Evolution of Conditional Entropy for Diffusion Dynamics on Graphs

2025-10-22

AI Summary

This work addresses the lack of rigorous theoretical characterization of the temporal evolution of conditional entropy in diffusion processes on graphs. We develop an analytical framework integrating continuous-time Markov chains with information theory. First, we formally define the conditional entropy of heat diffusion on graphs and rigorously prove that its evolution obeys both the first law of thermodynamics (an energy-conserving entropy balance) and the second law (monotonic non-decrease). We derive closed-form analytical solutions for complete, path, cycle, and Erdős–Rényi random graphs; numerically validate our results on Watts–Strogatz small-world networks; and establish an asymptotic theory for conditional entropy evolution on general graphs. These contributions provide a unified thermodynamic interpretation of network diffusion dynamics and fill a fundamental theoretical gap concerning entropy evolution on discrete structures.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

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.

Problem

• Modeling diffusion processes on graphs for network science applications
• Investigating thermodynamic laws and reversibility of network diffusion dynamics
• Developing mathematical framework for conditional entropy evolution in graphs

Innovation

• Introduces conditional entropy for heat diffusion on graphs
• Develops mathematical framework linking diffusion and information theory
• Obtains explicit evolution results for various graph types