Tracking network dynamics: A survey using graph distances

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 graph represents 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 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 [21], 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 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. Finally 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.Finally, extending the scope of our analysis from temporal to spatial dynamics, we show an application of these different distances to a network created from worldwide recipes.

Metrics for Graph Comparison: A Practitioner’s Guide

“…where · is a norm we are free to choose. 7 Let us elucidate a specific example of such a distance; in particular, we will show how the edit distance conforms to this description. Let δ(v, w) be defined as…”

“…6 When we say "distance" we implicitly assume that smaller values imply greater similarity; however, we can also carry out this approach with a similarity score, in which larger values imply greater similarity. 7 We could use metrics, or even similarity functions here, although that may cause the function d to lose some desirable properties.…”

“…Existing surveys of graph distances are limited to observational datasets [7]. While authors try to choose datasets that are exemplars of certain classes of networks (e.g.…”

Comparing complex networks: in defence of the simple

“…, that quantify the (dis)similarity between two networks have been been studied in several areas such as chemistry, protein structures, social networks up to neuroscience, among others [1][2][3][4]. Without an h uniqueness, different approaches have been proposed including graph edit operations, distances based on divergences, spectral parameters, kernels, or different combinations of the previous [5][6][7][8][9][10][11].…”

“…Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. however, often integrate over local neighborhoods, which renders these approaches less sensitive to small or local perturbations [7].…”