Evolution of Conditional Entropy for Diffusion Dynamics on Graphs

TL;DR

This paper introduces the concept of conditional entropy for diffusion processes on graphs, linking it to thermodynamics and providing explicit evolution results for various graph types, enhancing understanding of network diffusion dynamics.

Contribution

It presents a novel entropic measure for graph diffusion, establishes its thermodynamic laws, and derives explicit and asymptotic evolution results for different graph classes.

Findings

Conditional entropy satisfies thermodynamic laws.

Explicit evolution formulas for specific graph types.

Asymptotic behavior characterized for general networks.

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 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.