See Measures and metrics for networks

Katz centrality solves the problem posed above by giving all vertices a "free" centrality:

$\mathbf{x}=\alpha\mathbf{A}\mathbf{x}+\beta \mathbf{1}$ ....Eq. 2

or rearranging and setting $\beta=1$, because all we care is about relative centralities:

$\mathbf{x}=\beta (1-\alpha\mathbf{A})^{-1} \mathbf{1}=(1-\alpha\mathbf{A})^{-1} \mathbf{1}$

This is the Katz centrality. Often one computes this not by inverting the matrix (which requires $O(n^3)$ computations), but by iterating using Eq. 2 (which requires just $m$ multiplications per step (number of nonzero elements of $A$, often less steps overall).

A useful extension is to take $\beta \mathbf{1} \rightarrow \vec{\beta}$, i.e. give each node possibly a different weight maybe expressing some non-network importance

By Taylor expanding it we can see it is like Eigenvector centrality, but taking into account paths of all lengths, but with with a weight.