To quantify the fundamental evolution of time-varying networks, and detect abnormal behavior, one needs a notion of temporal difference that captures significant organizational changes between two successive instants. In this work, we propose a family of distances that can be tuned to quantify structural changes occurring on a graph at different scales: from the local scale formed by the neighbors of each vertex, to the largest scale that quantifies the connections between clusters, or communities. Our approach results in the definition of a true distance, and not merely a notion of similarity. We propose fast (linear in the number of edges) randomized algorithms that can quickly compute an approximation to the graph metric. The third contribution involves a fast algorithm to increase the robustness of a network by optimally decreasing the Kirchhoff index. Finally, we conduct several experiments on synthetic graphs and real networks, and we demonstrate that we can detect configurational changes that are directly related to the hidden variables governing the evolution of dynamic networks.