Intensity Profile Projection: A Framework for Continuous-Time Representation Learning for Dynamic Networks

TL;DR

This work introduces Intensity Profile Projection (IPP), a framework for learning continuous-time node trajectories in dynamic networks by estimating edge intensities and projecting intensity profiles onto a low-rank subspace. It unifies graph- and temporal-domain ideas to produce time-evolving embeddings with structural and temporal coherence, supported by non-asymptotic error bounds that quantify bias-variance trade-offs in histogram-based intensity estimation. Theoretical results hinge on Lipschitz intensity functions, spectral gaps, and coherence, and practical implementations leverage sparse SVD and quadrature for scalability. Empirically, IPP recovers coherent dynamic patterns in simulated bifurcating blocks and real school-contact data, offering interpretable trajectories that reflect timetables and class structures while enabling on-demand online inferences. The framework lays groundwork for downstream analyses (e.g., clustering, trend detection, and topology dynamics) and invites extensions to branching points and polarisation measures, with attention to bandwidth and dimension selection as practical trade-offs.

Abstract

We present a new representation learning framework, Intensity Profile Projection, for continuous-time dynamic network data. Given triples $(i,j,t)$, each representing a time-stamped ($t$) interaction between two entities ($i,j$), our procedure returns a continuous-time trajectory for each node, representing its behaviour over time. The framework consists of three stages: estimating pairwise intensity functions, e.g. via kernel smoothing; learning a projection which minimises a notion of intensity reconstruction error; and constructing evolving node representations via the learned projection. The trajectories satisfy two properties, known as structural and temporal coherence, which we see as fundamental for reliable inference. Moreoever, we develop estimation theory providing tight control on the error of any estimated trajectory, indicating that the representations could even be used in quite noise-sensitive follow-on analyses. The theory also elucidates the role of smoothing as a bias-variance trade-off, and shows how we can reduce the level of smoothing as the signal-to-noise ratio increases on account of the algorithm `borrowing strength' across the network.

Figures (10)

• Figure 1: A bias-variance trade-off. We simulate a network with common intensities $\lambda_{ij}(t) = 0.7 \times \{ 2 + \cos(t) \}$ for all $i,j$, and apply Intensity Profile Projection with a histogram intensity estimator with 5, 20, and 200 bins. In the 'bias' plots, the gray lines shows an estimand $X_i(t)$, while the blue lines shows its histogram approximation. The discrepancy between the gray line and the blue line corresponds the bias of the Intensity Profile Projection estimator. In the 'variance' plots, the blues lines are as in the 'bias' plots and the orange line show the estimate obtains using Intensity Profile Projection into one dimension. The discrepancy between the blue line and the orange line corresponds the variance of the Intensity Profile Projection estimator.
• Figure 2: One-dimensional PCA visualisation of the two-dimensional node representations, obtained using a collection of methods, for a network simulated from a bifurcating block model. Colours correspond to the community membership of the node.
• Figure 3: One-dimensional PCA visualisation of the 30-dimensional node representations for pairs of classes in the same year group. The solid lines show the average trajectory for each class, and the dashed line show one standard deviation above and below.
• Figure 4: Two-dimensional t-SNE visualisation of the 30-dimensional node representations of all pupils and teachers evaluated at 9:30am on Day 1, and 9:30am, 12:30pm and 3:30pm on Day 2.
• Figure 5: The first two dimensions of the spherical coordinates of the coordinates $\widehat{X}_i(t)$ using the histogram intensity estimator for times corresponding to the morning, lunchtime and afternoon across both days. The colours indicate classes with black points representing teachers.

Theorems & Definitions (20)

• Lemma 1
• Theorem 1
• Corollary 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5: Corollary 3.12 of bandeira2016sharp
• Lemma 6: Weyl's inequality
• Lemma 7
• Proposition 1