  Complex networks encountered in biology are often characterized by
significant structural diversity. Whether it be differences in the
three-dimensional structure of allosteric proteins, or the variation among the
micro-scale structures of organisms' cerebral vasculature systems, identifying
relationships between structure and function often poses a difficult challenge.
Here we showcase an approach to characterizing structure-function relationships
in complex networks applied in the context of flow networks tuned to perform
specific functions. Using persistent homology, we analyze flow networks tuned
to perform complex multifunctional tasks, answering the question of how local
changes in the network structure coordinate to create functionality at at the
scale of the entire network. We find that the response of such networks encodes
hidden topological features - sectors of uniform pressure - that are not
apparent in the underlying network architectures, Regardless of differences in
local connectivity, these features provide a universal topological description
for all networks that perform these types of functions. We show that these
features correlate strongly with the tuned response, providing a clear
topological relationship between structure and function and structural insight
into the limits of multifunctionality.
