  Many complex ecosystems, such as those formed by multiple microbial taxa,
involve intricate interactions amongst various sub-communities. The most basic
relationships are frequently modeled as co-occurrence networks in which the
nodes represent the various players in the community and the weighted edges
encode levels of interaction. In this setting, the composition of a community
may be viewed as a probability distribution on the nodes of the network. This
paper develops methods for modeling the organization of such data, as well as
their Euclidean counterparts, across spatial scales. Using the notion of
diffusion distance, we introduce diffusion Frechet functions and diffusion
Frechet vectors associated with probability distributions on Euclidean space
and the vertex set of a weighted network, respectively. We prove that these
functional statistics are stable with respect to the Wasserstein distance
between probability measures, thus yielding robust descriptors of their shapes.
We apply the methodology to investigate bacterial communities in the human gut,
seeking to characterize divergence from intestinal homeostasis in patients with
Clostridium difficile infection (CDI) and the effects of fecal microbiota
transplantation, a treatment used in CDI patients that has proven to be
significantly more effective than traditional treatment with antibiotics. The
proposed method proves useful in deriving a biomarker that might help elucidate
the mechanisms that drive these processes.
