Maxwells Demon: Controlling Entropy via Discrete Ricci Flow Over Networks

TL;DR

This paper introduces a novel method using discrete Ricci flow and feedback control to manipulate network entropy, aiming to enhance network robustness and resilience by leveraging geometric properties like Ricci curvature.

Contribution

It adapts Ricci flow control techniques to networks, linking geometry and entropy, and demonstrates stability analysis using Lyapunov methods for the first time.

Findings

Discrete Ricci flow can control network entropy effectively.

Lyapunov stability of Ricci flow control is established.

Preliminary results suggest potential for network robustness improvements.

Abstract

In this work, we propose to utilize discrete graph Ricci flow to alter network entropy through feedback control. Given such feedback input can reverse entropic changes, we adapt the moniker of Maxwells Demon to motivate our approach. In particular, it has been recently shown that Ricci curvature from geometry is intrinsically connected to Boltzmann entropy as well as functional robustness of networks or the ability to maintain functionality in the presence of random fluctuations. From this, the discrete Ricci flow provides a natural avenue to rewire a particular networks underlying geometry to improve throughout and resilience. Due to the real-world setting for which one may be interested in imposing nonlinear constraints amongst particular agents to understand the network dynamic evolution, controlling discrete Ricci flow may be necessary (e.g., we may seek to understand the entropic dynamics and curvature flow between two networks as opposed to solely curvature shrinkage). In turn, this can be formulated as a natural control problem for which we employ feedback control towards discrete Ricci-based flow and show that under certain discretization, namely Ollivier-Ricci curvature, one can show stability via Lyapunov analysis. We conclude with preliminary results with remarks on potential applications that will be a subject of future work.