Finding the resistance distance and eigenvector centrality from the network’s eigenvalues

Abstract

There are different measures to classify a network’s data set that, depending on the problem, have different success rates. For example, the resistance distance and eigenvector centrality measures have been successful in revealing ecological pathways and differentiating between biomedical images of patients with Alzheimer’s disease, respectively. The resistance distance measures an effective distance between two nodes of a network taking into account all possible shortest paths between them and the eigenvector centrality measures the relative importance of each node in a network. However, both measures require knowing the network’s eigenvalues and eigenvectors. Here, we show that we can closely approximate [find exactly] the resistance distance [eigenvector centrality] of a network only using its eigenvalue spectra, where we illustrate this by experimenting on resistor circuits, real neural networks (weighted and unweighted), and paradigmatic network models — scale-free, random, and small-world networks. Our results are supported by analytical derivations, which are based on the eigenvector–eigenvalue identity. Since the identity is unrestricted to the resistance distance or eigenvector centrality measures, it can be applied to most problems requiring the calculation of eigenvectors.