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To quantify the fundamental evolution of time-varying networks, and detectabnormal behavior, one needs a notion of temporal difference that capturessignificant organizational changes between two successive instants. In thiswork, we propose a family of distances that can be tuned to quantify structuralchanges occurring on a graph at different scales: from the local scale formedby the neighbors of each vertex, to the largest scale that quantifies theconnections between clusters, or communities. Our approach results in thedefinition of a true distance, and not merely a notion of similarity. Wepropose fast (linear in the number of edges) randomized algorithms that canquickly compute an approximation to the graph metric. The third contributioninvolves a fast algorithm to increase the robustness of a network by optimallydecreasing the Kirchhoff index. Finally, we conduct several experiments onsynthetic graphs and real networks, and we demonstrate that we can detectconfigurational changes that are directly related to the hidden variablesgoverning the evolution of dynamic networks.

The Resistance Perturbation Distance: A Metric for the Analysis of Dynamic Networks