Nonbacktracking Eigenvalues under Node Removal: X-Centrality and Targeted Immunization

. The non-backtracking matrix and its eigenvalues have many applications in network science and 5 graph mining, such as node and edge centrality, community detection, length spectrum theory, 6 graph distance, and epidemic and percolation thresholds. In network epidemiology, the reciprocal 7 of the largest eigenvalue of the non-backtracking matrix is a good approximation for the epidemic 8 threshold of certain network dynamics. In this work, we develop techniques that identify which 9 nodes have the largest impact on this leading eigenvalue. We do so by studying the behavior of 10 the spectrum of the non-backtracking matrix after a node is removed from the graph. From this 11 analysis we derive two new centrality measures: X -degree and X-non-backtracking centrality . We 12 perform extensive experimentation with targeted immunization strategies derived from these two 13 centrality measures. Our spectral analysis and centrality measures can be broadly applied, and will 14 be of interest to both theorists and practitioners alike. the perturbation of quadratic eigenvalue problems, with applications to the NB- eigenvalues of the stochastic block