Graph eigenvectors, fundamental weights and centrality metrics for nodes in networks

Double orthogonality in the set of eigenvectors of any symmetric graph matrix is exploited to propose a set of nodal centrality metrics, that is “ideal” in the sense of being complete, uncorrelated and mathematically precisely defined and computable. Moreover, we show that, for each node m, such a nodal eigenvector centrality metric reflects the impact of the removal of node m from the graph at a different eigenfrequency of that graph matrix. Fundamental weights, related to graph angles, are argued to be as important as the eigenvalues of the graph matrix. While the mathematical foundations of eigenvectors are crystal clear emphasizing its potential as an ideal set of nodal centrality metrics, the “physical” meaning of its application to graphs, the topological structure of a network, seems surprisingly opaque and, hence, constitutes a challenging question with fundamental significance for network science.