Eigenvector centrality of each node

The leading eigenvector of the (weighted) adjacency matrix of the undirected graph, scaled to a maximum of 1 (Bonacich 1972). It is found by power iteration on A + c I, c the largest weight, which has the same eigenvectors and whose leading eigenvalue is strictly dominant even on a bipartite graph (a co-offending network between two roles, a star), where plain power iteration oscillates. The iteration starts from 1 for every node and stops when no value changes by tol or more between two steps, or after maxit steps (then conv is FALSE).

Usage

ntevc(from, to, w = NULL, nodes = NULL, tol = 1e-13, maxit = 1e+05)

Arguments

from, to

Labels of the two ends of each edge: vectors of the same length (zero when nodes gives every node) with no missing values.

w

NULL, or one positive weight per edge (a tie's strength, such as the number of co-offences).

nodes

NULL, or every node's label once: the node order of the results and the way to include isolated nodes.

tol

The stopping tolerance: one number between 0 and 1.

maxit

The most steps: one whole number, at least 1.

Value

A list: cent, a data frame with node and evc; value, the leading eigenvalue; iter, the steps taken; conv, whether the tolerance was met.

Details

On a disconnected graph the leading eigenvector lives on the components with the largest leading eigenvalue; every other component gets 0 (each component is iterated and scaled on its own, and a component whose eigenvalue falls short of the largest by more than 1e-10 of it is set to 0). When several components share the largest eigenvalue each is scaled to a maximum of 1: any mix of them is an eigenvector too, so compare nodes within a component. A graph without edges gives 1 for every node. Graph rules as in ntdeg().

References

Bonacich, P. (1972). Factoring and weighting approaches to status scores and clique identification. Journal of Mathematical Sociology, 2(1), 113-120. doi:10.1080/0022250X.1972.9989806

Examples

ntevc(c("hub", "hub", "hub", "a"), c("a", "b", "c", "b"))
#> $cent
#>   node       evc
#> 1  hub 1.0000000
#> 2    a 0.8546377
#> 3    b 0.8546377
#> 4    c 0.4608111
#> 
#> $value
#> [1] 2.170086
#> 
#> $iter
#> [1] 33
#> 
#> $conv
#> [1] TRUE
#>