  In this paper, we consider to what degree the structure of a linear system is
determined by the system's input/output behavior. The structure of a linear
system is a directed graph where the vertices represent the variables in the
system and an edge (x,y) exists if x directly influences y. In a number of
studies, researchers have attempted to identify such structures using
input/output data. Thus, our main aim is to consider to what degree the results
of such studies are valid. We begin by showing that in many cases, applying a
linear transformation to a system will change the system's graph. Furthermore,
we show that even the graph's components and their interactions are not
determined by input/output behavior. From these results, we conclude that
without further assumptions, very few aspects, if any, of a system's structure
are determined by its input/output relation. We consider a number of such
assumptions. First, we show that for a number of parameterizations, we can
characterize when two systems have the same structure. Second, in many
applications, we can use domain knowledge to exclude certain interactions. In
these cases, we can assume that a certain variable x does not influence another
variable y. We show that these assumptions cannot be sufficient to identify a
system's parameters using input/output data. We conclude that identifying a
system's structure from input/output data may not be possible given only
assumptions of the form x does not influence y.
