Katz centrality
It was introduced by Leo Katz in 1953 and is used to measure the relative degree of influence of an actor (or node) within a social network.
[1] Unlike typical centrality measures which consider only the shortest path (the geodesic) between a pair of actors, Katz centrality measures influence by taking into account the total number of walks between a pair of actors.
[2] It is similar to Google's PageRank and to the eigenvector centrality.
Connections made with distant neighbors are, however, penalized by an attenuation factor
[4] Each path or connection between a pair of nodes is assigned a weight determined by
For example, in the figure on the right, assume that John's centrality is being measured and that
The weight assigned to each link that connects John with his immediate neighbors Jane and Bob will be
Similarly, the weight assigned to the connection between Agneta and John through Aziz and Jane will be
and the weight assigned to the connection between Agneta and John through Diego, Jose and Bob will be
The powers of A indicate the presence (or absence) of links between two nodes through intermediaries.
, mathematically: Note that the above definition uses the fact that the element at location
has to be chosen such that it is smaller than the reciprocal of the absolute value of the largest eigenvalue of A.
[5] In this case the following expression can be used to calculate Katz centrality: Here
[5] An extension of this framework allows for the walks to be computed in a dynamical setting.
[6][7] By taking a time dependent series of network adjacency snapshots of the transient edges, the dependency for walks to contribute towards a cumulative effect is presented.
The arrow of time is preserved so that the contribution of activity is asymmetric in the direction of information propagation.
Network producing data of the form: representing the adjacency matrix at each time
is a weighted count of the number of dynamic walks of length
can 'broadcast' and 'receive' dynamic messages across the network: Given a graph with adjacency matrix
is a nonnegative attenuation factor which must be smaller than the inverse of the spectral radius of
The original definition by Katz [8] used a constant vector
is constant the order induced on the nodes is identical.
[11] Katz centrality is more suitable in the analysis of directed acyclic graphs where traditionally used measures like eigenvector centrality are rendered useless.
[11] Katz centrality can also be used in estimating the relative status or influence of actors in a social network.
The work presented in [12] shows the case study of applying a dynamic version of the Katz centrality to data from Twitter and focuses on particular brands which have stable discussion leaders.
In neuroscience, it is found that Katz centrality correlates with the relative firing rate of neurons in a neural network.
[13] The temporal extension of the Katz centrality is applied to fMRI data obtained from a musical learning experiment in [14] where data is collected from the subjects before and after the learning process.
The results show that the changes to the network structure over the musical exposure created in each session a quantification of the cross communicability that produced clusters in line with the success of learning.
A generalized form of Katz centrality can be used as an intuitive ranking system for sports teams, such as in college football.
[15] Alpha centrality is implemented in igraph library for network analysis and visualization.
