Structured Systems of Nonlinear Equations

TL;DR

This paper investigates properties of structured nonlinear systems of equations, focusing on robustness and how it relates to the system's connection graph, introducing concepts like bottlenecks and methods to enhance robustness.

Contribution

It introduces a graph-based framework for analyzing robustness in structured nonlinear systems and presents a numerical method to identify and eliminate minimax bottlenecks.

Findings

Robust solutions depend on the structure of the connection graph.

Existence of a unique minimax bottleneck causes robustness failure.

Adding edges to the graph can remove the bottleneck and improve robustness.

Abstract

In a "structured system" of equations, each equation depends on a specified subset of the variables. In this article, we explore properties common to "almost every" system with a fixed structure and how the properties can be read from the corresponding connection graph. A solution $p$ of a system $F(p)=c$ is called robust if it persists despite small changes in $F$. We establish methods for determining robustness that depends on the structure, as expressed in the properties of the corresponding directed graph of the structured system. The keys to understanding linear and nonlinear structured systems are subsets of variables that we call forward and backward bottlenecks. In particular, when robustness fails in a structured system, it is due to the existence of a unique "backward bottleneck", that we call a "minimax bottleneck". We present a numerical method for locating the minimax bottleneck. We show how to remove it by adding edges to the graph.