The modeling of diffusion processes on graphs is the basis for many network science and machine learning approaches. Entropic measures of network-based diffusion have recently been employed to investigate the reversibility of these processes and the diversity of the modeled systems. While results about their steady state are well-known, very few exact results about their time evolution exist. Here, we introduce the conditional entropy of heat diffusion in graphs. We demonstrate that this entropic measure satisfies the first and second laws of thermodynamics, thereby providing a physical interpretation of diffusion dynamics on networks. We outline a mathematical framework that contextualizes diffusion and conditional entropy within the theories of continuous-time Markov chains and information theory. Furthermore, we obtain explicit results for its evolution on complete, path, and circulant graphs, as well as a mean-field approximation for Erd\"os-R\'enyi graphs. We also obtain asymptotic results for general networks. Finally, we experimentally demonstrate several properties of conditional entropy for diffusion over random graphs, such as the Watts-Strogatz model.

翻译：图扩散过程的建模是许多网络科学与机器学习方法的基础。基于网络扩散的熵度量最近被用于研究这些过程的可逆性以及建模系统的多样性。尽管其稳态结果已广为人知，但关于其时间演化的精确结果却非常有限。本文引入了图热扩散的条件熵。我们证明该熵度量满足热力学第一与第二定律，从而为网络上的扩散动力学提供了物理解释。我们构建了一个数学框架，将扩散与条件熵置于连续时间马尔可夫链与信息论的语境中。此外，我们获得了该熵在完全图、路径图、循环图上的显式演化结果，以及Erd\"os-R\'enyi图的平均场近似解。我们还推导了一般网络的渐近结果。最后，我们通过实验验证了随机图（如Watts-Strogatz模型）上扩散条件熵的若干性质。