Duality and Nonlinear Graph Laplacians

TL;DR

This paper introduces an iterative algorithm for solving nonlinear Laplacian systems efficiently, extending linear Laplacian solvers to nonlinear cases with promising performance on sparse graphs.

Contribution

It presents a novel nonlinear extension of Laplacian solvers, including a new energy function and duality analysis, enabling faster solutions for nonlinear systems.

Findings

Algorithm solves nonlinear Laplacian systems in nearly quadratic time for sparse graphs.

Performance surpasses standard linear equation solvers in certain cases.

Provides theoretical tools for extending spectral analysis to nonlinear contexts.

Abstract

We present an iterative algorithm for solving a class of \\nonlinear Laplacian system of equations in $\tilde{O}(k^2m \log(kn/\epsilon))$ iterations, where $k$ is a measure of nonlinearity, $n$ is the number of variables, $m$ is the number of nonzero entries in the graph Laplacian $L$, $\epsilon$ is the solution accuracy and $\tilde{O}()$ neglects (non-leading) logarithmic terms. This algorithm is a natural nonlinear extension of the one by of Kelner et. al., which solves a linear Laplacian system of equations in nearly linear time. Unlike the linear case, in the nonlinear case each iteration takes $\tilde{O}(n)$ time so the total running time is $\tilde{O}(k^2mn \log(kn/\epsilon))$. For sparse graphs where $m = O(n)$ and fixed $k$ this nonlinear algorithm is $\tilde{O}(n^2 \log(n/\epsilon))$ which is slightly faster than standard methods for solving linear equations, which require approximately $O(n^{2.38})$ time. Our analysis relies on the construction of a nonlinear "energy function" and a nonlinear extension of the duality analysis of Kelner et. al to the nonlinear case without any explicit references to spectral analysis or electrical flows. These new insights and results provide tools for more general extensions to spectral theory and nonlinear applications.