Title: Tracking network dynamics: a survey of distances and similarity metrics,
Abstract: From longitudinal biomedical studies to social networks, graphs have emerged
as a powerful framework for describing evolving interactions between agents in
complex systems. In such studies, after pre-processing, the data can be
represented by a set of graphs, each representing a system's state at different
points in time. The analysis of the system's dynamics depends on the selection
of the appropriate analytical tools. After characterizing similarities between
states, a critical step lies in the choice of a distance between graphs capable
of reflecting such similarities. While the literature offers a number of
distances that one could a priori choose from, their properties have been
little investigated and no guidelines regarding the choice of such a distance
have yet been provided. In particular, most graph distances consider that the
nodes are exchangeable and do not take into account node identities. Accounting
for the alignment of the graphs enables us to enhance these distances'
sensitivity to perturbations in the network and detect important changes in
graph dynamics. Thus the selection of an adequate metric is a decisive --yet
delicate--practical matter.
In the spirit of Goldenberg, Zheng and Fienberg's seminal 2009 review, the
purpose of this article is to provide an overview of commonly-used graph
distances and an explicit characterization of the structural changes that they
are best able to capture. We use as a guiding thread to our discussion the
application of these distances to the analysis of both a longitudinal
microbiome dataset and a brain fMRI study. We show examples of using
permutation tests to detect the effect of covariates on the graphs'
variability. Synthetic examples provide intuition as to the qualities and
drawbacks of the different distances. Above all, we provide some guidance for
choosing one distance over another in certain types of applications.