  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 with 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 fairly accurate estimates. 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. Our results may also be useful in designing
group-based inoculation strategies to control epidemics.
