  The non-backtracking matrix and its eigenvalues have many applications in
network science and graph mining, such as node and edge centrality, community
detection, length spectrum theory, graph distance, and epidemic and percolation
thresholds. Moreover, in network epidemiology, the reciprocal of the largest
eigenvalue of the non-backtracking matrix is a good approximation for the
epidemic threshold of certain network dynamics. In this work, we develop
techniques that identify which nodes have the largest impact on the leading
non-backtracking eigenvalue. We do so by studying the behavior of the spectrum
of the non-backtracking matrix after a node is removed from the graph. From
this analysis we derive two new centrality measures: X-degree and
X-non-backtracking centrality. We perform extensive experimentation with
targeted immunization strategies derived from these two centrality measures.
Our spectral analysis and centrality measures can be broadly applied, and will
be of interest to both theorists and practitioners alike.
