On the α-nonbacktracking centrality for complex networks: Existence and limit cases

Nonbactracking centrality was introduced as an attempt to correct some deficiencies of eigenvector centrality. In this work the α-nonbacktracking centrality is introduced as an extension that interpolates between the nonbacktracking centrality of the edges of a directed network and the eigenvector centrality of the corresponding directed line graph. The existence of this new [Formula presented]-nonbacktracking centrality is proved in terms of the connectivity of the original network. We prove that the limit of the [Formula presented]-nonbacktracking centrality when [Formula presented] decreases to zero exists and is well defined. Moreover, it coincides with the nonbacktracking centrality when this measure is defined. With the same techniques we also prove the convergence of PageRank vectors to the eigenvector centrality vector when the damping factor tends to 1.