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

Abstract

Several expressions for the $j$-th component $\left( x_{k}\right)_{j}$ of the $k$-th eigenvector $x_{k}$ of a symmetric matrix $A$ belonging to eigenvalue $λ_{k}$ and normalized as $x_{k}^{T}x_{k}=1$ are presented. In particular, the expression \[ \left( x_{k}\right)_{j}^{2}=-\frac{1}{c_{A}^{\prime}\left( λ_{k}\right) }\det\left( A_{\backslash\left\{ j\right\} }-λ_{k}I\right) \] where $c_{A}\left( λ\right) =\det\left( A-λI\right) $ is the characteristic polynomial of $A$, $c_{A}^{\prime}\left( λ\right) =\frac{dc_{A}\left( λ\right) }{dλ}$ and $A_{\backslash\left\{ j\right\} }$ is obtained from $A$ by removal of row $j$ and column $j$, suggests us to consider the square eigenvector component as a graph centrality metric for node $j$ that reflects the impact of the removal of node $j$ from the graph at an eigenfrequency/eigenvalue $λ_{k}$ of a graph related matrix (such as the adjacency or Laplacian matrix). Removal of nodes in a graph relates to the robustness of a graph. The set of such nodal centrality metrics, the squared eigenvector components $\left( x_{k}\right)_{j}^{2}$ of the adjacency matrix over all eigenvalue $λ_{k}$ for each node $j$, is 'ideal' in the sense of being complete, \emph{almost} uncorrelated and mathematically precisely defined and computable. Fundamental weights (column sum of $X$) and dual fundamental weights (row sum of $X$) are introduced as spectral metrics that condense information embedded in the orthogonal eigenvector matrix $X$, with elements $X_{ij}=\left( x_{j}\right)_{i}$. In addition to the criterion {\em If the algebraic connectivity is positive, then the graph is connected}, we found an alternative condition: {\em If $\min_{1\leq k\leq N}\left( λ_{k}^{2}(A)\right) =d_{\min}$, then the graph is disconnected.}