What does it mean for Katz Centralities to "diverge" In Mark Newman's Networks book, 2010 edition, page 173, he explains some mathematical details behind the Katz Centrality measure:

In matrix terms, Eq. (7.8) can be written x = αAx + β1, (7.9) where 1
is the vector (1, 1, 1 . . .). Rearranging for x, we find that x = β(I
− αA)^{−1} · 1. As we have said, we normally don’t care about the
absolute magnitude of the centrality, only about which vertices have
high or low centrality values, so the overall multiplier β is
unimportant. For convenience we usually set β = 1, giving x = (I −
αA)^{−1} · 1. (7.10) This centrality measure was first proposed by Katz
in 1953 [169] and we will refer to it as the Katz centrality. The Katz
centrality differs from ordinary eigenvector centrality in the
important respect of having a free parameter α, which governs the
balance between the eigenvector term and the constant term in Eq.
(7.8). If we wish to make use of the Katz centrality we must first
choose a value for this constant. In doing so it is important to
understand that α cannot be arbitrarily large. If we let α → 0, then
only the constant term survives in Eq. (7.8) and all vertices have the
same centrality β (which we have set to 1). As we increase α from zero
the centralities increase and eventually there comes a point at which
they diverge. This happens at the point where (I − αA)^{−1} diverges in
Eq. (7.10), i.e., when det(I − αA) passes through zero. Rewriting this
condition as det(A − α^{−1}I) = 0, (7.11) we see that it is simply the
characteristic equation whose roots α^{−1} are equal to the eigenvalues
of the adjacency matrix. As α increases, the determinant first
crosses zero when α^{-1} = κ1, the largest eigenvalue of A, or
alternatively when α = 1/κ1. Thus, we should choose a value of α less
than this if we wish the expression for the centrality to converge

Now, what does it mean for centralities to diverge when the characteristic equation of the adjacency matrix is bigger than zero? I do not understand why we should pick a value of α < 1/κ1 when in a footnote to the above paragraph we read:

Formally one recovers finite values again when one moves past 1/κ1 to higher α, but in practice
these values are meaningless. The method returns good results only for α < 1/κ1.


By diverging, Newman really means that if $\alpha$ is equal to the inverse of any eigenvalues of $A$, then some or all values of the vector $x$ may tend to infinity, i.e. diverge. You can see this as, in this case, when doing $\frac{1}{(I-\alpha A)}$ you are "dividing by zero" somewhere.
One should pick a value $\alpha<\kappa_1^{-1}$ because in this case, as $|\alpha A|<1$, we can express $(I-\alpha A)^{-1}$ as the Neumann series of $\alpha A$:
$$
(I-\alpha A)^{-1}=I + \alpha A + \alpha^2 A^2 + \alpha^3 A^3 + ...
$$
Hence
$$
x = \left(\sum_{\ell=0}^\infty\alpha^\ell A^\ell\right)\vec{1}.
$$
This gives a nice interpretation of the Katz centrality of a node $i$ as the weighted sum of all walks emanating from $i$, with the count for walks of length $\ell$ (given by $A^\ell$) weighted by a factor $\alpha^\ell$. This also shows you that all values of the Katz centrality will be positive (this is not guaranteed when $\alpha > \kappa_1^{-1}$).
