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 ${\displaystyle A_{i,j}}$, Katz centrality is defined as follows:

${\displaystyle {\vec {x}}=(I-\alpha A^{T})^{-1}{\vec {e}}-{\vec {e}}\,}$

where ${\displaystyle e_{j}}$ is the external importance given to node ${\displaystyle j}$, and ${\displaystyle \alpha }$ is a nonnegative attenuation factor which must be smaller than the inverse of the spectral radius of ${\displaystyle A}$. The original definition by Katz ^[1] used a constant vector ${\displaystyle {\vec {e}}}$. Hubbell^[2] introduced the usage of a general ${\displaystyle {\vec {e}}}$.

Half a century later, Bonacich and Lloyd^[3] defined alpha centrality as

${\displaystyle {\vec {x}}=(I-\alpha A^{T})^{-1}{\vec {e}}\,}$

which is essentially identical to Katz centrality. More precisely, the score of a node ${\displaystyle j}$ differs exactly by ${\displaystyle e_{j}}$, so if ${\displaystyle {\vec {e}}}$ 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:

${\displaystyle x_{i}={\frac {1}{\lambda }}A_{i,j}^{T}x_{j}}$

where ${\displaystyle x_{i}}$ is the amount of influence that node ${\displaystyle i}$ carries, ${\displaystyle A_{i,j}}$ is the adjacency matrix and ${\displaystyle \lambda }$ 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 ${\displaystyle i}$ receives at every round is encoded in ${\displaystyle e_{i}}$. The process described above should now stop when

${\displaystyle x_{i}=\alpha A_{i,j}^{T}x_{j}+e_{i}\,}$

where ${\displaystyle \alpha }$ is a constant that trades off the importance of external influence against the importance of connection. When ${\displaystyle \alpha =0}$ only the external influence matters. When ${\displaystyle \alpha }$ 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 ${\displaystyle x}$, obtaining the following equation:

${\displaystyle x=(I-\alpha A^{T})^{-1}e\,}$

Applications[]

Alpha centrality is implemented in igraph library for network analysis and visualization.^[4]

Notes and references[]