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Spectral embedding of inhomogeneous Poisson processes on multiplex networks

Joshua Corneck, Edward A. K. Cohen, Francesco Sanna Passino

TL;DR

This work proposes a model for multiplex network data observed in continuous-time and establishes two-to-infinity norm consistency and asymptotic normality for spectral-embedding-based estimation of the model parameters as both network size and time resolution increase.

Abstract

In many real-world networks, data on the edges evolve in continuous time, naturally motivating representations based on point processes. Heterogeneity in edge types further gives rise to multiplex network point processes. In this work, we propose a model for multiplex network data observed in continuous-time. We establish two-to-infinity norm consistency and asymptotic normality for spectral-embedding-based estimation of the model parameters as both network size and time resolution increase. Drawing inspiration from random dot product graph models, each edge intensity is expressed as the inner product of two low-dimensional latent positions: one dynamic and layer-agnostic, the other static and layer-dependent. These latent positions constitute the primary objects of inference, which is conducted via spectral embedding methods. Our theoretical results are established under a histogram estimator of the network intensities and provide justification for applying a doubly unfolded adjacency spectral embedding method for estimation. Simulations and real-data analyses demonstrate the effectiveness of the proposed model and inference procedure.

Spectral embedding of inhomogeneous Poisson processes on multiplex networks

TL;DR

This work proposes a model for multiplex network data observed in continuous-time and establishes two-to-infinity norm consistency and asymptotic normality for spectral-embedding-based estimation of the model parameters as both network size and time resolution increase.

Abstract

In many real-world networks, data on the edges evolve in continuous time, naturally motivating representations based on point processes. Heterogeneity in edge types further gives rise to multiplex network point processes. In this work, we propose a model for multiplex network data observed in continuous-time. We establish two-to-infinity norm consistency and asymptotic normality for spectral-embedding-based estimation of the model parameters as both network size and time resolution increase. Drawing inspiration from random dot product graph models, each edge intensity is expressed as the inner product of two low-dimensional latent positions: one dynamic and layer-agnostic, the other static and layer-dependent. These latent positions constitute the primary objects of inference, which is conducted via spectral embedding methods. Our theoretical results are established under a histogram estimator of the network intensities and provide justification for applying a doubly unfolded adjacency spectral embedding method for estimation. Simulations and real-data analyses demonstrate the effectiveness of the proposed model and inference procedure.
Paper Structure (39 sections, 20 theorems, 142 equations, 3 figures)

This paper contains 39 sections, 20 theorems, 142 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and related literature
  3. Matrix and norm notation
  4. Probabilistic asymptotic notation
  5. Multiplex inhomogeneous Poisson process random dot product graphs
  6. Defining the MIPP-DPG
  7. A spectral estimation procedure for the MIPP-DPG
  8. Theoretical results
  9. Asymptotic consistency
  10. Central limit theorem
  11. Simulation studies
  12. Block structure MIPP-DPG
  13. Discontinuous MIPP-DPG
  14. Application to a global air transportation network
  15. Conclusion and discussion
  16. ...and 24 more sections

Key Result

Proposition 1

Under Assumptions ass:bound_and_int and ass:moment_stab, as $N \to \infty$, we have for all $i \in [d]$, and $\sigma_{d+1}(\bar{\boldsymbol{\Lambda}}) = 0$. Furthermore, we have that $\kappa(\tilde{\boldsymbol{X}}),\ \kappa(\boldsymbol{Y}),\ \mu(\tilde{\boldsymbol{X}}),\ \mu(\boldsymbol{Y}) = \mathcal{O}(1)$, where $\kappa$ and $\mu$ denote the condition number and incoherence para

Figures (3)

  • Figure 1: Estimated and true latent positions for the simulation study described in Section \ref{['sec:sim1']}, for different values of the number of nodes $N$ and number of bins $M$.
  • Figure 2: Estimated and true latent positions for the simulation study described in Section \ref{['sec:sim2']}, for different values of the number of nodes $N$ and number of bins $M=50$.
  • Figure 3: t-SNE embeddings of the inferred airport latent vectors, shown (a) after stacking the dynamic position across time and (b) for two representative layers. Each point represents an airport in the 2D t-SNE space. Points are coloured by continent (left column) and by inferred cluster membership (right column).

Theorems & Definitions (21)

  • Definition 1: Doubly unfolded adjacency spectral embedding, DUASE; baum2025
  • Proposition 1: Behaviour of latent position matrices
  • Theorem 1: Two-to-infinity norm bound
  • Theorem 2: Studentised asymptotic normality
  • Lemma 1: Poisson tail bound
  • Theorem 3: Wedin's $\sin\Theta$ Theorem; e.g. Theorem 2.9 in chen_2021
  • Theorem 4: Matrix Bernstein; e.g. Theorem 1.6.2 in tropp_2015
  • Theorem 5: Subexponential Bernstein; e.g. Corollary 2.9.2 in vershynin_2018
  • Proposition 2: Poisson tail bound
  • Proposition 3
  • ...and 11 more