Conditional Entropy of Heat Diffusion on Temporal Networks

Diffusion-based information-theoretic approaches provide new theoretical and practical tools to study complex networks. So far, they have not been generalized to temporal networks. In this work, we show that common entropic measures based on modeling diffusion on graphs, such as the entropy rate and the spectral entropy, do not generalize straightforwardly to temporal networks due to the process's non-stationarity and temporal asymmetries. Instead, we propose the conditional entropy of heat diffusion as an entropic measure for continuous-time temporal networks and study its properties. 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 provide an upper bound and suggest a lower bound on its evolution and explain how discrepancies from it arise due to asymmetric temporal paths. 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. Finally, we show how to use detected change points to guide community detection on targeted sub-intervals, improving the quality and interpretability of the clustering results.