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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.19441 |
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| _version_ | 1866914107847868416 |
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| author | Koovely, Samuel Bovet, Alexandre |
| author_facet | Koovely, Samuel Bovet, Alexandre |
| contents | 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 Erdös-Rényi 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19441 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Evolution of Conditional Entropy for Diffusion Dynamics on Graphs Koovely, Samuel Bovet, Alexandre Dynamical Systems Information Theory Probability Data Analysis, Statistics and Probability 37A35 (Primary), 60J27, 94C15, 80M60 (Secondary) 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 Erdös-Rényi 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. |
| title | Evolution of Conditional Entropy for Diffusion Dynamics on Graphs |
| topic | Dynamical Systems Information Theory Probability Data Analysis, Statistics and Probability 37A35 (Primary), 60J27, 94C15, 80M60 (Secondary) |
| url | https://arxiv.org/abs/2510.19441 |