Spectral embedding of inhomogeneous Poisson processes on multiplex networks

Technical Summary: Spectral Embedding of Inhomogeneous Poisson Processes on Multiplex Networks

Problem Statement
Real-world network data often exhibit two critical complexities: heterogeneity in connection types (multiplexity) and evolution in continuous time. While static multiplex networks and discrete-time dynamic networks have been extensively studied, there is a lack of rigorous statistical frameworks for continuous-time multiplex networks where edge interactions are modeled as point processes. Existing continuous-time approaches often separate from the Random Dot Product Graph (RDPG) framework, which provides powerful theoretical guarantees for latent position estimation. This work addresses the gap by proposing a model for multiplex network data observed in continuous time, where edge intensities are driven by low-dimensional latent positions.

Methodology
The authors introduce the Multiplex Inhomogeneous Poisson Process Dot-Product Graph (MIPP-DPG). In this model:

1. Data Structure: Observations are quadruples $(i_k, j_k, \ell_k, t_k)$ representing a directed interaction from node $i$ to node $j$ in layer $\ell$ at time $t$.
2. Latent Positions: Each node $i$ possesses two types of latent positions:
   • A dynamic, layer-agnostic position $X_i(t) \in \mathbb{R}^d$ that evolves over time but is shared across all layers.
   • A static, layer-dependent position $Y_{\ell i} \in \mathbb{R}^d$ specific to each layer $\ell$.
3. Intensity Function: The intensity $\lambda_{\ell ij}(t)$ of the inhomogeneous Poisson process governing the edge $(i, j)$ in layer $\ell$ is defined as the inner product of these latent positions:
   $$\lambda_{\ell ij}(t) = X_i(t)^\top Y_{\ell j}$$
4. Estimation Procedure: To estimate these latent positions, the authors propose a discretization approach:
   • The continuous time interval $[0, 1]$ is divided into $M$ bins.
   • A histogram estimator is constructed for the edge intensities within each bin, resulting in a block matrix $\hat{\Lambda}$.
   • The Doubly Unfolded Adjacency Spectral Embedding (DUASE) is applied to $\hat{\Lambda}$. This involves stacking the adjacency matrices of all layers and time bins into a large matrix and performing a singular value decomposition (SVD) to recover estimates $\hat{X}$ and $\hat{Y}$.

Key Theoretical Contributions
The paper establishes rigorous asymptotic properties for the DUASE estimator applied to the MIPP-DPG as the number of nodes $N$, the number of time bins $M$, and the number of layers $L$ grow.

1. Consistency (Theorem 1): The authors prove row-wise two-to-infinity norm consistency for the estimated latent positions. Specifically, they show that the estimated positions converge to the true positions (up to a linear transformation) with an error bound that decays as $N \to \infty$.

   • The bound accounts for both the variance (stochastic error from the Poisson process) and the bias (introduced by time discretization).
   • Crucially, the theory demonstrates that even though the data is discretized into bins, the estimator asymptotically recovers the continuous-time nature of the latent positions, provided the number of bins $M$ grows at a rate slower than $N$ (specifically $M = o(N^{1/2}/\log^4 N)$).
   • Two bias assumptions are considered: (i) Rotation-free Lipschitz continuity and (ii) Subspace Lipschitz continuity, the latter being a weaker, rotation-invariant condition on the dynamics of the latent space.

2. Asymptotic Normality (Theorem 2): The paper establishes a Central Limit Theorem (CLT) for the studentized vector of differences between the estimated and true latent positions. This result provides a theoretical foundation for constructing confidence intervals and performing hypothesis testing on the latent positions. The CLT holds under mild conditions regarding the rank and density of the intensity matrix.

Empirical Results
The proposed method is validated through simulations and a real-world application:

• Simulations: The authors simulate block-structured MIPP-DPGs. Results confirm that the DUASE estimator successfully recovers both the dynamic trajectories $X_i(t)$ and the static layer-specific positions $Y_{\ell j}$. The simulations also demonstrate that the method remains effective even when the strict Lipschitz continuity assumptions are violated (e.g., in the presence of discontinuities), provided the network size $N$ is sufficiently large.
• Real-World Application: The model is applied to a global air transportation network (OpenSky Network, April 2021), treating different aircraft models as layers. The spectral embedding successfully recovers latent structures that align with geographic regions (continents) while also identifying functional clusters (e.g., intercontinental hubs vs. regional airports) that transcend simple geography.

Significance and Claims
The authors claim that this work provides the first rigorous theoretical justification for applying spectral embedding methods to continuous-time multiplex networks modeled as inhomogeneous Poisson processes.

• Bridging Frameworks: The work connects the theory of continuous-time point processes with the established RDPG framework, allowing for the use of spectral methods in continuous time.
• Discretization Justification: A key contribution is the proof that simple histogram-based discretization, followed by DUASE, is sufficient to recover continuous-time latent structures asymptotically.
• Downstream Utility: The asymptotic normality result is highlighted as particularly significant for downstream tasks, such as community detection, as it theoretically supports the use of Gaussian mixture models for clustering the embedded positions.

The paper concludes that while the method relies on Lipschitz continuity for theoretical guarantees, empirical evidence suggests robustness to violations of these assumptions. Future work is suggested to explore scenarios where the network structure itself is unknown and to investigate more complex point process models like self-exciting processes.