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, the data typically consists of a set of graphs representing a system’s state at different points in time or space. The analysis of the system’s dynamics depends on the selection of the appropriate tools. In particular, after specifying properties characterizing similarities between states, a critical step lies in the choice of a distance 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. However, these distances’ sensitivity to perturbations in the network’s structure and their ability to identify important changes are crucial to the analysis, making the selection of an adequate metric 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. To see how this translates in real-life situations, we use as a guiding thread to our discussion the application of these distances to the analysis a longitudinal microbiome study — as well as on synthetic examples. Having unveiled some of traditional distances’ shortcomings, we also suggest alternative similarity metrics and highlight their relative advantages in specific analysis scenarios. Above all, we provide some guidance for choosing one distance over another in certain types of applications. Finally, we show an application of these different distances to a network created from worldwide recipes. Tracking Network Dynamics: a review of distances and similarity metrics

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