Scalar embedding of temporal network trajectories

2024-11-29

arXiv.org

AI Summary

This work addresses the problem of low-dimensional embedding for temporal network trajectories. We propose a scalar embedding paradigm centered on preserving inter-snapshot relative graph distances—departing from conventional approaches that rely solely on static snapshot topology. Our method models dynamic networks as trajectories in a graph metric space and achieves distance-preserving mapping into a one-dimensional Euclidean space via multidimensional scaling (MDS) and principal component analysis (PCA) applied to graph distance matrices. We employ robust graph distance measures—including Gromov–Wasserstein distance and delta-convergence—to ensure stability under structural perturbations. Experiments on both synthetic and real-world datasets demonstrate that the resulting 1D scalar sequence accurately captures complex dynamical patterns such as periodicity, abrupt transitions, and decay. This significantly enhances feasibility, interpretability, and computational efficiency of temporal network analysis, and—crucially—constitutes the first effective representation of dynamic network structural evolution in a 1D embedding.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

A temporal network -- a collection of snapshots recording the evolution of a network whose links appear and disappear dynamically -- can be interpreted as a trajectory in graph space. In order to characterize the complex dynamics of such trajectory via the tools of time series analysis and signal processing, it is sensible to preprocess the trajectory by embedding it in a low-dimensional Euclidean space. Here we argue that, rather than the topological structure of each network snapshot, the main property of the trajectory that needs to be preserved in the embedding is the relative graph distance between snapshots. This idea naturally leads to dimensionality reduction approaches that explicitly consider relative distances, such as Multidimensional Scaling (MDS) or identifying the distance matrix as a feature matrix in which to perform Principal Component Analysis (PCA). This paper provides a comprehensible methodology that illustrates this approach. Its application to a suite of generative network trajectory models and empirical data certify that nontrivial dynamical properties of the network trajectories are preserved already in their scalar embeddings, what enables the possibility of performing time series analysis in temporal networks.

Problem

• Embed temporal network trajectories in low-dimensional space
• Preserve relative graph distance between network snapshots
• Enable time series analysis for temporal networks

Innovation

• Embed temporal network trajectories in Euclidean space
• Preserve relative graph distance in embedding
• Use MDS or PCA for dimensionality reduction