Editing Centrality (section)

===Radial-volume centralities exist on a spectrum===
The characterization by walk structure shows that almost all centralities in wide use are radial-volume measures. These encode the belief that a vertex's centrality is a function of the centrality of the vertices it is associated with. Centralities distinguish themselves on how association is defined.

Bonacich showed that if association is defined in terms of [[Glossary of graph theory terms#walk|walks]], then a family of centralities can be defined based on the length of walk considered.<ref name="Bonacich1987"/> [[#Degree centrality|Degree centrality]] counts walks of length one, while [[#Eigenvector centrality|eigenvalue centrality]] counts walks of length infinity. Alternative definitions of association are also reasonable. [[Alpha centrality]] allows vertices to have an external source of influence. Estrada's subgraph centrality proposes only counting closed paths (triangles, squares, etc.).

The heart of such measures is the observation that powers of the graph's adjacency matrix gives the number of walks of length given by that power.  Similarly, the matrix exponential is also closely related to the number of walks of a given length. An initial transformation of the adjacency matrix allows a different definition of the type of walk counted. Under either approach, the centrality of a vertex can be expressed as an infinite sum, either

:<math>\sum_{k=0}^\infty A_{R}^{k} \beta^k </math>

for matrix powers or

:<math>\sum_{k=0}^\infty \frac{(A_R \beta)^k}{k!}</math>

for matrix exponentials, where

* <math>k</math> is walk length,
* <math>A_R</math> is the transformed adjacency matrix, and
* <math>\beta</math> is a discount parameter which ensures convergence of the sum.

Bonacich's family of measures does not transform the adjacency matrix. [[Alpha centrality]] replaces the adjacency matrix with its [[resolvent formalism|resolvent]]. Subgraph centrality replaces the adjacency matrix with its trace. A startling conclusion is that regardless of the initial transformation of the adjacency matrix, all such approaches have common limiting behavior. As <math>\beta</math> approaches zero, the indices converge to [[#Degree centrality|degree centrality]]. As <math>\beta</math> approaches its maximal value, the indices converge to [[#Eigenvector centrality|eigenvalue centrality]].<ref name=Benzi2013/>