Centrality and Network Analysis: A Perturbative Approach to Dynamical Importance

The purpose of this paper is to investigate methods for analyzing networks from an algebraic perspective. The main focus will be on the dominant eigenpair of the adjacency matrix representing a graph, as many different centrality measures for networks are cast in terms of eigenvalue problems. We will see how it is affected by small perturbations to the graph and also propose methods for estimating these changes. This analysis will be justified from the ground up, only assuming basic elements of linear algebra.