Generalized Euclidean Measure to Estimate Network Distances

Estimating the distance covered by a propagation phenomenon on a network is an important task: it can help us estimating the infectiousness of a disease or the effectiveness of an online viral marketing campaign. However, so far the only way to make such an estimate relies on solving the optimal transportation problem, or by adapting graph signal processing techniques. Such solutions are either inefficient, because they require solving a complex optimization problem; or fragile, because they were not designed with this problem in mind. In this paper, we propose a new generalized Euclidean approach to estimate distances between weighted groups of nodes in a network. We do so by adapting the Mahalanobis distance, incorporating the graph's topology via the pseudoinverse of its Laplacian. In experiments we see that this measure returns intuitive distances which agree with the ones a human would estimate. We also show that the measure is able to recover the infection parameter in an epidemic model, or the activation threshold in a cascade model. We conclude by showing that the measure can be used in online social media settings to identify fast-spreading behaviors. Our measure is also less computationally expensive.