  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.
