  The nonbacktracking matrix, and the related nonbacktracking centrality (NBC)
play a crucial role in models of percolation-type processes on networks, such
as non-recurrent epidemics. Here we study the localization of NBC in infinite
sparse networks that contain an arbitrary finite subgraph. Assuming the local
tree-likeness of the enclosing network, and that branches emanating from the
finite subgraph do not intersect at finite distances, we show that the largest
eigenvalue of the nonbacktracking matrix of the composite network is equal to
the highest of the two largest eigenvalues: that of the finite subgraph and of
the enclosing network. In the localized state, when the largest eigenvalue of
the subgraph is the highest of the two, we derive explicit expressions for the
NBCs of nodes in the subgraph and other nodes in the network. In this state,
nonbacktracking centrality is concentrated on the subgraph and its immediate
neighbourhood in the enclosing network. We obtain simple, exact formulas in the
case where the enclosing network is uncorrelated. We find that the mean NBC
decays exponentially around the finite subgraph, at a rate which is independent
of the structure of the enclosing network, contrary to what was found for the
localization of the principal eigenvector of the adjacency matrix. Numerical
simulations confirm that our results provide good approximations even in
moderately sized, loopy, real-world networks.
