Compute Eigenvector Centrality

[Rochelle Zhang]

Dear Statalisters,

Please find attached the formula (I could not type formula here ) for Eigenvector centrality of node i as Ci

I have matrix A and computed lambda (eigenvalue) and eigenvector C,

I want to ask how to compute the Eigenvector centrality Ci , i.e. to sum over all the product of Aij *Cj for each i, and i not equal to j.

Thank you,

Rochelle

[Sergiy Radyakin]

Unless centrality and eigenvector is the same thing (which is not) don't use the same letter (C) to denote both.
Post formula as an image: Not every mobile phone has MSWord installed on it.
The way I see it, the problem is not to sum something over something, the problem is to solve a system of linear equations (your C_i-s depend on C_j-s, which creates recursive definition). You have n equations and n+1 unknowns (lambda is the plus one). You say you have lambda, good, then it is n linear equations and n unknowns. Should be a unique solution.
Sergiy.

[Mike Lacy]

The -netsis- package (package st0248 from http://www.stata-journal.com/) includes the calculation of eigenvector centrality among its many network analysis features. I have not used this package, but perhaps it might solve the problem here.

Regards,
Mike Lacy

[Phil Schumm]

If you type search centrality, you'll find Hirotaka Miura's library for network analysis. I believe that includes a function to compute eigenvector centrality. Also, if you want to enter math in your posts, you may do so using MathJax.

[Rochelle Zhang]

Many thanks to Sergiy, Mike and Phil !!! I downloaded google doc from Hirotaka Miura's website and will see if I can figure out how his ado file works.

By the way, Sergiy, do you have some reference on how to solve the n linear equation in stata?

Warm regards,

Rochelle

[Sergiy Radyakin]

By the way, Sergiy, do you have some reference on how to solve the n linear equation in stata?

Hm, this tutorial suggests that inverting the matrix of coefficients might be the key.
Stata has a bunch of tools to do that.
Best, Sergiy

[Rochelle Zhang]

Hi Sergiy,
I just saw your last posting. The link to "bunch of tools" is not working. Do you mind resending it?

Thanks a lot,

Rochelle

[Joe Canner]

Rochelle,

I can't read your Word document, so I don't know if the following applies to your situation or not...

In general, solving a system of n linear equations in n unknowns is very easy using matrices. As outlined in the tutorial Sergiy posted if you have equations in the form:

a₁x + b₁y + c₁z + ... = d₁
a₂x + b₂y + c₂z + ... = d_2
...
a_nx + b_ny + c_nz + ... = d_n


and you set up to matrices A and B, such that A is a square matrix containing the a, b, and c coefficients, and B is an nx1 matrix contain the d coefficients, then the solution (values of x, y, and z) can be found using one of the solvers Sergiy tried to link to. In this particular case (square matrix), the best choice is the lusolve() function:

: X=lusolve(A,B)

You can find information about the other solvers in the M-4 "Solvers" section of the Mata Reference Manual. Note that you can also solve this system of equations using X=A^-1B, but the manual recommends against computing the inverse of A, as the method used by lusolve() is more efficient.

Regards,
Joe

[Sergiy Radyakin]

Hi Sergiy,
I just saw your last posting. The link to "bunch of tools" is not working. Do you mind resending it?

Thanks a lot,

Rochelle

http://www.stata.com/manuals13/m-4solvers.pdf