  Statisticians and quantitative neuroscientists have actively promoted the use
of independence relationships for investigating brain networks, genomic
networks, and other measurement technologies. Estimation of these graphs
depends on two steps. First is a feature extraction by summarizing measurements
within a parcellation, regional or set definition to create nodes. Secondly,
these summaries are then used to create a graph representing relationships of
interest. In this manuscript we study the impact of dimension reduction on
graphs that describe different notions of relations among a set of random
variables. We are particularly interested in undirected graphs that capture the
random variables' independence and conditional independence relations. A
dimension reduction procedure can be any mapping from high dimensional spaces
to low dimensional spaces. We exploit a general framework for modeling the raw
data and advocate that in estimating the undirected graphs, any acceptable
dimension reduction procedure should be a graph-homotopic mapping, i.e., the
graphical structure of the data after dimension reduction should inherit the
main characteristics of the graphical structure of the raw data. We show that,
in terms of inferring undirected graphs that characterize the conditional
independence relations among random variables, many dimension reduction
procedures, such as the mean, median, or principal components, cannot be
theoretically guaranteed to be a graph-homotopic mapping. The implications of
this work are broad. In the most charitable setting for researchers, where the
correct node definition is known, graphical relationships can be contaminated
merely via the dimension reduction. The manuscript ends with a concrete
example, characterizing a subset of graphical structures such that the
dimension reduction procedure using the principal components can be a
graph-homotopic mapping.
