Localization of nonbacktracking centrality on dense subgraphs of sparse networks

TL;DR

This paper investigates how nonbacktracking centrality localizes around finite subgraphs within sparse networks, providing explicit formulas and demonstrating that NBC concentrates near the subgraph when its eigenvalue dominates.

Contribution

It offers a theoretical framework for understanding NBC localization in infinite sparse networks with finite subgraphs, including explicit formulas and conditions for localization.

Findings

NBC localizes on the subgraph when its eigenvalue is the largest

Explicit formulas for NBC in localized states are derived

NBC decay is exponential and independent of the network's structure

Abstract

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.