M.E.J. Newman's Networks Book defines the Katz Centrality as:
$$ x = \left(\mathbf{I} - \alpha A \right)^{-1}\mathbf{1} $$
, wheras in - P. Bonacich, P. Lloyd (2001), Eigenvector-like measures of centrality for asymmetric relations, Social Networks - the same centrality definition is called Alpha-Centrality. Also there a Katz Centrality is mentioned, which is defined as:
$$ x = \left( \left(\mathbf{I} - \alpha A \right)^{-1} - \mathbf{I} \right)\mathbf{1} $$
Is there meanwhile a canonical naming for these centrality measures established?
Both formulas mean the same, and it is a simple case of alpha centrality. The first formula is derived in such way: the matrix of discounted total numbers of all paths (walks) is calculated as a^{0}A^{0}+a^{1}A^{1}+a^{2}A^{2}+a^{3}A^{3}+... Here elements (i,j) of the matrix A^1 are numbers of 1-step paths connecting i and j, elements of A^2 are numbers of 2-step paths, etc. But A^0=I in the sum is written only for commodity; really there are no 0-step paths. That is why in the second formula this term is excluded: (-I). In other words, the difference between the indices given by the two formulas for each node i is x(i)-x(i)=1.