  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.
