Estimating the distance covered by a spreading event on a network can lead to a better understanding of epidemics, economic growth, and human behavior. There are many methods solving this problem-which has been called Node Vector Distance (NVD)-for single layer networks. However, many phenomena are better represented by multilayer networks: networks in which nodes can connect in qualitatively different ways. In this article, we extend the literature by proposing an algorithm solving NVD for multilayer networks. We do so by adapting the Mahalanobis distance, incorporating the graph's topology via the pseudoinverse of its Laplacian. Since this is a proper generalization of the Euclidean distance in a complex space defined by the topology of the graph, and that it works on multilayer networks, we call our measure the Multi Layer Generalized Euclidean (MLGE). In our experiments, we show that MLGE is intuitive, theoretically simpler than the alternatives, performs well in recovering infection parameters, and it is useful in specific case studies. MLGE requires solving a special case of the effective resistance on the graph, which has a high time complexity. However, this needs to be done only once per network. In the experiments, we show that MLGE can cache its most computationally heavy parts, allowing it to solve hundreds of NVD problems on the same network with little to no additional runtime. MLGE is provided as a free open source tool, along with the data and the code necessary to replicate our results.
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