(This article was first published on

Leo Spizzirri does an excellent job of providing mathematical intuition behind eigenvector centrality. As I was reading through it, I found it easier to just work through the matrix operations he proposes using R. You can find his paper here: https://www.math.washington.edu/~morrow/336_11/papers/leo.pdf**Econometric Sense**, and kindly contributed to R-bloggers)My R code follows. This is for the example using degree centrality which made the most sense to me. For some reason, in the last section, for very large k there is an issue with computing the vector, I think it is stemming from my definition of B_K. The same issue occurs with large k using the function MM. It could be that the values get too large for R to handle? Regardless, as pointed out on page 7, even at k = 10 that ck approaches the eigenvector.

# *------------------------------------------------------------------

# | PROGRAM NAME: R_BASIC_SNA

# | DATE: 4/9/12

# | CREATED BY: MATT BOGARD

# | DATE: 11/5/12

# | PROJECT FILE: P:\R Code References\SNA

# *----------------------------------------------------------------

# | PURPOSE: COMPANION CODE TO Justification and Application of

# | Eigenvector Centrality by Leo Spizzirri

# | https://www.math.washington.edu/~morrow/336_11/papers/leo.pdf

# *------------------------------------------------------------------

# specify the adjacency matrix

A <- matrix(c(0,1,0,0,0,0,

1,0,1,0,0,0,

0,1,0,1,1,1,

0,0,1,0,1,0,

0,0,1,1,0,1,

0,0,1,0,1,0 ),6,6, byrow= TRUE)

EV <- eigen(A) # compute eigenvalues and eigenvectors

max(EV$values) # find the maximum eigenvalue

# get the eigenvector associated with the largest eigenvalue

centrality <- data.frame(EV$vectors[,1])

names(centrality) <- "Centrality"

print(centrality)

B <- A + diag(6)

EVB <- eigen(B) # compute eigenvalues and eigenvectors

# they are the same as EV(A)

# define matrix M

M <- matrix(c(1,1,0,0,

1,1,1,0,

0,1,1,1,

0,0,1,1),4,4, byrow= TRUE)

# define function for B^k for matrix M

MM <- function(k){

n <- (k-1)

B_K <- C

for (i in 1:n){

B_K <- B_K%*%M

}

return(B_K)

}

MM(2) # M^2

MM(3) # M^3

# define c

c <- matrix(c(2,3,5,3,4,3))

# define c_k for matrix B

ck <- function(k){

n <- (k-2)

B_K <- B

for (i in 1:n){

B_K <- B_K%*%B

}

c_k <- B_K%*%c

return(c_k)

}

# derive EV centrality as k -> infinity

library(matrixcalc)

# k = 5

ck(5)/frobenius.norm(ck(5))

# k = 10

ck(10)/frobenius.norm(ck(10))

print(v0)

# k = 100

ck(100)/frobenius.norm(ck(100))

To

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