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

The nonbacktracking 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. In network epidemiology, the reciprocal of the largest eigenvalue of the nonbacktracking 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 this leading eigenvalue. We do so by studying the behavior of the spectrum of the nonbacktracking matrix after a node is removed from the graph. From this analysis we derive two new centrality measures: X-degree and X-nonbacktracking 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.