Geodesics of Dynamic Graphs for Regime Change Detection
Authors: William Cappelletti, Étienne Voutaz, Pascal Frossard
Abstract
Traditional change point detection in dynamic networks assumes abrupt transitions between stationary states, overlooking scenarios of continuous evolution which arise in most real-world applications, such as social networks or physical systems. We address this gap by formally defining regimes as periods of coherent dynamics in temporal graphs, which we characterize as trajectories along geodesics in a suitably defined graph space. This original perspective allows us to define regime changes as significant drifts in dynamics, either toward new trajectories or with pace changes. We leverage graph regression methods to measure the cumulative distance of sequences of observed graphs from the estimated geodesics between their endpoints, in the relevant graph space, which we can combine with change point detection algorithms. We present experiments on dynamic networks, with changing trajectories and varying speeds, in which we outperform state of the art change point detection models. Then, we analyse mobility data during the Covid-19 pandemic, and show that our assumptions on regular network evolution lead to change points that are more aligned to external events compared to the outcomes of baseline methods. Our work is the first to model and detect changes between evolving regimes in graph space, providing a realistic and powerful tool for analyzing complex temporal graph data.
A central challenge in dynamic network analysis is to represent temporal evolution in a way that is both geometrically meaningful and statistically identifiable. One approach embeds a sequence of network snapshots as trajectories in a Euclidean space and relates these trajectories to node embeddings. In multilayer and unfolded spectral constructions, however, node embeddings and their underlying latent positions are identifiable only up to general linear transformations. Although this ambiguity preserves edge probabilities, it can distort geometry and invalidate distance based temporal comparisons at both the trajectory and node-levels. We develop Multiscale Euclidean Network Trajectories (MENT), a framework for multiscale temporal trajectories based on second-moment geometry. By imposing an isotropic normalization on the anchor latent positions, we reduce the relevant ambiguity to orthogonal transformations and prevent distortion of the second-moment geometry. In this canonical representation, we define a trace variation distance and mode-wise variation distances along orthogonal directions, and use multidimensional scaling to obtain low-dimensional trajectories of time points at both global and mode-wise levels. The resulting trajectories support interpretation and inference. They admit mode-wise decompositions, support attribution of global and mode-wise temporal changes to nodes, and enable change point detection through 1D trajectories. We prove consistency of the proposed unfolded spectral embedding and of the induced temporal trajectories. Experiments on two synthetic and two real dynamic networks illustrate stable and interpretable recovery of temporal structure and show strong performance against existing change point detection baselines.
Many complex systems can be modeled by temporal networks, whose organization often evolves through distinct structural phases. Detecting the change points that delimit these phases is both important and challenging. In this work, we extend the conditional entropy of heat diffusion from static graphs to temporal networks and study its properties. We provide an upper bound and explain how discrepancies from it arise from the presence of asymmetric temporal paths. Moreover, we show that this quantity is monotone in time, yielding an information-theoretic analog of the second law of thermodynamics for inhomogeneous diffusion on temporal networks. We then introduce a local version of conditional entropy, designed to probe diffusion over finite temporal windows, and show that it provides an informative signal for change-point detection in continuous-time temporal networks. We evaluate the proposed methodology on synthetic benchmarks, including comparative experiments with existing nonparametric baselines in the snapshot setting, and then apply it to a real-world temporal contact network from a French primary school. Finally, we show how to use detected change points to perform community detection on targeted sub-intervals, improving the quality and interpretability of the clustering results.
Inspired by graph-based methodologies, we introduce a novel graph-spanning algorithm designed to identify changes in both offline and online data across low to high dimensions. This versatile approach is applicable to Euclidean and graph-structured data with unknown distributions, while maintaining control over error probabilities. Theoretically, we demonstrate that the algorithm achieves high detection power when the magnitude of the change surpasses the lower bound of the minimax separation rate, which scales on the order of nd. Our method outperforms other techniques in terms of accuracy for both Gaussian and non-Gaussian data. Notably, it maintains strong detection power even with small observation windows, making it particularly effective for online environments where timely and precise change detection is critical.
Katerina Papagiannouli, Yang-wen Sun, Vladimir Spokoiny