulations and demonstrate the application of DNCMD to understand edge functions in Drosophila larval connectome data, as well as to analyze temporal patterns in dynamic trade networks.

Dynamic networks clustering via mirror distance

TL;DR

This work introduces Dynamic Network Clustering through Mirror Distance (DNCMD), a method that clusters multiple dynamic networks by comparing their Euclidean mirrors—low-dimensional representations of network evolution derived from latent-position changes. It provides rigorous guarantees of exact recovery under deterministic and random latent-position models, with error rates that scale favorably with network sparsity and time horizon. The approach combines adjacency spectral embedding, CMDS, Procrustes alignment, and hierarchical clustering, and it is validated through simulations and real data analyses of Drosophila connectomes and international trade networks. The results demonstrate that mirror-based clustering captures meaningful evolutionary patterns, enabling discovery of key structural differences across dynamic networks and offering scalable, distributed computation advantages for large-scale data.

Abstract

The classification of different patterns of network evolution, for example in brain connectomes or social networks, is a key problem in network inference and modern data science. Building on the notion of a network's Euclidean mirror, which captures its evolution as a curve in Euclidean space, we develop the Dynamic Network Clustering through Mirror Distance (DNCMD), an algorithm for clustering dynamic networks based on a distance measure between their associated mirrors. We provide theoretical guarantees for DNCMD to achieve exact recovery of distinct evolutionary patterns for latent position random networks both when underlying vertex features change deterministically and when they follow a stochastic process. We validate our theoretical results through numerical simulations and demonstrate the application of DNCMD to understand edge functions in Drosophila larval connectome data, as well as to analyze temporal patterns in dynamic trade networks.

Paper Structure

This paper contains 25 sections, 11 theorems, 64 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations
  3. Model
  4. Algorithm
  5. Theoretical Results
  6. Theoretical results for random latent positions
  7. Simulation Results
  8. Error in mirror estimate
  9. Clustering performance
  10. Real Data Experiments
  11. Drosophila larval connectome data
  12. Trade dynamic networks
  13. Conclusion and Discussion
  14. Additional Numerical Results
  15. Additional analysis for Drosophila larval connectome
  16. ...and 10 more sections

Key Result

Theorem 1

If two dynamic networks $i,j\in[m]$ share the same probability matrices, i.e. $\mathbf{P}^{(i)}_t=\mathbf{P}^{(j)}_t$ for all $t\in[T]$, they then have the same distance matrix $\mathbf{D}^{(i)}=\mathbf{D}^{(j)}$.

Figures (12)

  • Figure 1: Dendrogram of dynamic networks of neurons for the removal of each single edge, obtained using the DNCMD algorithm. For each removed edge we have $11$ replicates.
  • Figure 2: Examples for dynamic networks.
  • Figure 3: Empirical estimate rates for $\frac{1}{\sqrt{T}}\min_{\mathbf{O}\in\mathcal{O}_r}\|\hat{\mathbf{M}}\mathbf{O}-\mathbf{M}\|_F$ as $T$ or $n$ changes. Left panel: vary $T\in\{20,30,40,50,70,100\}$ while fixing $n=100$. Right panel: vary $n\in\{50,100,200,400,800\}$ while fixing $T=10$. The results are averaged over $100$ independent Monte Carlo replicates.
  • Figure 4: Sample means and $90\%$ empirical confidence intervals of adjusted Rand index for DNCMD and the k-means algorithm based on $100$ independent Monte Carlo replicates. We tune $n\in\{20,30,40,80,120,200\}$. Other detailed settings can be found in Section \ref{['sec:performance']}.
  • Figure 5: Schematic of the process including the learning of an association and its extinction.
  • ...and 7 more figures

Theorems & Definitions (20)

  • Definition 1: Random dot product graph athreya2018statistical
  • Remark 1: Orthogonal nonidentifiability in RDPGs
  • Definition 2: CMDS embedding to dimension $r$
  • Remark 2: Orthogonal nonidentifiability in CMDS embeddings
  • Theorem 1
  • Theorem 2
  • Definition 3: Adjacency spectral embedding
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • ...and 10 more