The Cactus Criterion: When Nonlinear Hodge Theory Reduces to Linear on Graphs

TL;DR

This paper introduces a nonlinear selector on graphs based on a specific edge potential, showing it reduces to the linear Hodge projector precisely on cactus graphs, with explicit computations and a Newton method.

Contribution

It characterizes when the nonlinear Hodge-like operator coincides with the linear one on graphs, identifying cactus graphs as the key condition.

Findings

The nonlinear selector is real analytic and matches the Hodge projector to first order.

For cactus graphs, the nonlinear selector equals the linear Hodge projector on all of C^1(G).

A Newton method for computing the nonlinear selector is provided.

Abstract

Let $G$ be a finite connected simple graph with a chosen orientation of its edges. For the edge potential $\psi(t)=\cosh t-1,$ we minimize $\sum_{e\in E^\to}\psi(z_e)$ over each affine class $\omega+dC^0(G)\subset C^1(G)$. The minimizer is the unique representative satisfying the nonlinear coclosed equation $\delta\sinh z=0,$ and hence defines a nonlinear selector $\Picc:C^1(G)\to C^1(G).$ We show that $\Picc$ is real analytic, identify its image as $\imop \Picc=\Mcc=\operatorname{arsinh}(\ker\delta),$ and compute its differential as a weighted Hodge projector. In particular, $\Picc$ agrees with the ordinary Hodge projector $\PiH$ to first order at the origin, and the first nonlinear correction is cubic. Our main global theorem is a graph-theoretic criterion: for every admissible edge potential -- even, $C^2$, strictly convex, and non-quadratic -- the associated nonlinear selector coincides with $\PiH$ on all of $C^1(G)$ if and only if $G$ is a cactus graph. Finally, we work out the two-triangle graph, the smallest connected simple obstruction, and record a self-concordant Newton method for computing $\Picc$.