Alpha centrality

In graph theory and social network analysis, alpha centrality is a measure of centrality of nodes within a graph. It is an adaptation of eigenvector centrality with the addition that nodes are imbued with importance from external sources.

Definition

Given a graph with adjacency matrix the alpha centrality is defined as follows:

where is the external importance given to node , and is a parameter.

Motivation

To understand alpha centrality one must first understand Eigenvector Centrality. An intuitive process to compute eigenvector centrality is to give every node a starting random positive amount of influence. Each node then splits its influence evenly and divides it amongst its outward neighbors, receiving from its inward neighbors in kind. This process repeats until everyone is giving out as much as they're taking in and the system has reached steady state. The amount of influence they have at this steady state is their eigenvector centrality. Computationally this process is called the power method. We know that this process has converged when the vector of influence changes only by a constant as follows.