From longitudinal biomedical studies to social networks, graphs have emerged as essential objects for describing evolving interactions between agents in complex systems. In such studies, after pre-processing, the data are encoded by a set of graphs, each representing a system’s state at a different point in time or space. The analysis of the system’s dynamics depends on the selection of the appropriate analytical tools. In particular, after specifying properties 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 to 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—ignoring node “identities.” Alignment of the graphs according to identified nodes 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 et al.’s seminal 2009 review [Found. Trends Mach. Learn. 2 (2010) 129–233], this article provides an overview of commonly-used graph distances and an explicit characterization of the structural changes that they are best able to capture. We show how these choices affect real-life situations, and we use these distances to analyze both a longitudinal microbiome dataset and a brain fMRI study. One contribution of the present study is a coordinated suite of data analytic techniques, displays and statistical tests using “metagraphs”: a graph of graphs based on a chosen metric. Permutation tests can uncover the effects of covariates on the graphs’ variability. Furthermore, synthetic examples provide intuition as to the qualities and drawbacks of the different distances. Above all, we provide some guidance on choosing one distance over another in different contexts. Finally, we extend the scope of our analyses from temporal to spatial dynamics and apply these different distances to a network created from worldwide recipes.