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

Abstract: Several expressions for the $j$-th component $\left( x_{k}\right)<em>{j}$ of the $k$-th eigenvector $x</em>{k}$ of a symmetric matrix $A$ belonging to eigenvalue $\lambda_{k}$ and normalized as $x_{k}<sup>{T}x_{k}=1$ are presented. In particular, the expression [ \left( x_{k}\right){j}^{2}=-\frac{1}{c{A}^{\prime}\left( \lambda_{k}\right) }\det\left( A_{\backslash\left{ j\right} }-\lambda_{k}I\right) ] where $$c_{A}\left( \lambda\right) =\det\left( A-\lambda I\right)$$ is the characteristic polynomial of $A$, $$c_{A}^{{\prime}\left(} \lambda\right) =\frac{dc_{A}\left( \lambda\right) }{d\lambda}$$ 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 $\lambda_{k}$ 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 $\left( x_{k}\right)<em>{j}<sup>{2}$ of the adjacency matrix over all eigenvalue $\lambda</em>{k}$ 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 $X_{ij}=\left( x_{j}\right)<em>{i}$. In addition to the criterion {\em If the algebraic connectivity is positive, then the graph is connected}, we found an alternative condition: {\em If $$\min</em>{1\leq k\leq N}\left( \lambda_{k}^{{2}(A)\right)} =d_{\min}$$, then the graph is disconnected.}