Approximating nonbacktracking centrality and localization phenomena in large networks

TL;DR

This paper introduces a degree-class-based approximation method for nonbacktracking centrality in large networks, effectively capturing localization phenomena and reducing computational complexity.

Contribution

The authors propose a novel, smaller matrix approximation for nonbacktracking centrality that accounts for degree correlations, improving analysis of large networks.

Findings

Approximation accurately estimates nonbacktracking eigenvalues and eigenvectors.

Method effectively captures localization phenomena in large networks.

Performance is good when networks have few short cycles.

Abstract

Message-passing theories have proved to be invaluable tools in studying percolation, non-recurrent epidemics and similar dynamical processes on real-world networks. At the heart of the message-passing method is the nonbacktracking matrix whose largest eigenvalue, the corresponding eigenvector, and the closely related nonbacktracking centrality play a central role in determining how the given dynamical model behaves. Here we propose a degree-class-based method to approximate these quantities using a smaller matrix related to the joint degree-degree distribution of neighbouring nodes. Our findings suggest that in most networks degree-degree correlations beyond nearest neighbour are actually not strong, and our first-order description already results in accurate estimates, particularly when message-passing itself is a good approximation to the original model in question, that is when the number of short cycles in the network is sufficiently low. We show that localization of the nonbacktracking centrality is also captured well by our scheme, particularly in large networks. Our method provides an alternative to working with the full nonbacktracking matrix in very large networks where this may not be possible due to memory limitations.