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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 Maxwell’s 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 network’s 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. 

At the core of understanding dynamical systems is the ability to maintain and control the systems behavior that includes notions of robustness, heterogeneity, and/or regime-shift detection. Recently, to explore such functional properties, a convenient representation has been to model such dynamical systems as a weighted graph consisting of a finite, but very large number of interacting agents. This said, there exists very limited relevant statistical theory that is able cope with real-life data, i.e., how does perform analysis and/or statistics over a “family” of networks as opposed to a specific network or network-to-network variation. Here, we are interested in the analysis of network families whereby each network represents a “point” on an underlying statistical manifold. To do so, we explore the Riemannian structure of the tensor manifold developed by Pennec previously applied to Diffusion Tensor Imaging (DTI) towards the problem of network analysis. In particular, while this note focuses on Pennec definition of “geodesics” amongst a family of networks, we show how it lays the foundation for future work for developing measures of network robustness for regime-shift detection. We conclude with experiments highlighting the proposed distance on synthetic networks and an application towards biological (stem-cell) systems. 

This paper studies a distributed reinforcement learning problem in which a network of multiple agents aim to cooperatively maximize the globally averaged return through communication with only local neighbors. An asynchronous multi-agent actor-critic algorithm is proposed for possibly unidirectional communication relationships depicted by a directed graph. Each agent independently updates its variables at “event times” determined by its own clock. It is not assumed that the agents’ clocks are synchronized or that the event times are evenly spaced. It is shown that the algorithm can solve the problem for any strongly connected graph in the presence of communication and computation delays.