Gradient and Passive Circuit Structure in a Class of Non-linear Dynamics on a Graph

TL;DR

This paper explores a class of non-linear dynamics on graphs, revealing a gradient structure under detailed balance, linking passive circuits, Markov chains, and energy functions, with implications for convergence and network synthesis.

Contribution

It introduces a gradient formulation for non-linear graph dynamics under detailed balance, connecting passive network synthesis with Markov chains and energy functions.

Findings

Gradient form of dynamics under detailed balance

Equivalence of passivity and convexity of energy functions

Relationship between Markov chains and passive circuits

Abstract

We consider a class of non-linear dynamics on a graph that contains and generalizes various models from network systems and control and study convergence to uniform agreement states using gradient methods. In particular, under the assumption of detailed balance, we provide a method to formulate the governing ODE system in gradient descent form of sum-separable energy functions, which thus represent a class of Lyapunov functions; this class coincides with Csisz\'{a}r's information divergences. Our approach bases on a transformation of the original problem to a mass-preserving transport problem and it reflects a little-noticed general structure result for passive network synthesis obtained by B.D.O. Anderson and P.J. Moylan in 1975. The proposed gradient formulation extends known gradient results in dynamical systems obtained recently by M. Erbar and J. Maas in the context of porous medium equations. Furthermore, we exhibit a novel relationship between inhomogeneous Markov chains and passive non-linear circuits through gradient systems, and show that passivity of resistor elements is equivalent to strict convexity of sum-separable stored energy. Eventually, we discuss our results at the intersection of Markov chains and network systems under sinusoidal coupling.