Updating Katz centrality by counting walks

TL;DR

This paper introduces efficient algorithms for updating Katz centrality after node or edge removal, using new theoretical concepts and providing bounds on network communicability changes, validated through extensive experiments.

Contribution

It presents novel explicit formulas and algorithms for updating Katz centrality efficiently after network modifications, based on the concept of -avoiding first-passage walks.

Findings

Algorithms effectively update Katz centrality post-removal.

Explicit formulas quantify walk loss due to removals.

Numerical experiments validate theoretical bounds.

Abstract

We develop efficient and effective strategies for the update of Katz centralities after node and edge removal in simple graphs. We provide explicit formulas for the ``loss of walks" a network suffers when nodes/edges are removed, and use these to inform our algorithms. The theory builds on the newly introduced concept of $\cF$-avoiding first-passage walks. Further, bounds on the change of total network communicability are also derived. Extensive numerical experiments on synthetic and real-world networks complement our theoretical results.