Duality and nonlinear graph Laplacians

Abstract

In this paper we show that a recent nearly linear time algorithm for solving a system of equations arising from a graph Laplacian can be extended to a large class of nonlinear systems of equations, based on a nonlinear generalization of the graph Laplacian. This result follows from a nonlinear generalization of the notable duality between graph Laplacians and electrical flows. Beyond providing a fast algorithm for a class of nonlinear sets of equations, our work highlights the robustness of spectral graph theory and suggests new directions for research in spectral graph theory.Specifically, we present an iterative algorithm for solving a class of nonlinear system of equations arising from a nonlinear generalization of the graph Laplacian in O˜(k2mlog⁡(kn/ϵ)) iterations, where k is a measure of nonlinearity, n is the number of variables, m is the number of nonzero entries in the generalized graph Laplacian L, ϵ is the solution accuracy and O˜() neglects (non-leading) logarithmic terms. This algorithm is a natural nonlinear extension of the one in Kelner et al. (2013), which solves a linear Laplacian system of equations in nearly linear time. Unlike the linear case, in the nonlinear case each iteration takes O˜(n) time so the total running time is O˜(k2mnlog⁡(kn/ϵ)). For sparse graphs with m=O(n) and fixed k this nonlinear algorithm is O˜(n2log⁡(n/ϵ)) which is slightly faster than standard methods for solving linear equations, which require approximately O(n2.38) time.