Alpha centrality

In graph theory and social network analysis, alpha centrality is an alternative name for Katz centrality. It 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 formula_1, Katz centrality is defined as follows:
where formula_3 is the external importance given to node formula_4, and formula_5 is a nonnegative attenuation factor which must be smaller than the inverse of the spectral radius of formula_6. The original definition by Katz
used a constant vector formula_7. Hubbell
introduced the usage of a general formula_7.
Half a century later, Bonacich and Lloyd defined alpha centrality as 
which is essentially identical to Katz centrality. More precisely, the score of a node formula_4 differs exactly by formula_3, so if formula_7 is constant the order induced on the nodes is identical.
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:
where formula_14 is the amount of influence that node formula_15 carries, formula_1 is the adjacency matrix and formula_17 happens to be the principal eigenvalue.
Alpha centrality enhances this process by allowing nodes to have external sources of influence. The amount of influence that node formula_15 receives at every round is encoded in formula_19. The process described above should now stop when
where formula_5 is a constant that trades off the importance of external influence against the importance of connection. When formula_22 only the external influence matters. When formula_5 is very large then only the connectivity matters, i.e. we reduce to the eigenvector centrality case.
Rather than perform the iteration described above we can solve this system for formula_24, obtaining the following equation:
Applications.
Alpha centrality is implemented in igraph library for network analysis and visualization.