  Message-passing theories have proved to be invaluable tools in studying
percolation, non-recurrent epidemics and similar dynamical processes on
real-world networks. At the heart of the message-passing method is the
nonbacktracking matrix whose largest eigenvalue, the corresponding eigenvector,
and the closely related nonbacktracking centrality play a central role in
determining how the given dynamical model behaves. Here we propose a
degree-class-based method to approximate these quantities using a smaller
matrix related to the joint degree-degree distribution of neighbouring nodes.
Our findings suggest that in most networks degree-degree correlations beyond
nearest neighbour are actually not strong, and our first-order description
already results in accurate estimates, particularly when message-passing itself
is a good approximation to the original model in question, that is when the
number of short cycles in the network is sufficiently low. We show that
localization of the nonbacktracking centrality is also captured well by our
scheme, particularly in large networks. Our method provides an alternative to
working with the full nonbacktracking matrix in very large networks where this
may not be possible due to memory limitations.
