  Network-theoretic tools contribute to understanding real-world system
dynamics, e.g., in wildlife conservation, epidemics, and power outages. Network
visualization helps illustrate structural heterogeneity; however, details about
heterogeneity are lost when summarizing networks with a single mean-style
measure. Researchers have indicated that a hierarchical system composed of
multiple metrics may be a more useful determinant of structure, but a formal
method for grouping metrics is still lacking. We develop a hierarchy using the
statistical concept of moments and systematically test the hypothesis that this
system of metrics is sufficient to explain the variation in processes that take
place on networks, using an ecological systems example. Results indicate that
the moments approach outperforms single summary metrics and accounts for a
majority of the variation in process outcomes. The hierarchical measurement
scheme is helpful for indicating when additional structural information is
needed to describe system process outcomes.
